d I n
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dvances I n A n a l y s i s ings Pro cedings of the 4th International ISAA Congress York University, Toronto, Canada
11-16 August 2003
Editors H. G. W. Begehrt fREIE uNIVERSITAT bERLIN, gERMANY r. p. gILBERT uNIVERSITY OF dELAWARE, uSa m. e. muldoonoo n YorUniversity, Canada M. W. Wong York University, Canada
World Scientific NEW J E R S E Y. London
Singapore . Beijing . Shanghai > Hong Kong . Taipi. Chennai
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ADVANCES IN ANALYSIS Proceedings of the 4th ISAAC Congress Copyright Q 2005 by World Scientific Publishing Co. Pte.Ltd.
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PREFACE The Fourth Congress of the International Society for Analysis, Applications and Computation (ISAAC) was held at York University in Toronto from August 11, 2003 to August 16, 2003. It is a sequel to the series of events in Delaware 1997, Fukuoka 1999 and Berlin 2001. The Toronto Congress was supported by the Academic Initiative Fund of the Faculty of Arts, National Sciences and Engineering Research Council of Canada grants held by some members of the Department of Mathematics and Statistics and the Office of the Vice-president Academic of York University. There were nine plenary lectures and seventeen special sessions representing most major themes in analysis, its applications and computation. Immediately after the Congress, all speakers were asked to write up their lectures for publication in a volume in the ISAAC series. Plenary speakers were asked to submit their papers to the editors. Speakers in special sessions were directed to send their papers to their session organizers. The editors and the organizers of special sessions handled the papers received and sent them out for peer review. Thus, all papers in this volume are refereed. They are culled from plenary lectures and special sessions, which we list as follows. 1. Plenary Lectures Speakers: R. Askey, P. Greiner and S. Thangavelu 2. Approximation Theory Organizer: A. Kushpel
3. Banach Spaces of Analytic Functions Organizers: R. Aulaskari and J. Taskinen 4. Boundary Value Problems and Integral Equations Organizer: P. A. Krutitskii 5. Complex and Functional Analytic Methods in Partial Differential Equations Organizers: H. G. W. Begehr and A. Cialdea 6. Harmonic Analysis and Partial Differential Equations Organizers: D.-C. Chang, G. Dafni, A. Fraser, J. Tie and M. W. Wong 7. Hemivariational Inequalities and Applications Organizer: S. Carl 8. Hyperbolic Problems: Degeneracies, Nonlinearities and Global Existence Organizers: M. Ressig and D. Del Santo
vi
Preface
9. Inverse Problems Organizers: R. P. Gilbert, A. Wirgin and Y . Xu 10. Nonlinear Analysis and Applications Organizers: M. R. Singh, S. P. Singh and B. Watson 11. Orthogonal Polynomials and Special Functions Organizer: M. E. Muldoon 12. Reproducing Kernels and Related Topics Organizers: D. Alpay, J. A. Ball, T. Ohsawa and S. Saitoh 13. Time-Frequency Analysis, Wavelets and Applications Organizer: J. S. Walker 14. Toeplitz-Like Strutures in Analysis and Applied Sciences Organizers: B. Silbermann and N. Vasilevski 15. Value Distributions of Complex Functions, Generalizations and Related Topics Organizers: G. Barsegian and A. Escassut
In spite of the continuing SARS crisis in Toronto during the spring and the summer of 2003, the Toronto Congress was held as scheduled and was well attended by mathematicians from all over the world. Many of the papers presented at the Congress were not submitted to this proceedings; two plenary lectures and the papers originating from the special session on pseudo-differential operators were published in Professor Israel Gohberg’s Birkhauser series in operator theory. Nevertheless, most of the traditional strengths of ISAAC are represented in this volume. Moreover, it should be pointed out that main-stream analysis such as orthogonal polynomials, special functions, harmonic analysis and PDE are new features of the Toronto Congress and hence this volume. Two of the three plenary lectures in this volume are highly readable and comprehensive surveys on active and modern areas in analysis: geometric analysis and PDE (P. Greiner), and harmonic analysis on Lie groups (S. Thangavelu). The plenary lecture by R. Askey is an interesting, albeit more speialized, piece of mathematics on determinants and orthogonal polynomials. On the recommendation of Professor Richard Askey, Dr. Olga Holtz was invited to submit a paper as a valuable supplement to his plenary lecture. The broad coverage of topics in this volume allows it to be a useful source of information to mathematicians, scientists and engineers who have basic knowledge of mathematical analysis and numerical methods at the first or second year graduate level in a North American university. It is envisaged that the plenary lectures and the papers in the special sessions are helpful to a broad spectrum of graduate students in mathematical sciences looking for research topics.
LEE LORCH: CITATION FOR HONORARY LIFE MEMBERSHIP IN ISAAC By resolution of the ISAAC Board of Directors, meeting during ISAAC 2003 an Honorary Life Membership in ISAAC was granted to Lee Lorch, Professor Emeritus of Mathematics at York University. The following citation was read by Professor Victor Burenkov of Cardiff University at the formal presentation of this Membership on August 16, 2003: Lee Lorch is the author or coauthor of about 80 articles in mathematics published in the past 60 years. About a month from his 88th birthday, he is still publishing with articles to appear and submitted. Professor Lorch has made contributions to several areas of interest to members of ISAAC including Real Analysis, Special Functions, Ordinary Differential Equations, Summability Theory, Approximation Theory, and Fourier Analysis. His work has appeared in such leading journals as Acta Mathematica, Acta Mathematica Hungarica, American Journal of Mathematics, Canadian Journal of Mathematics, Communications on Pure and Applied Mathematics, Duke Mathematical Journal, Pacific Journal of Mathematics, Proceedings of the American Mathematical Society, and SIAM Journal on Mathematical Analysis. Professor Lorch has contributed much to the study of the order of magnitude and asymptotic expansion of the Lebesgue constants for various expansions. Another theme, in which Lee and his coauthor Peter Szego can be said to have started a new field was in higher monotonicity properties (regular sign behaviour of higher derivatives and differences) of Sturm-Liouville functions. Some of the interesting conjectures from their first paper (1963) have not been settled yet. One of Professor Lorch’s strengths has been the ability to take an old method such as the Sturm comparison theorem and extract some valuable new information by its use. Professor Lorch was honoured by Fellowship of the Royal Society of Canada in 1968. He has been active in the service of mathematics through the committees of the Natural Sciences and Engineering Research Council of Canada, the Canadian Mathematical Society and the American Mathematical Society. But perhaps he has stood out most prominently as a champion of the social responsibility of scientists in combating war and discrimination. He has constantly championed the rights of women and minorities to decent educational and career opportunities.
viii
Lee Lorch: Citation for Honorary Life Membership in ISAAC
Fig. 1. Lee Lorch
- photo courtesy of Mary Kennedy
Professor Lorch made important contributions in developing contacts between western and eastern mathematicians. Not only did he visit the former Soviet Union several times in the sixties and seventies but also he organized visits of prominent Russian mathematicians to Canada; this was quite an achievement at that time. Professor Lorch actively supported mathematicians from developing countries. At the International Congress of Mathematicians in Warsaw in 1983, he organized a large meeting that discussed the problems of mathematicians from developing countries in great detail and gave an impulse to action. Professor Lorch: The Society for Analysis, its Applications and Computation salutes you for your distinguished mathematical contributions, your valiant struggles on behalf of the disadvantaged, and your continuing efforts for world peace and understanding between peoples, and is delighted to ask you to accept an Honorary Life Membership in the Society.
CONTENTS Preface..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lee Lorch: Honorary Life Membership in ISAAC
v
. . . . . . . . . . vii
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Plenary Lectures 1. Evaluation of Sylvester Type Determinants using Orthogonal Polynomials R.Askey.. . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. Geometric Analysis on SubRiemannian Manifolds
0. Calin, D . C . Chang and P . Greiner . . . . . . . . . . . . . 17 3. A Survey of Hardy Type Theorems S . Thangavelu . . . . . . . . . . .
. . . . . . . . . . . . . .
39
Approximation Theory 4. Optimal sk-Spline Approximation of Sobolev’s Classes on the 2-Sphere C . Grandison and A. Kushpel
. . . . . . . . . . . . . . . . . 71
5. slc-Spline Approximation on the Torus A . Kushpel . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Entropy Numbers of Sobolev and Besov Classes on Homogeneous Spaces A . Kushpel and S. Tozoni . . . . . . . . . . . .
81
. . . . . . . 89
7. Operator Equations and Best Approximation Problems in Reproducing Kernel Hilbert Spaces with Tikhonov Regularization S. Saitoh, T . Matsuum and M . Asaduzzaman . . . . . . . . . 99
Contents
x
Banach Spaces of Analytic Functions 8. Finite Fourier Transforms and the Zeros of the Riemann &Function G . Csordas and C.C. Yang . . . . . . . . . . . . . . . . . . 9.
Spaces and Harmonic Majorants E . Ramirez de Arellano, L . F . Resbndis 0. and L . M . Tovar S. . . . . . . . . . . . . . . . . .
109
B P , Qp
. . . . . . .
10. The (t, a)-Lattice and Decomposition Theory for Function Spaces Irfan Ul-haq and Zhijian W u . . . . . . . . . . . . . . .
121
. . 139
Boundary Value Problems and Integral Equations 11. On a Generalization of the N. A. Davydov Theorem 0 . F. Germs and M . Shapiro . . . . . . . . . . . .
. . . . . 159
12. Dual Integral Equations Method for Some Mixed Boundary Value Problems J. M. Rappoport . . . . . . . . . . . . . . . . . . . . . . .
167
13. One Parameter-Dependent Nonlinear Elliptic Boundary Value Problems Arising in Population Dynamics K . Umezu . . . . . . . . . . . . . . . . . . . . . . . . . .
177
Complex and Functional Analytic Methods in Partial Differential Equations 14. Combined Integral Representations H . Begehr . . . . . . . . . . . . . . . . . . . . . . . . .
187
15. Remarks on Quantum Differential Operators R . Carroll . . . . . . . . . . . . . . . . . .
197
. . . . . . . .
16. The Inverse Monodromy Problem in a Class of Knizhnik-Zamolodchikov Equations V . Golubeva . . . . . . . . . . . . . . . . . .
. . . . .
211
Contents
xi
17. Reduction of Two Dimensional Neumann and Mixed Boundary Value Problems to Dirichlet Boundary Value Problems M . Jahanshahi . . . . . . . . . . . . . . . . . .
. . . . . .
221
18. Lipschitz Type Inequalities for a Domain Dependent
Neumann Eigenvalue Problem for the Laplace Operator Pier Domenico Lamberti and Massimo Lanza de Cristoforis . . . . . . . . . . . . .
. .
227
19. Half Robin Problems for the Dirac Operator in the Unit Ball of IWm (m 2 3 )
Zhenyuan Xu . . . . . . . . . . . . . . . . . . . . . . . . .
235
Harmonic Analysis and Partial Differential Equations 20. Problems Related to the Analytic Representation of Tempered Distributions C. Carton-Lebrun . . . . . . . . . . . . . . . . . .
. . . .
245
21. Strong Unique Continuation for Generalized Baouendi-Grushin Operators N . Garofalo and D. Vassilev . . . . . . . . . . . . . . . . . 255 22. Schoenberg's Theorem for Positive Definite Functions on Heisenberg Groups J. Kim and M. W . W o n g . . . . . . . . . . . . . . . .
. . . 265
23. Weak and Strong Solutions for Pseudo-Differential Operators M . W . Wong . . . . . . . . . . . . . . . . . . . . . . . . .
275
Hemivariat ional Inequalities and Applications 24. Comparison Results for Quasilinear Elliptic Hemivariational Inequalities S. Carl . . . . . . . . . . . . . . . . . . . . . . . . . . . .
285
25. Hemivariational Inequalities Modeling Dynamic Viscoelastic Contact Problems with Friction S . Migdrski . . . . . . . . . . . . . . . . . . . . . . . . . .
295
xii
Contents
26. Sensitivity Analysis for Generalized Variational and Hemivariational Inequalities B . S. Mordukhovich . . . . . . . . . . . . . . . . .
. . . .
305
Hyperbolic Problems: Degeneracies, Nonlinearit ies and Global Existence 27. A Smoothing Property of Schrodinger Equations and a Global Existence Result for Derivative Nonlinear Equations M . Sugimoto and M . Ruzhansky . . . . . . . . . . . . . . . 315 28. Existence and Blow up for a Wave Equation with a Cubic Convolution K.Tsutaya.. . . . . . . . . . . . . . . . . . . . . . . .
.
321
29. Zeros and Signs of Solutions for Some Reaction-Diffusion Systems H. Uesaka . . . . . . . . . . . . . . . . . . . . . . . . . .
327
Inverse Problems 30. Constructions of Approximate Solutions for Linear Differential Equations by Reproducing Kernels and Inverse Problems M . Asaduzzaman, T. Matsuura and S . Saitoh . .
. . . . . .
31. Ultrasound as a Diagnostic Tool to Determine Osteoporosis J . L. Buchanan, R . P . Gilbert and Y . X u . . . . . . . . . 32. Remarks on Quantum KdV R.Carroll . . . . . . . . .
.
335
345
. . . . . . . . . . . . . . . . .
355
33. A Mathematical Model of Ductal Carcinoma in Situ and Its Characteristic Patterns Y.Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . .
365
Nonlinear Analysis and Applications 34. Classical Dynamics of Quantum Variations M . Kondratieva and S . Sadov . . . . . .
. . . . . . . . . . 375
Contents
xiii
Orthogonal Polynomials and Special Functions 35. On the Zeros of a Transcendental Function M . V. DeFazio and M . E . Muldoon . . . .
. . . . . . . . . .
385
36. Evaluation of Sylvester Type Determinants Using Block-Triangularization
0. Holtz . . . . . . . . . . . . . . . . . . . . . . . . . . . 37. Square Summability with Geometric Weight for Classical Orthogonal Expansions D. Karp . . . . . . . . . . . . . . . . . . . . . . . . . . 38. The First Positive Zeros of Cylinder Functions and of Their Derivatives L. Lorch . . . . . . . . . . . . . . . . . . . . . . . .
395
.
407
. . .
423
. . .
429
Reproducing Kernels and Related Topics 39. Bergman Kernel for Complex Harmonic Functions on Some Balls K. hjita . . . . . . . . . . . . . . . . . . . . . . . .
40. Applications of Reproducing Kernels to Best Appoximations, Tikhonov Regularizations and Inverse Problems S. Saitoh . . . . . . . . . . . . . . . . . . . . . . . . . . .
439
41. Equality Conditions for General Norm Inequalities in Reproducing Kernel Hilbert Spaces A. Y a m a d a . . . . . . . . . . . . . . . . . . . . . . .
447
. . .
Time-Frequency Analysis, Wavelets and Applications 42. Comparing Multiresolution SVD with Other Methods for
Image Compression R. Ashino, A. Morimoto, M . Nagase and R . Vaillancourt
.
43. Non-Translation-Invariancein Principal Shift-Invariant Spaces J . A . Hogan and J. D . Lakey . . . . . . . . . . . . . . . . .
457 471
44. Time-Frequency Spectra of Music
J . S. Walker and A . J . Potts . . . . . . . . . . . . . . . . .
487
xiv
Contents
Toeplitz-Like Structures in Analysis and Applied Sciences 45. Dynamics of Spectra of Toeplitz Operators S . Grudsky and N . Vasilevski . . . . . . . . . . . . . . . . .
495
46. Operator Equalities for Singular Integral Operators and Their Applications
0.Karelin . . . . . . . . . . . . . . . . . . . . . . . . . .
505
Value Distributions of Complex Functions, Generalizations and Related Topics 47. On Sets of Range Uniqueness for Entire Functions M . T . Alsugaray . . . . . . . . . . . . . . . . . .
. . . . .
515
48. Analytic Mappings in the Tree M u Z t ( K [ z ] ) K . Boussaf, A . Escassut and N . Mainetti . . . . . . . . . .
519
49. Using Level Curves t o Count Non-Real Zeros of f" S.Edwards.. . . . . . . . . . . . . . . . . . . .
531
. . . . .
50. The Functional Equation P ( f ) = Q ( g ) in a p A d i c Field A . Escassut and C . C. Yang . . . . . . . . . . . . .
. . . . 539
51. Conjectures and Counterexamples in Dynamics of Rational Semigroups R. Stankewits, T. Sugawa and H . Sumi . . . . . . . . . .
. . 549
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials Richard Askey Department of Mathematics University of Wisconsin Madison WI 53706 USA
[email protected]
Summary. A very attractive determinant evaluation stated by Sylvester is proven in two different ways; an elementary argument using row and column operations, and by the use of a specific set of orthogonal polynomials, symmetric Krawtchouk polynomials. The second argument is generalized to a number of other sets of orthogonal
polynomials.
1 Introduction Over 50 years ago Mark Kac published a paper, “Random Walk and the Theory of Brownian Motion” [7],which continued earlier work of Smoluchowski and others. Kac’s paper was awarded the Chauvenet prize. When the Mathematical Association of America published two books of Chauvenet prize winners, Kac was given the opportunity t o comment on this paper [8]. The last part of his comments were: Finally, an oddity. From a paper by Istvan Vincze, Uber das Ehrenfestsche Model1 des Warmeubertragung, Archiv der Mathematik, XV (1964) 394-400, I have learned that the eigenvalues of P(m(n;1) (actually 2RP(mln;1))were surmised by Sylvester. Sylvester’s paper - a record in brevity - is reproduced below in its entirety:
T H ~ O R ~ MSUR E LES D~TERMINANTS [Nouvelles Annales de Mathkmatiques, XIII. (1854), p. 305.1
2
Richard Askey Soient les d6terminants
la loi de formation est kvidente; effectuant, on trouve
A,
A2
- 1 2 , A(X2
(A2
-
- 2 7 , (A2 - 12)(A2
- 3 7 , A(A2 - 2”(P
l 2 > ( A 2 - 32)(A2 - 52), A(A2 - 2’)(A2
- 421 7
1,
- 42)(A2 - 62
et ainsi de suite. There is no indication that Sylvester had a general proof and the mystery is why did he publish the note. Perhaps a sleuth with a historical bent will find a solution. The present paper has three goals. The first is to show that while Sylvester did not include a proof of this determinant evaluation, there are proofs which he could easily have given. One will be given here, and Olga Holtz 161, has found another one. The second reason for this paper is to give another proof of Sylvester’s claim, and show how this argument can be used to evaluate a few other determinants. This argument will also give easy constructions of the right and left eigenvectors of the upper left hand corner minors. In his paper, Kac claimed to have had some trouble finding the left eigenvectors. The third reason is to suggest that someone rewrite Kac’s paper making the use of orthogonal polynomials explicit, rather than implicit as they were in many places in “71.
2 A Proof of Sylvester’s Determinant Result Sylvester’s determinant is the tridiagonal determinant
(2.1)
The evaluation of Dn(z) can be done using elementary row and column operations. For simplicity, when n = 4 we have
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials
2-33-xx-33-2 3 x -2 3 -x x -3 ~ ~ 1 3
[
=
2-3 0 3 2+1
0 1
(
3
alternately subtract ~andaddlower x 1 rows
0
add the column to the Ieft of each column
(x - 3)03(x + 1 ) .
In general, x 1 0 0 0 N x 2 0 0 ON-1 z 3 0 DN+l(X) = 0 0 N - 2 ~4
0
... 2xN 01 x
0
... ... ...
X-N N-x X-N N-x N x+I-N N-z X-N 0 N-1 x + ~ - N N-x 0 0 N-2 x + ~ - N
()2.2)
... 2 ~ - I N - x 0 1 x
0
X-N 0 0 0 N x+l 1 0 0 N - 1 ~ + 1 2 0 0 N - 2 ~ + 1
... 0
... ... ...
0
* * *
2~+1N-1 0 1 x+l
4
Richard Askey
This gives
D ~ N ( x=) (x2 - 12)(x2- 32)-
(x2- ( 2 N - 1)2)
(2.3)
and
D2N+l(z)= z ( X 2
- 22). ' ' (X2 - (2N)2).
(2.4)
This determinant along with two related ones was given as a problem in [3]. The other two determinants are also tridiagonal, and are 1 0 0.s. 0 2-2 2 0--- 0 0 -(N-l)z-43
X
-N AN+&)
=
0 0
0 0 (2.5)
...
0 0
0 0
0 0
0
o...
N
-22-2(N-1) -1
X-2N
and
N ( a - 1) x-1 2a 0 ... 0 0 ( N - l ) ( ~ - l )~ - 2 3 ~ . * *0
...
0 0
0 0
0 0
0 0
(2.6)
0 - . * 2 ( -~ 1) x - ( N - 1) N U 0 ... 0 a-1 2-N
The values of A N + ~ ( and x ) B N + ~ ( are x ) given as
A N + ~ (= x )(X - N)"-l N
BN+~(= z)
H[x+ ( N
- 2j)a - N
+j ]
(P.290)
(2.7)
(p.290).
(2.8)
j=O
An outline of how to evaluate A N + ~ ( was x ) given on page 229. This outline was adapted to obtain the evaluation of D N + ~ ( xgiven ) above. It is easy to see that the evaluation of B N + ~ ( xgives ) the evaluation of A N + ~ ( xReplace ). x by ~ / 2 multiply , each row by 2 and let a = To get Sylvester's determinant from B N + ~ ( xreplace ), x by ax, divide each row by a , and let a + 00.
i.
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials
5
3 Orthogonal Polynomials and Tridiagonal Matrices Another Proof of Sylvester’s Claim To connect the determinants AN(^), B N ( ~and ) D N ( ~and ) find other determinants whose eigenvalues can be determined, it is useful to connect them with orthogonal polynomials. Let {p,(X(z))}, n = 0 , 1 , . . . ,N be a set of polynomials with p,(X(z)) a polynomial of degree n in X(z). Usually X(z) is taken to be 2, but we will be interested in some cases which can be more attractively described for other choices of X(z). Usually N is taken to be infinity, but here it will be finite. {p,(X(z))} is orthogonal when ~p,(X(Z))pm(X(Z))da(5)
= 0, m
# n iN,
=h,>O,
(34
m=nLN,
with da(z) a nonnegative measure. Some of the theory of orthogonal polynomials can be worked out when this restriction on da(z) is removed, and replaced by h, # 0, n = 0 , 1 , . . . , N . If {p,(X(x))) is orthogonal, then
+ bnpn(X(z)) + cnp,-l(X(z)),
X(z)I)n(X(z)) = anp,+l(X(z))
(3.2)
with p-l(X(z))= 0, po(X(z))a constant which is often taken to be 1,a,, b,, c, real, (3.3)
(3.4)
(3.5) (3.5)
and thus a,-lc, > 0, n = 1,2,. . . ,N . See Szego [ll].Formulas for specific sets of orthogonal polynomials are given in [9]. The recurrence relation can be rewritten in various forms, using
Pn(X(4)
= tnm(X(z>).
(3.6)
In monic form, it is X(Z)T,(X(Z))
= Tn+l(X(X))
.-1(X(z)) = 0 ,
+ brim(+)) + an-lc,rn-l(x(z>) TO(+))
(3.7)
= 1, tn+1 = t n / a n .
Another form which will be useful is m. e. muldoon
6
Richard Askey
In this case tn+l = ~ + ~ t , / u , .
.(3.9)
To see this, expand using the last row or column. Sylvester had
He claimed that T N + ~ ( x )vanishes on an arithmetic progression of step size two which is symmetric to the origin. To tie this up with more standard notation for a set of polynomials called symmetric Krawtchouk polynomials, a shift and rescaling of z will be made, so that the resulting polynomials K N + ~ ( z ;N ) will vanish when z = 0, 1,...,N . We will treat the more general Krawtchouk polynomials which are orthogonal on x = 0,1, . . . ,N with respect to the finite binomial distribution. To do this it is useful to introduce the usual notation for hypergeometric functions. The shifted factorial ( u ) k is defined by
i,
( ~ ) k= U(U
= 1,
+ 1).. . (U + k - l),k = 1 , 2 , . . . , k = 0.
(3.10)
The generalized hypergeometric series, or hypergeometric series, is (3.11)
Krawtchouk polynomials Kn(z;p , N ) are defined by (3.12)
Ordinarily, the restriction bj # 0, -1,. . . , is made for j = 1,.. . ,q, since a zero will arise in the denominator because (-n)k = O , k = n + l , n + 2 , . .
However, this restriction has to be relaxed to include Krawtchouk and some other important sets of orthogonal polynomials. The added restrictions of
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials
7
n, z = 0, 1,. . . , N are made to get around the problem of dividing by zero. When only n = 0 , 1 , . . . ,N is assumed, the series is
2 k=O
(-n)k(-z)k
(-N)kk!
(:)
where the new sum starts when k = N
k
+ another sum
+ 1 and can be written as
00
(-n)k+N+I
(-z)k+N+I
k=O
-
(-n>,(l)N-n(-+V+l (- N )N (1)N+ 1PN+' (I"
N+l-n,N+l-z
N+2
;-). 1
P
When 2 = 0,1,. . . ,N , this sum vanishes because of the factor (-z)~+1.Thus, while K,(z; p , N ) is a polynomial of degree n in z for n = 0,1,. . . , N , it is (3.13) which is not the same as the above extended version of zJ71 ( - n , -z;
-N
'>
P
unless z = 0,1,. . . , N . Krawtchouk polynomials satisfy the recurrence relation
(3.14)
with
a, = p ( N - n), c, = n(1- p ) , a,
+ b, + c, = 0,
(3.15)
the last since K,(O;p, N ) = 1. See [9] for this and other facts about Krawtchouk polynomials. The orthogonality relation is
(3.16)
The determinant representation (3.9) when p =
3 is
8
Richard Askey -N - x 2
1 2 0
N 2
0
0
... 0
0
0
N2 - x - N 2- 1
0
...
0
0
0
N2 - x N-2 ... 2
0
0
0
z2
.
(3.17)
... 0
0
0
0
... 2 2
0
0
0
0
...
0
n-1
7
-N2 - x
The polynomial P,(x; !j,N ) is a multiple of Kn(-x; $, N ) , so to show Sylvester’s claim is true, it is sufficient to show that K N + ~ ( x ; P N,) vanishes when x = 0 , 1 , . . . ,N , and to check the coefficient of the highest power of x . We will do this for K N + ~ ( x ; N P ,) by showing that the recurrence relation (3.14)(3.15) holds when n = N and x = 0 , 1 , . . . ,N . What needs to be shown is that (3.18)
- x K N ( x ; P , N ) = -N(1 - p ) [ K N ( x ; p N , ) -KN-l(x;P,N)]
when x = 0 , 1 , . . . ,N . The left hand side is N
(-.)k k=O
k=O
k!
).(!
= - x (1 -
:)
(3.19)
when x = O , l , . . . , N by the binomial theorem. Notice this is no longer a polynomial in x , which might seem strange since the left hand side of (3.19) is a polynomial of degree N 1 in x . However, (3.19) has only been shown to be true for x = 0,1,. . . ,N , and it is not true for all real x . The right hand side of (3.18) is
+
-w -PI -
-
c
[ ( - N ) k - (1 - N ) k ] ( - x ) k
(-
k=l
k 2 (t)i
xN(1-p) p(-N)
4 1 -P>
N ) k k!
k=l
(;)k
(1 - N ) k - l [ - N - ( - N + k ) ] ( l - Z ) k - l (1 - N ) k - l k !
(l -k! x)k
= -x
-
(;)
IC-l
!JZ,
k=l
which shows that (3.18) holds when x = 0 , 1 , . . . ,N . (3.14) cannot hold for all x when n = N since the left hand side is a polynomial of degree N 1 while the right hand side is a polynomial of degree N since AN = 0. Now that we have the evaluation of the determinant for general Krawtchouk polynomials, it is possible to use it to derive an evaluation of B N + ~ ( z )De. fine B,(x; a , N ) by the recurrence relation implicit in the definition (2.6) of
+
B N + 1 (XI
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials
9
Bn+l(z;a,N ) = (z - n)B,(z; a,N ) - an(a - 1)(N+ 1 - n)B,-l(z; a,N ) or
zB,(z; a,N ) = Bn+l(z; a,N)+nB,(z; a,N)+a(a-l)n(N+l-n)B,-l(z;
a, N ) .
(3.20)
To match this up with (3.14), we rewrite (3.14) explicitly
-zK,(z;p, N ) = p ( N - n)Kn+l(z;p,N )
+ p p - 1).
- p ~ ] K , ( z ; pN, ) + (1 - p)nKn-l(z;p, N ) .
First, replace z by z/(2p - 1) in (3.20) and multiply by 2p - 1. Next subtract pNB,(x; a,N ) and then replace z by z p N . This gives for these modified polynomials:
+
(3.21)
Then replace Cn(z) by knDn(z)to get (3.22)
Choose k, so that
kn,-1 - a ( N -
- n).
kn
Then (3.23)
+ (2p - 1)(u - l)nD,-1(z). Let (2p - 1)a = p . Then (2p - l ) ( a - 1) = 1 - p , so (3.23) is zDn(X) =
+
(pN - n)Dn+l(z) [ ( 2 p- 1). +(I - p)nDn-.l(z).
- pN]D,(z)
(3.24) (3.24)
Finally, set Dn(-s) = K,(z)..The result is (3.14) for K,(z;p,N). Tom Koornwinder showed me how t o make this reduction. The determinant for K N + ~ (pz, ;N ) is
10
Richard Askey (3.25)
-X
+p N
(1-p)
0
pN -z+pN+(1-2p) 2 ( 1 -PI
0
...
0
0
P(N-1)
...
0
0
...
0
0
-z+pN+2(1-2p)
.. 0
0
... N ( 1 - p ) --2
+ p N + N ( 1 - 2p) I = (--2)N+1.
While orthogonality with respect to a positive measure only holds when 0 < p < 1, (3.25) continues to hold for all complex p since the determinant is a polynomial in p .
4 Orthogonal Polynomials and Tridiagonal Matrices -
Other Examples There are other sets of orthogonal polynomials which are orthogonal on a finite set of points for which both the orthogonality relation and the three term recurrence relation are known explicitly. Some can be represented as generalized hypergeometric series, while others are basic hypergeometric series. Basic hypergeometric series will be described later after the remaining hypergeometric series examples are given. Proofs will not be given except for the most general example, since they are all similar to the proof given in Section 3, and the most general example contains all of the others as special or limiting cases. The first hypergeometric examples beyond Krawtchouk polynomials in terms of simplicity of the recurrence coefficients are dual Hahn polynomials. Instead of linear functions of n, their recurrence coefficients are quadratic in n. As hypergeometric series they are (4.1)
where X(x) = -x(x
+ y + 6 + 1).
(4.2)
Their recurrence relation is (4.3) with
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials
11
+ + 1)(N- n)
a, = ( n y
c,=(N+l+b-n)n b, = -a,
- c, = -N(2n
(4.4)
+ y + 1) + n(2n + y - 6 ) .
The Sylvester type determinant is (4.5)
...
0
2(N
+ b - 1)
X(X)
+N
y +5)
-2(y - 6
+ 4)
I
..
I
0
The special case when y = b = 0 is very attractive. It is a symmetric tridiagonal matrix with N , 2 ( N - l),. . . ,k ( N - k l),. . . as the off diagonal terms and --z(x 1) N 2k(N - k) on the diagonal. The value of the determinant is (-l)N+l(-x - N - l ) 2 ~ + 2 .The special case y 6 = -1 gives - z2). an interesting value for the determinant: The next examples are the Hahn polynomials. These are dual to the dual Hahn polynomials, interchanging n and x. They are polynomials of degree n in z, but the recurrence coefficients are rational functions which are not polynomials.
+ + +
+
nz,(j2
+
(4.6)
The three term recurrence relation is (4.7)
with
12
Richard Askey
+ p + l ) ( n + a + 1)(N - n ) + + p + 1)(2n+ a + P + 2 ) n(n + a + P + N + l ) ( n + P) c,= ( 2 n + a + P)(2n + + p + 1 )
a, =
( n+ a (2n a
(4.8)
b, = -a, - c,.
The determinant is messy to write explicitly, so this is left t o the interested reader. The value is ( - z ) N + ~ when X(z) = --2 is used in (3.9). The special case a = /3 gives a more attractive determinant, since
+ + 1)(N- n )
a, =
(n 2a 2(2n
c, =
n(n 2 a N 1) 2(2n 2 a 1)
+2 a + 1)
+ + + + +
(4.9)
b, = N / 2 .
The most general of the classical hypergeometric orthogonal polynomials are called Racah polynomials. They can be given as -n,n
+a +p+
+ +
1, --5,-2 +y
+
+6 + 1;l)
(4.10)
+ +
where X(z) = -z(z+y 6 1) and one of a 1,p 6 1 or y + 1 is - N . We will take p 6 1 = -N and remove 6, but other choices lead to similar formulas. When y --+ 00, one gets the Hahn polynomials, and when a + 00, one gets the dual Hahn polynomials. One can obtain Krawtchouk polynomials from either Hahn or dual Hahn polynomials by a similar limiting procedure. From Hahn polynomials, let (a + p ) / a = l / p and let a and p go to infinity. The recurrence relation for Racah polynomials is
+ +
(4.11))
with a, =
c, =
(n
+ a + 1)(n+ + p + l ) ( n + y + 1)(N- n) (an+ a + p + 1 ) ( 2 n + + p + 2 ) Q
(2n
+ a + P ) ( 2 n + a + p + 1)
(4.12)
b, = -a, - c,.
Again, the special case a = P is attractive. The value of the determinant with X(x) = -z(z y 6 1) is ( - z ) ~ + l ( x y 6 1 ) ~ + 1 .
+ + +
+ + +
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials
13
Basic hypergeometric series are series C C k with C k + l / C k a rational function of q k . The analogue of the shifted factorial ( a ) k is
k = 1 , 2 ,.",
( a ; q ) k= ( l - a ) ( l - a q ) - . . ( l - a q k - i ) ,
(4.13) =1
,
k=O.
The simplest basic hypergeometric series is (4.14)
Ordinarily, q is assumed to satisfy 141 < 1, but since we will be dealing with polynomials this is not necessary. Limiting cases sometimes use (4.15) and Hahn [5] introduced a notation to handle this case. See Gasper and Rahman [4, (1.2.22)] for this notation. We will not need it although it could be used for some of the special cases. q-Racah polynomials are defined by (4.16)
X(X) = -(1- q-=)(l
(4.17)
- q"+'cd).
We assume that one of aq, bdq or cq is q - N , and as was done for Racah polynomials, assume that bdq = q-N and remove d. The three term recurrence relation is
with a, =
(1 - abq"+l)(l - aq"+')(l - q"-N)(l - cqn+') (1 - ubq2"+1)(1 - ubq2n+2)
7
b, = -a, - I&. The value of the determinant is (4.20)
14
Richard Askey
To show this we need to show that the recurrence relation (4.18) holds for n = N when x = 0 , 1 , . . . ,N and qx+'cd = q"-Nc/b = q - k , k = 0,1,. . . ,N . This was shown in [2] when z = 0 7 1 , .. . ,N , so it suffices to show this is true when q" = bqN-'/c, k = 0 , 1 , . . , ,N . This will follow when we show that
cq-N (1 - q N ) ( l - b q N ) ( l - $qN)(-abqN+l) -(1 - abq2N)(1- abq2N+1) b
This is true when q" = bqN/c, i.e. when k = 0, since both sides vanish then. For the other values the right hand side can be changed using formula (3.6) in [2] t o replace the difference by a set of factors times (4.22)
In both the 4'p3 in (4.21) and in (4.22), the series terminate because of q-" and ql-", k = 1 , 2 , .. . ,N , so the series become 3 ~ 2 ' ~In . both cases the series have the following form (4.23)
and so can be summed. See [1,(10.10.3)], [4,(1.7.2)]. A routine calculation shows that the two sides are equal. This along with a check that the coefficient of the highest powers are equal shows that the determinant (3.9) with n = N has the value (-l)N+l (q-"; q)N+1(q"+'Cd; q)N+1. (4.24) There are many special cases which could be written, but none are as attractive as the one Sylvester stated, so they will not be listed here. The needed formulas are given in [9].
5 Eigenvectors Returning to the determinant (3.9) and the recurrence relations (3.2), (3.7) and (3.8), we can give both right and left eigenvectors for the matrix
Evaluation of Sylvester Type Determinants Using Orthogonal Polynomials
--bo
uo
0
O
c1
-bl
a1
0
0
c2
-bz
a2
..-
0 0 0
0 0 0
0 0 0
-
...
0 0
0 0
0 0
=
0 0
c,-1
-bn-l
0
15
M.
U,-l
-b,
C,
is equivalent t o ztj(X)
with co = 0. If
tj(-z)
ZPj
= ~ j t j + l ( z) bjtj(z)
+
cjtj-l(z)
(5.3)
= (-l)jpj(z), then p j ( ~ satisfies )
(4 = U j P j + l ( . ) + b j P j (4+ CjPj- 1 (4
which is the recurrence relation satisfied by p j ( than X(X) for simplicity. For the left eigenvectors for the matrix M.
TM
~ )Here .
(5.4)
we have used x rather
= XT
(5.5)
is equivalent t o z t j ( z ) = Cj+ltj+i(X) - b j t j ( X )
+uj-itj-i(~),
(5.6)
which is related to (3.8) in the same way that (5.3) is related t o (3.2). The eigenvectors are these vectors when rn+l(z)= 0. There are n + l real values by a well known theorem about polynomials which are orthogonal with respect t o a positive measure. When n = N , we have found the zeros explicitly for the polynomials treated above, Racah and q-Racah polynomials and some of their special cases or limits. For a treatment of this section see Wilf [12]. There are papers by Cayley and Painvin which are related to the work in this paper. See Thomas Muir, “The Theory of Determinants, Volume 2”, pages 429-434.
References 1. G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge Univ. Press, Cambridge, 1999.
16
Richard Askey
2. R. Askey and J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols, SIAM J. Math. Anal. 10 (1979), 1008-1016. 3. D. K. Faddeev and I. S. Sominskii, Problems in Higher Algebra, translated by J. L. Brenner, Freeman, San Francisco, 1965. 4. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, 1990. 5. W. Hahn, Beitrage zur Theorie der Heineschen Reihen, Math. Nachr. 2 (1949) 340-379. 6. 0. Holtz, Evaluation of Sylvester type determinants using blocktriangularization, ISAAC 2003 Proceedings, . 7. M. Kac, Random walk and the theory of Brownian motion, American Mathematical Monthly, 54 (1947) 369-391. 8. M. Kac, Appendix to [7]. In J. C. Abbott, editor, The Chauvenet Papers, vol. I, MAA, Washington, D.C. 1978, 276-277. 9. R. Koekoek and R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, h t t p ://aw .twi .t u d e l f t .nl/Nkoekoek/askey/ 10. J. J. Sylvester, Nouvelles Annales de Mathkmatiques, XI11 (1854), 305, Reprinted in Collected Mathematical Papers, vol. 11, 28. 11. G. Szego, Orthogonal Polynomials, fourth edition, American Mathematical Society, 1975. 12. H. Wilf, Mathematics for the Physical Sciences, Wiley, New York, 1962, reprinted, Dover, New York, 1978.
Geometric Analysis on SubRiemannian Manifolds Ovidiu Calin’, Der-Chen Chang2, and Peter Greiner3 Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197 USA ocalin0emunix.emich.edu
Department of Mathematics Georgetown University Washington DC 20057 USA changQmath.georgetown.edu
Department of Mathematics University of Toronto Toronto ON M5S 3G3 Canada
[email protected]
Summary. We discuss analytic and geometric results on a step 2k pseudoconvex hypersurface in C 2 ,with special emphasis on complex Hamiltonian mechanics, which yields an integral form for the fundamental solution of the sub-Laplacian.
1 Introduction We propose a possible structure for geometrically invariant formulas for the fundamental solutions, heat kernels, and wave kernels of partial differential operators of the form
where XI,. . . , X, are m linearly independent vector fields on M,, an n-dimensional real manifold with m 5 n. The subspace T x spanned by XI, . . . , X, is called the horizontal subspace, and its complement is referred t o as the missing directions. We note that T x = T M , if and only if A x is elliptic. When T x # TM,, the operator is non-elliptic. However, if we assume X satisfies Chow’s condition [13]:“the horizontal vector fields X and their brackets span TM,”,
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
18
then Hormander’s theorem [19] implies that AX is subelliptic and hence, hypoelliptic. The number given by the minimum number of brackets necessary to generate T M , plus 1is referred to as the “step” of the operator A x . In particular, an elliptic operator is step 1,one bracket generators are step 2, and everything else is referred to as higher step. To illustrate the proposed structure, we shall discuss a family of operators for which “explicit” fundamental solutions given in geometric terms are available. To avoid technical complications, we shall work in the three dimensions, ~ 1 , ~ 2 ,= t ) (z, t), with 2 vector fields,
x=(
The differential operator one wants to invert is
+
A x is step 2 at points xs x; # 0, and step 2k otherwise. The fundamental solution K(x,y) of A x is the distribution solution of
where the derivation is taken with respect to the x-variable. We shall look for K(x,y) in the form
where the function g is a solution of the Hamilton-Jacobi equation
given by a modified action integral of a complex Hamiltonian problem. The associated energy E --- - ag
37
is the first invariant of motion, and the volume element v is the solution of the second order transport equation
(4) where
Geometric Analysis on SubRiemannian Manifolds
19
is differentiation along the bicharacteristic curve. Equation (4)can be reduced to order 1 when k = 1 since Ax(Ew) = 0 in that case. Let
denote the Hamiltonian, where
El,
& and 0 are dual variables to z1, 2 2 and
t respectively. The complex bicharacteristics are solutions of the Hamiltonian system of differential equations
f
=H ~ ,
e=-H~,
Xj =H
tj
~ ~ ,=- H ~ ~ ,j =i,2
with the non-standard boundary conditions z(0) = 50,
z(7) = z,
t(7)= t ,
e = -i.
(7)
The energy E is given by
and the modified action g is given by
We note that t , the “missing direction”, must be treated separately. The fundamental solution K ( x ,y) defined by (2) has a simple geometric interpretation. The operator A x has a characteristic variety in the cotangent bundle T * M , given by H = 0. Over every point x E R3,this is a line, parametrized by 0 E (-9m), (1 =
-2kz2lzl
2k-20
,
&. = 2kz1lzl2k-20 .
Consequently, K may be thought of as the (action)-’ summed over the characteristic variety with measure Ev.When A x is elliptic, its characteristic variety is the zero section, so we do get simply l/distance, as expected. When A x is sub-elliptic, 7 g behaves like the square of a distance function, even though it is complex. In the rest of the paper, we shall discuss analytic and geometric problems related to the operator A x , the modified complex action g and the volume element u. This article is one of a series (see [ S ] , [9], [lo],and [ll]),whose aim is to study the subRiemannian geometry induced by the operator A x and its analytic consequences. The paper is based on lectures given by the second and the third author at the Fourth ISAAC Congress which was held at York University, Toronto, Canada, August 11-16, 2003. The authors take great pleasure in expressing their thanks to the organizing committee, especially Professor Man Wah Wong for the invitations and the warm hospitality they have received while they visited York University.
20
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
2 The Complex Action
&.Hamilton’s function is
The Laplace operator in Rn is d = - Cj”=,
H(E) = r:
+ ,t; + . . + r;. ’
A differential operator is elliptic if the Hamiltonian function H(J) = 0 0. Hamilton’s equations for the bicharacteristics are
%
6=
where s E [0, r ] ,with x(0) = 0 and x ( r ) = x. Thus, for j = 1,.. . ,n, 5 j ( s ) = 2&
and &(s) = 0,
so & ( s ) = c j , a constant, and ?j((s)
= 2cj
~ j ( 0=) 0 Xj(7) = x j
or Xj(.)
=+
Xj(S) = 2cjs
+d j ,
=+
dj = O , xcj =+ c . - - 27’
xcj = -s, 7
“j
<j(S)
= -,
27
Definition 2.1 x(s) = (xl(s),. . . ,x,(s)) connecting xo to x in tame 7, is called a geodesic, i f it is the projection of a bicharacteristic curve onto the base. According to the above calculations the geodesics are straight lines in the Euclidean case. The action integral is
which yields the distance, and is the solution of the Hamilton-Jacobi equation
In this case, the heat kernel for the Laplace operator d can be written as
Geometric Analysis on SubRiemannian Manifolds
21
We are interested in subelliptic operators. The operator one wants to invert is 1
Ax
=
Z(X?+ Xi)
where the vector fields are defined by (1). The Hamiltonian A x is
where (1, (2 and 8 are dual variables to 2 1 , 22 and t , respectively. The bicharacteristics are solutions of the Hamilton’s system Xj
=Htj,
(j
= -Hzj,
j = 1,2
and
t=He,
e=-H,=O
with boundary conditions. The momentum 8 is a constant along bicharacteristics. Let ( x ( s ) , ( ( s ) )s, E [O,+r] be a solution for the Hamilton’s system given by the Hamiltonian (1) with the non-standard boundary conditions
2(o) = zo,
+)
= X,
t(7) = t ,
e = -i.
(2)
Definition 2.2 The complex action associated to this complex bicharacteristic is
The formal relation between the complex action and the classical action S is g =
s - it(0)
with
where B(s) = B = -i. The Lagrangian is given by 2
L
= C(j(s)ij(s) - it - H ( x ( s ) , ( ( s ) ) . j=l
A direct computation yields
22
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
The Euler-Lagrange system is
with boundary conditions (2). Multiplying the first equation by second by k2 and adding yields iilk1+ 22k2 = 0 , i.e. 1 -(kf 2
k1
and the
+ 5;) = E ,
where E is the “energy”. Multiplying the first equation of the system ( 4 ) by and the second equation by 21, yields
22
ii1z2= 4 k 2 e ( ~ ( 2 k - 2 k 2 z 2 i 5 2 ~ l= - 4 k 2 e ) ~ ) 2 k - 2 k l z l .
Because of rotational symmetry, we may rewrite the Lagrangian in terms of polar coordinates z1 = r cos 4, 2 2 = r sin 4. Subtracting the above two equations, one has ii1z2- 2 2 ~=14e(k
+ 1)1Z)2k(i2z2+ *lzl).
This is equivalent to
+ 1)r2’(222i2 + 2 = 2 e ( k + l)r2k(zf+ 2;)
(?I22 - k2zl)’ = 2 e ( k
= 2e(k
~ ~ 5 ~ )
+ i)r2k(r2>’
(
= 28 ( r 2 ) k + 1 ) .
and there is a constant 0 , the “angular momentum”, given by k I x 2- i221 = 2er2@+l)
+ n.
3 The Complex Action g The Hamiltonian in polar coordinates In this paragraph we shall write the Hamiltonian
in polar coordinates. The associated Lagrangian is given by (3). In polar coordinates (4,r ) the Lagrangian is
L
=
1 2
-(P
+ r”2)
- i(t + 2kT-2”).
Geometric Analysis on SubRiemannian Manifolds
23
Let p4, pry pt be the momenta associated with the coordinates (6, r, t.
dL
p --=+i+=p,,
-
a+
Legendre transform yields the Hamiltonian
+ +
H = ~ 4 4pr+ ptt
-L
Proposition 3.1 The Hamiltonian (1) has the following form in polar coordinates (4, r ) :
The Hamiltonian does not depend on the variables t and conserved quantities:
4. This yields the
pt = -Ht = 0 ==+ pt = 0 = -i (constant), p+ = -H+ = 0 r 2 (4. - 2kir2k-2) = C (momentum constant). a
The eiconal equation
We want to solve the Hamilton-Jacobi equation in polar coordinates
Separating variables,
then (3) yields
U’(r) + H = 0,
which yields
U ( r ) = -Er
+ ,B,
since H = E = constant along the bicharacteristics, and ,B is a parameter. Consider E as a parameter. Substituting in (3) yields an eiconal equation for
W
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
24
or.
We assume that d W / d $ = 0, and d V / d t = 0. Then
V ( t )= 6% + a , where a is a constant. The equation ( 5 ) becomes a first order ODE
dW - = &J2E dr
- 4k2927-4k-2.
Taking 0 = -i and the positive sign,
dW = J2E
ar
+ 4k2r4k-2.
Suppose the geodesic starts at r(0) = ro. Then W depends on parameters E and ro,
W ( r ,ro, E ) = =
1: lr
J2E
+ 4k2u4k-2 d u
d2E
+ 4k2u4k-2 d u -
iTo+ J2E
4k2u4k-2 d u
=I-lo. First,
I=
J' 0
2E + 4k2u4"' du d2E + 4 k 2 ~ 4 k - 2
I'+
d2E + = ~ E T 4k2J. = 2E
1 du 4k2~4k-2
+ 4k2
lT
J2E
U4k-2
+ 4 k 2 ~ 4 k - 2d u
Next [3] and [ll]give
Lemma 3.2 Let r ( s ) = J x : ( s )
Therefore,
+ x;(s) and r ( r ) = r. Then
Geometric Analysis on SubRiemannian Manifolds
25
Replacing r by ro yields an expression for 10
Summing up
+ p + et +
- r o / w + a + p ) .
+
Let go = g ( r 0 , 0). Then the constant Q ,f?= go. Taking go = 0 we recover the following result of Beals, Gaveau and Greiner [3].
Theorem 3.3 The complex action is given by
Corollary 3.4 The complex action starting at the origin i s given by
In the step 2 case the energy E depends on r(7) = r ,
Solving for E yields
E=
2r2 sinh2(2r)
Corollary 3.5 W h e n k = 1, the complex action starting f r o m the origin i s given by g = -it r2coth(2r) = -it (xf xi) coth(2r).
+
+
+
26
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
Proof. Setting Ic = 1 in Corollary 3.4 yields
Using (8) we find
+
E r2 1 sinh2(2r) -+r2= +r2 = r 2 sinh2(2r) sinh2( 2 7 ) = r2
cosh2(2r) sinh2( 2 7 )
= r2 coth2(2r), so = -it
+ r2 coth(2r).
4 SubRiemannian Geometry SubRiemannian geometry starts with Carathkodory’s formalization of thermodynamics [12],where the quasi-static adiabatic processes are related to the integral curves of a Darboux model. Recall that the classical action from the
the integral of the Lagrangian along the bicharacteristic with boundary conditions: z(0) = 0, x ( r ) = z, t ( 0 ) = 0, t ( r )= t. Here we fix t o = t ( O ) , so 6 cannot be chosen arbitrarily; it is a real constant along the path. In the case of the complex action, the boundary condition t ( 0 ) = 0 is replaced by 0 = -i. The Hamiltonian H is homogeneous of degree 2 with respect to ((1, (2, e), so
where HOis the constant value of the Hamiltonian along the bicharacteristics. The Hamilton- Jacobi equation may be rewritten as
so
Geometric Analysis on SubRiemannian Manifolds
27
Definition 4.1 The modified complex action function f is f ( x ,t , 4 = T
d X ,t,
4,
(1)
where g ( x ,t , r ) is the complex action.
We shall show that the critical points of f with respect to r yield the lengths of the (real) subRiemannian geodesics. This was shown first by Beals, Gaveau and Greiner (see [l]and [2]) for the step 2 case. The general results of this section were obtained by the authors (see [lo]).The geodesics starting from the origin are described by Theorems 3.1 and 3.2 in 191 (see also [l]and [161): Theorem 4.2 There are finitely many geodesics that join the origin t o ( 5 ,t ) i f and only i f x # 0. These geodesics are parametrized by the solutions $I of:
There is exactly one such geodesic af and only if:
It1 < P(C1)1x12k, where c1 is the first singularity o f p . The number of geodesics increases without bound as --+ 00. If 0 5 $1 < . . < $JN are the solutions of (2), then there are exactly N geodesics, and their lengths are given by
&
where (2k-lW
2k-2
( v )dv] 2k
+ p ( $ J ) )s i n m ( ( 2 k - 1)q) 2k
k2'((1
(4)
Theorem 4.3 The geodesics that join the origin to a point (0, t ) have lengths
with the constants M and Q expressed an terms of beta functions
M = B(-
1 4k-2'2
For each length, ,s
sl.
") '
the geodesics of that length are parametrized by the circle
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
28
Differentiating (l),yields
af dr
a(rg) = g 67
-
ag + r-dT
=g - (g
+ i t ( 0 ) )= -it(O).
For t ( 0 ) # 0, we obtain a modified form of (2) as follows ([S] and [ll]):
J=$ -
t - t ( 0 ) = -P(J)1x12k,
$01
and therefore
Since
= -28r2k-2 ([9] and
[lo]),one has
Now (6) implies
Proposition 4.4 Let x # 0 . rc is a critical point for the modified complex action f ( x , t , r ) if and only if
is a solution of the equation
Also Theorem 4.2 implies
Proposition 4.5 Let action f (2, t , r ) . The numbers
r1, . . . r,
1
be the critical points of the modified complex
73
= 2i
satisfy the equation
r2k-2(s) ds,
j = 1,.. . ,m,
t p =P(-bh
(9)
and for each [j we have a geodesic connecting ( x , t ) t o the origin. The length of the geodesic parametrized by [ j is
l j
Geometric Analysis on SubRiemannian Manifolds
29
Consequently, the critical points T~ of the modified complex action function give the lengths of the geodesics. Here we just mention two special cases: the step 2 and the step 4 cases. 0 Step 2 case: the Heisenberg group Here lc = 1. The vector fields (1)
These vector fields are left-invariant with respect to the Heisenberg translations X O Y = (x1 + Y l , x 2 + Y 2 , t + s + 2 [ X 2 Y l - x l Y 2 ] ) , where x = (51, z2, t ) and y = (yl,92, s). By Theorem 4.2 one knows that there are finitely many geodesics that join the origin to (x,t ) if and only if x # 0 (see also [l]and [IS]).These geodesics are parametrized by the solutions 4 Of
The number of geodesics increases without bound as f$ -+ 00. < $N are the solutions of equation (lo), there are N If 0 5 $1 < geodesics joining origin and their lengths are given by
where v(x) = and
X2
x
+ sin2x - sin xcos x ’
X cL(4= & - cot x.
When x = 0, the geodesics that join the origin to a point ( 0 , t ) have lengths e1,e2,. .
.,
em2 = rnTlt(.
For each length em, the geodesics of that length are parametrized by the circle 91.
When z # 0, Fj = 2i7j and rj is purely imaginary by (7). The lengths of the geodesics between the origin and (x,t), z # 0, are given by (3) and ( l l ) ,
30
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
II
~~
1
I 15
Figure 1: The graph o f p an step 2 case. Figure 2: The graph of v in step 2 case. 0
Step 4 case Here k = 2. The vector fields (1) are x 1
a + 4221x12 -,8 at
=-
8x1
x 2=
a
- 4x+( 8x2
2 8
-. at
There is no group law on IR3 underlying these vector fields. Again, by Theorem 4.2 (see also results in [7] and [S]), one concludes that there are finitely many geodesics that join the origin to ( z , t ) if and only if x # 0. These geodesics are parametrized by the solutions of:
+
There is exactly one such geodesic if and only if (ti < P ( c I ) I ~ ~ , where c1 is the first critical point of p. The number of geodesics increases It1 without bound as 1. 14 --f 00. If 0 5 +1 < . . < +N are the solutions of equation (13), there are exactly N geodesics, and their lengths are given by
i%= v w m )(it1
[ J’lZ sin-;
where u(x) =
16(1
and 2
44 = 3
+ iXi4),
(14)
(v)dv] 4
+ p(x)) sinQ(3z) ’
J’Fsin$ (v)dv A x
When x = 0, one may apply Theorem 4.3 to conclude that the geodesics that join the origin to a point (0, t) have lengths el, i 2 , . . . , where
Geometric Analysis on SubRiemannian Manifolds
with M = B
(i, ;).
For each length
em, the
31
geodesics of that length are
parametrized by the circle S1. From Proposition 3.4 and (13), we may find all the critical points of the modified complex action and hence c j ; readers may consult paper [9] for a detailed discussion. Our results are contained in Proposition 4.6 Let rj denote the critical points of the modified complex action f ( r )= rq(7).Setting ( j = f ( i r j ) ,the lengths of the geodesics between the origin and the point ( q t ) , 1x1 # 0 are given by
Here the function f has the following expression:
with u = 24/33'14 z and am-lw = s n - ' ( f i - 1,k').
The functions E , sd, am, sn and
04
are discussed in Lawden [20].
5 The Fundamental Solution and the Heat Kernel As we mentioned in section 1, the fundamental solution
has a simple geometric interpretation. The operator A x has a characteristic variety in the cotangent bundle T * M , given by H = 0. Over every point x E M,, this is a line, parametrized by B E (-m, m ) , = - 2 k ~ ~ 1 ~ 1 ~ ~ -j2 ~=e 2, k ~ ~ 1 ~ 1 ~ ~ - ~ e .
Consequently, K may be thought of as the (action)-' summed over the characteristic variety with measure Ev. When A x is elliptic, its characteristic variety is the zero section, so we do get simply lldistance, as expected. When A x is sub-elliptic, r g behaves like the square of a distance function, even though it is complex.
32
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
0 Step 2 case In this case, the operator A x is left-invariant with respect to the Heisenberg translations. Thus we may set y = 0 in K(x,y). From Corollary 2.5, we know that g(x,0,T ) = g(x,7) = (z: z;) coth(27) - it. In this case, the volume element v is the solution of the following transport equation -L dV - x ( x j g ) ( x j v ) (Axg)v = 0.
+
a7
+
+
j=l
Moreover, the energy E and the volume element v can be calculated explicitly: E(X,7)=
&I
--
ar
=
2Izl2 sinh2(27)
Change the variable of integration
1 sinh(2~) 47r2 1x12 .
V(X,T)= --
and 7
t o g:
where
g& = lim g = f ( z2l T-+*OO
+ x2) - it. 2
In this notation
v = --eisr/2
1
47r2 J
’
where we made a branch cut
C = (-oo - it,g-]U [g+,-it
+ oo),
with upper and lower directed edges denoted by C+ and C-, respectively. Integrals of u/g vanish on upper and lower semicircles as their radii increase without bound. Therefore we can integrate on a “dumbbell” with waist a t g h . Inside this domain v/g has a simple pole at g = 0. Hence
Geometric Analysis on SubRiemannian Manifolds
Figure 3: The "dumbbell" with waist at gk.
hence
1
- --
/
=i c-
dg
v-
9
=
i -K(x,O),
=
and therefore we obtain a closed form for (3)
33
34
Ovidiu Calin, Der-Chen Chang, and Peter Greiner 1
1
This is the Folland-Stein formula of [14]and [15]on the Heisenberg group. Unfortunately this simple formula is just a lucky coincidence of too much symmetry, and on general non-isotropic Heisenberg groups the fundamental solutions are given in the form (2). The reason is that the fundamental solution must include all the distances, which necessitates the use of g , and the summation over all the distances means integration on 7. Note that the change of variables T t g is reminiscent of classical calculations in actionangle coordinates. 0 The higher step case, k > 1 In this case, there is no group structure and the complex bicharacteristics run between two arbitrary points y and x. We obtain 2 invariants of the motion, the energy E and the angular momentum 0. One cannot calculate them explicitly, but we know their analytic properties, and g (see Theorem 3.3) and v may be found in terms of E and 0. We state the result as follows.
Theorem 5.1 For k > 1, the fundamental solution K ( z , y , t - s) of A x has the following invariant representation,
where the second order transport equation (4) may be reduced t o an EulerPoisson-Darboux equation and solved explicitly as a function of E and 0. Namely, @r/2 2)
= --
2n3k
(A+ - g)-* (A-
-
,)-a
F(P+,P-),
where
-
,A+ = A- =
( ~ +9 z ; ) +~ ( y t + Y , ” ) -~ i(t -
and
with 0, = lim 0. T-Cttoo
Here F i s a hypergeometric function of 2 variables, 0
rl
rl I
I
- -
S)
0+ + g+,
=-
k
Geometric Analysis on SubRiemannian Manifolds
35
When k = 2, the fundamental solution K ( x ,y , t - s) has the following simple form: i K ( z , y , t - s) = 2.rr2dlog h7 where
h ( P , P )=
11 -P2l - i ( P + P ) 1 lP(2
+
1
with
1
.a=-(( ICf 2
+ x ; ) 2 + (y,” +
+ i(t - s ) ) .
This formula first appeared in [17]. Next we write the heat kernel associated to the sub-Laplacian A x on the Heisenberg group in terms of the “distance function” f = r g as follows (see [I] and [S]):
where
Here
f(z, t ,T ) = rg(x,t ,T ) = -irt
+ ~ ( 2 +: 2;)c o t h ( 2 ~ )
is the complex action and
V ( r )=
27 sinh(27)
is the Van Vleck determinant. Using the modified complex action function f (2,t , r ) , one may discuss the small time behavior of the heat kernel. Then we have the following theorems.
Theorem 5.2 Given a fixed point ( x , t ) , 2 # 0 , let 9, denote the solution of equation (2) in the interval [O,.rr/2). T h e n the heat kernel o n H1 has the following small time behavior:
where Q(x,t)=
9c
J[l - 29, coth(28,)](s?
and d , denotes the Carnot- Carathe‘odory distance.
+ 2:) ’
36
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
Theorem 5.3 At point ( O , t ) , t
# 0, we have
One also has
Theorem 5.4 T h e heat kernel Pu(x,t) o n HI has the following sharp upper bound:
There exist no explicit heat kernel for a higher step heat operator as yet. For the examples of this paper we are looking for a heat kernel of the form
where f = r g . turns out to be a constant of motion, just like = -E is; i.e., a constant on the bicharacteristics. Then V is a solution of
2
+
r(T -;-j;Ag)x
av - ZafA X V
= 0,
(4)
where T is derivation along the bicharacteristic curve which is defined by (5). The equation (4)may be put in the following form:
This should be compared to the equation for the volume element of ( 2 ) which is a solution of av d g (T A x g ) - - -AXW = 0 . d r a7 As we mentioned earlier, the above equation may be reduced to an EulerPoisson-Darboux equation by a clever choice of coordinates. To find a higher step heat kernel we need a solution of equation (5). Equation (6) suggests that one may try to find such a solution as a perturbation of the volume element of the fundamental solution.
+
6 On the Geodesics of a Step 3 Operator The examples we have discussed above are all even steps. Now let us turn to a step 3 example. The vector fields
XI
1 , a + -x 2 d Z 3 7 8x1
d
=-
x2=-
a
8x2 induce a subRiemannian geometry on R3, which is step 3 if 2 2 = 0, and step 2 everywhere else. The following results are derived in [18].
Geometric Analysis on SubRiemannian Manifolds
Theorem 6.1 Given a point
P(xl,x2,x3)
37
o n the surface
1 2 -x1x2, 6 there is a unique subRiemannian geodesic between the origin and the point P whose projection o n the ( X I , xa)-plane is a straight line with length l =
x3 =
d-.
Theorem 6.2 Every point P ( x l , x 2 , x 3 ) is connected to the origin b y at least one subRiemannian geodesic. The number of subRiemannian geodesics connecting P(x1, x2,x3) to the origin is finite if and only if x2 # 0 . W h e n 2 2 = 0 , every point of the ‘(canonical submanifold” {(XI, 0 , 0 ) , x1 # 0} is connected t o the origin by a continuous infinity of geodesics, while every point of its complement {(x1,O,xg), x3 # 0} is connected to the origin by a discrete infinity of geodesics. On the Heisenberg group the t-axis is the canonical submanifold which is formally analogous to the x3-axis of (l),hence Theorem 6.2 is surprising. To bolster this fact, we have evidence that the canonical submanifold of (1) at step 2 points, i.e., when x2 # 0, is a curve which in the limit x2 -+0 becomes the 21-axis. The canonical submanifold seems to yield the missing direction with the largest amount of symmetry.
Final Remarks (1). The results in this paper are known for n-dimensional non-isotropic Heisenberg groups, where the group law is
xoy=
[+ 21
1
n
~ 1* * ,
9
x2n
+ ~ 2 nt ,+ s - 2 1
aj(xjyj+n - ~ j x j + l ) 7
j=1
with aj > 0, j = 1,.. . ,n. The complex action g and the volume element are given by n
aj(xj”
g(x, t , 7) = j=1
+ ~ j ” + ~coth(aj.r) ) - it,
21
n
V(T)
=
2aj sinh(2aj.r) ’ j=1
see [2]and [6]. (2). Such invariant formulas also yield full Hadamard-Kodaira expansions for the parametrix of step 2 subelliptic operators on Heisenberg manifolds. A Heisenberg manifold is an odd dimensional manifold together with a subbundle of the tangent bundle of one lower dimension, and with the first bracket generating property. See [2]. (3). For elliptic operators which degenerate quadratically along submanifolds, such formulas explain the Gevrey class properties of their fundamental solution; see [4].
38
Ovidiu Calin, Der-Chen Chang, and Peter Greiner
References 1. R. Beals, B. Gaveau, and P.C. Greiner, Hamilton-Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures Appl. 79 (7)(2000),633-689. 2. R. Beals, B. Gaveau and P.C. Greiner, Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians, I, 11, 111, Bull. Sci. Math., 21(1997), 1-36,97-149,195-259. 3. R. Beals, B. Gaveau and P.C. Greiner, On a geometric formula for the fundamental solution of subelliptic Laplacians, Math. Nachr., 181(1996), 81-163. 4. R. Beals, B. Gaveau, P.C. Greiner and Y. Kannai, Exact fundamental solutions for a class of degenerate elliptic operators, Comrn. PDEs., 24[3-4](1999), 719742. 5. R. Beals and P.C. Greiner, Calculus o n Heisenberg manifolds, Ann. Math. Studies #119, Princeton University Press, Princeton, New Jersey, 1988. 6. C. Berenstein, D.C. Chang and J. Tie, Laguerre Calculus and Its Applications in the Heisenberg Group, AMS/IP series in advanced mathematics #22, International Press, Cambridge, Massachusetts, 2001. 7. 0.Calin, Ph. D. Thesis, University of Toronto, 2000. 8. 0. Calin, D.C. Chang and P.C. Greiner, On a Step 2(k + 1) subRiemannian manifold, to appear in the J. Geometric Analysis, (2004). 9. 0.Calin, D.C. Chang and P.C. Greiner, Real and complex Hamiltonian mechanics on some subRiemannian manifolds, to appear in Asian J. Math., (2004). 10. 0. Calin, D.C. Chang and P.C. Greiner, Geometric mechanics on a family of pseudoconvex hypersurfaces, manuscript. 11. 0. Calin, D.C. Chang and P.C. Greiner, Analysis and Geometry o n Pseudoconvex Hypersurfaces, (Book in preparation). 12. C. CarathCodory, Untersuchungen iiber die Grundlagen der Thermodynamik, Math. Ann., 67 (1909),93-161. 13. W.L. Chow, Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939),98-105. 14. G.B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. SOC.,79(1973), 373-376. 15. G.B. Folland and E.M. Stein, Estimates for the f& complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27(1974), 429-522. 16. B. Gaveau, Principe de moindre action, propagation de la chaleur et estimCes souselliptiques sur certains groupes nilpotents, Acta Math., 139 (1977),95-153. 17. P. Greiner, A fundamental solution for a non-elliptic partial differential operator, Can. J. Math., 31(1979), 1107-1120. 18. P. C. Greiner and 0. Calin, On subRiemannian Geodesics, Analysis and Applications, 1, #3(2003), 289-350. 19. L. Hormander, Hypoelliptic second order differential equations, Acta Math., 119 (1967),147-171. 20. D. F. Lawden, Elliptic functions and Applications, Applied Math. Sciences 80, Springer-Verlag, 1989.
A Survey of Hardy Type Theorems S. Thangavelu Statist ics-Mathematics Division Indian Statistical Institute 8th Mile, Mysore Road Bangalore 560 059 India veluma0isibang.ac.in
Summary. A classical theorem of Hardy on Fourier transform pairs characterise the heat kernel associated to the standard Laplacian on R". In recent years analogues of this result has been studied in various contexts auch as Riemannian symmetric spaces, semisimple and nilpotent Lie groups. In this survey we present some of the important results that have been proved in this direction.
1 Introduction In 1933, G. H. Hardy published a cute little theorem on Fourier transform pairs. For a function f E L1(R) let
-ca
be its Fourier transform. Hardy proved:
Theorem 1.1 Suppose the functions f and
lf(x>I 5 c e-a"',
f
satisfy the conditions
~ f ( y >5l c e-by2
f o r all x,y E R where a , b > 0. Then f = 0 i f ab > and f (x)= c e-ax2 i f ab = Moreover, when ab < there are infinitely many linearly independent functions satisfying both conditions.
i.
a,
As Hardy mentions in the introduction t o his paper, the result originates from a remark of Norbert Wiener to the effect that a pair of functions f and cannot both be very small. Thus Hardy's theorem (when ab > is an instance of an uncertainty principle for the Fourier transform. And the case ab = f is a characterisation of the Gaussian. The examples for the case ab < were provided by the Hermite functions.
p
i)
40
S. Thangavelu
This theorem of Hardy remained fairly unknown restricted only to experts in Fourier analysis for about forty years until Dym and McKean brought it to a wider audience in their book 'Fourier series and integrals' published in 1972. It remained as an isolated result for another two decades until Alladi Sitaram and his student M. Sundari started their investigation on finding analogues of Hardy's theorem on certain Lie groups. From then on several people got interested in proving Hardy type theorems and now there is a formidable amount of work on this topic mostly done by students and colleagues of Sitaram in India. Our aim in this three part article is to survey some of the important results that have been proved in this area and also to mention some of the open problems. In part I we deal with Euclidean spaces. Parts I1 and 111 will be devoted to semisimple and nilpotent groups respectively.
Part I: Euclidean Spaces
2 Fourier Transforms on R" For a function f E L1(R"), its Fourier transform is defined by
The n-dimensional version of Theorem 1.1 appeared first in a 1995 paper of Sitaram et a1 [5] where they have also considered Hardy's theorem for the Heisenberg and Euclidean motion groups. It is convenient to state the ndimensional Hardy's theorem in terms of the heat kernel p t ( x ) associated to the Laplacian A . Recall that for t > 0
p t ( x >= (4nt)-4e-hI"la. The following theorem was proved in [5] using the Radon transform.
Theorem 2.1 Suppose the functions f and
If (41I c
P S ( 4 '
f^ satisfy the conditions
If^(E)I I c $t(O
f o r all x , < E R" where s , t > 0 . Then f = 0 if s < t ;f = c p t i f s = t and when s > t there are infinitely many linearly independent functions satisfying both conditions. For a function f on R",its Radon transform Rf is a function on R x S"-l given by
R f ( t ,w ) =
J f ( t w +u)du
W l
A Survey of Hardy Type Theorems
where du is the (n - 1)-dimensional Lebesgue measure on w' complement of Rw.Then it is well-known that
/
41
, the orthogonal
00
Rf(t,w)e-ixtdt = (27r)?f(Xw).
-ca
Using this one reduces Theorem 2.1 to the one dimensional case. Hardy proved his theorem by a clever application of Phragmen-Lindelof maximum principle. The crux of the matter is to prove the following result on entire functions. Lemma 2.2 Let F ( z ) be an entire function of one complex variable satisfying
IF(.>! 5 c ( l +
Izl)"ealS(z)12,
z Ec
and
I F ( ~ 5) IC ( I + Ix\)me-+12 , x € R where a > 0. Then F ( z ) = p(z)e-az2 where p is a polynomial of degree at most m. When f satisfies the estimate given in Theorem 2.1 it is easy to show that f can be extepded as an entire function of C E C satisfying the estimate 5 c eklS(c)l . Then by appealing to Lemma 2.2 one establishes Hardy's theorem. There is also an n-dimensional analogue of Lemma 2.2 and hence one can directly prove Theorem 2.1 without appealing to, the Radon transform. In view of Lemma 2.2 the hypothesis on f and f in Theorem 2.1 can be replaced by
[)(fI
If (4I c (1 + 14>"Ps(x>, If(t>l6 c (1+ Irl>"a(c) and the conclusion when s = t is f (x)= q(x)pt(x) where q is a polynomial of degree 5 m. The most optimal version of Hardy's theorem for the Fourier transform on R was proved by Pfannschmidt [6] in 1996.
Theorem 2.3 Suppose for a, b > 0 and 2,y
I f I).(
ER
6 P(,) cax2 , If(y)l 5 Q(Y) cby2
where the functions P and Q satisfy
i;
a,
f
when ab = f and are entire functions Then f = 0 whenever ab > with f ( z ) = p(z> e - a z 2 , f < z >= q(z> e-bz2 where p and q are entire functions of order 2 and of minimal type.
42
S. Thangavelu
The proof of this theorem is quite involved. It uses properties of the indicator of an entire function and also the notion of proximate orders. An exact analogue of Theorem 2.3 is still unknown on R". However, with an extra condition on P and Q the following result can be easily proved.
Theorem 2.4 Suppose for a , b > 0 and z,( E R"
If(.)[
5 P(.) e--a1212,If"(E)l 5 Q(E) e-blEIZ
where the functions P and Q satisfy
Further assume that log P and log Q are subadditive. Then f = 0 whenever ab > $ and when ab =
a,
where the functions p and q do not grow faster than P and Q.
The subadditivity of log P and log Q allow us to estimate the Radon transform Rf and its one dimensional Fourier transform. But then Theorem 2.3 can be applied to reach the conclusions of Theorem 2.4. Without the subadditivity, Theorem 2.3 is still unknown. Note that the assumptions in Theorem 2.3 are the best possible since faster growth assumptions on P and Q will make the theorem not correct.
3 Hardy's Theorem for the Homogeneous Space R" = M ( n ) / S O ( n ) Any careful reader is sure to observe that Hardy's theorem looks as though it is a result in complex analysis. Except the definition of Fourier transform we have not used any other tool from harmonic analysis. This is more so with Theorem 2.3 as the reader can easily convince himself/herself that an enormous amount of complex analysis has gone into the proof but nothing at all from Fourier analysis. But the whole situation changes drastically if we take a different point of view of the Fourier transform. Instead of thinking of R" as a locally compact abelian group, we can consider it as the homogeneous space G / K where G = M ( n ) is the Euclidean motion group acting on R" by translations and rotations coming from K = SO(n).In this set up Fourier transform is viewed not as a function of E E R" but as a function of ( X , w ) E R+ x Thus
p
j ( ~ , w >= j ( ~ w = > (27r)-5
J e-ixw.z f(+z. R"
A Survey of Hardy Type Theorems
We replace the pointwise estimates on f and ditions:
43
f by the following integral con-
(i)
(ii) for r, X 2 0 and ask: What can we say abdut f under these conditions? To answer this question one requires a lot of harmonic analysis: spherical harmonics, Bessel functions, Hecke-Bochner formula and all such tools. The space L2(Sn-’) has an orthonormal basis {Ym,j: 1 5 j 5 dm,m = 0,1,2,. . .} consisting of spherical harmonics. Now, if we let &,j(X)
=
s
f*(xw)Ymj(W)dW
Sn-1
then by Parseval’s identity for spherical harmonic expansions
Using the formula
with z = 1x1~’and J , standing for Bessel function of order a we obtain
where f m j ( r ) is the (m,j)-th spherical harmonic coefficient of f . The above means, in view of Hecke-Bochner’s identity that, X-mFmj(X) is the ( nf 2 m ) dimensional Fourier transform of the radial function g m j ( z ) = IzI-mfmj(lzl). The conditions on f and f imply Igmj(4I
5 c e -a1x12
7
Iimj(0I
< c e-bIEi2 -
i,
and hence for ab > gmj = 0 and as this is true for all m and j we conclude f = 0. And when ab = gmj(x) = ~z~-mf,j(lz\> = cmj e--alx12.
But this is not compatible with the estimate on f ( z ) unless cmj = 0 for all m # 0. Thus f is radial and equals a constant multiple of e--alz12. More generally we have
44
S. Thangavelu
Theorem 3.1 Suppose f and
lsn-l
J
(ii)
f satisfy
5 cmj A m ( l + A ) N
f(Xw)Y,j(w)dw
e-bX2.
(ii) where the functions P and'Q are such t h a i
Then f = 0 whenever ab
f(.)
>
a and when ab a
=P ( 4 e
=
-alzI2 7
f(E)
= q(()
e-blE12
where p and q are such that their L2-norms o n 1x1 = r do not grow faster than
P ( r ) and Q ( r ) respectively. The proof is similar to that of the previous theorem and uses HeckeBochner formula and Poisson integral representation of Bessel functions. We need to appeal to Theorem 2.3 to complete the proof.
A Survey of Hardy Type Theorems
45
4 Hardy's Theorem for the Motion Group So far we have considered the Euclidean Fourier transform on R" but now we go one step further and look at the group Fourier transform on the motion group M ( n ) . This group is the semidirect product of R" with K = S O ( n ) and functions on R" can be thought of as right K-invariant functions on M ( n ) . Let M = S O ( n - 1) considered as a subgroup of K leaving the point e l = (1,0,. . .O) fixed. Then all the irreducible unitary representations of M ( n ) relevant for the Plancherel theorem are parametrised (up to unitary equivalence) by pairs (X,a) where X > 0 and u E h;r, the unitary dual of M . For a function f E L 1 ( M ( n ) its ) Fourier transform f is the operator valued function
f^(X, 0) =
1
f (2, Jc)m,,(a:,k)dx dlc.
M(")
Here (z,k ) E R" x K and dlc stands for the Haar measure on K . ? ~ x , ~k() x , are unitary operators on certain Hilbert spaces H ( K , a ) of functions on K and consequently f ( X , a ) is a bounded linear operator on H ( K ,a). When f E L1nL2(M(n))it is known that f(X, u)is Hilbert-Schmidt. We let Ilf(X, u ) I / ~ s to stand for its Hilbert-Schmidt operator norm. With these notations Sundari [9] proved the following result in 1998.
Theorem 4.1 Suppose f and f satisfy the following conditions: (i) If (x,k)l I c P S ( + (z, k ) E M ( n ) (ii) \ I ~ ( xu>l~Hs , I c e--tA2, (A, u >E R+ x Il;r where p , is the heat kernel f o r A o n R". Then f = 0 whenever s < t . Note that the Laplacian A is M ( n ) invariant; in fact, any differential operator on R" which is invariant under the action of M ( n ) is a polynomial in A . We have measured the decay of f and f in terms of the heat kernel associated to A . Though the formulation of the above theorem requires a knowledge of the representations T A , ~ ,the proof does not involve harmonic analysis on M ( n ) in any nontrivial way. However, the case s = t requires a detailed analysis. The action of T A , ~ ( Z ,k) on a function (p coming from H ( k , u) is given by rA,,(z,
for x E
k)(p(u) = e i A ( ~ ~ u - ' . e l ) ( p ( u k )
R" and k,u E K . From this it follows that for (p, ?1, E H ( k , a )
M(n) K
It can be shown that when f is a right K-invariant function on M ( n ) , f(X2u) = 0 unless u is the trivial representation, in which case the action of f(X, u) on the constant function (po(k) = 1 is given by
46
S. Thangavelu
Here f^ on the right hand side is the Euclidean Fourier transform of f on R". Thus Theorem 4.1 includes Hardy's theorem for R". The Fourier transform f(X, a)can also be computed explicitly for functions of a certain form.
Lemma 4.2 Let f E L 1 ( M ( n ) )be of the f o r m f ( z , u ) = g(Iz.1) P(z1 h(u) where P is a solid harmonic of degree m. Then f o r every X > 0,a E M and 'p E H ( k , a ) we have f(X, a) p ( k ) = XmG(X) P ( k - l e l )
where G(A) is the ( n
s
K
h(u)'p(ku)du
+ 2m) dimensional Fourier transform of g(1xl).
A proof of this lemma uses several results from Euclidean Fourier analysis. First of all it requires the Hecke-Bochner formula which says that if f(z)= g ( 1x1) P ( x ) with P , a solid harmonic of degree rn then
j(0= ( - i ) m p ( oG(ltl>
+
where G(ltl) is the ( n 2m) dimensional Fourier transform of g(lz1). It also needs Funk-Hecke formula which deals with spherical harmonic expansions of zonal functions. If a is a function on the interval ( - 1 , l ) this formula says that
s
a ( z ' . y ' ) Ymj(y')dy'=
cmj
Ymj(z').
s*-1
The coefficients cmj are given in terms of ultraspherical polynomials. More precisely, if Gz-'(t) are ultraspherical polynomials of type (2 - l),then
Cmj
r ( m+ l ) T ( n- 2 ) w,-~ = r ( m +n - 2)
1
/ a ( t ) G z - l ( t ) ( l - t2)*dt. -1
Let us write A, for the Laplacian on R" and p r for the associated heat kernel. Fractional powers of A, can be defined using Fourier transform as ((-A,)CYf)A(()= IEI2"f^(E). With these notations we now state the most general form of Hardy's theorem on M ( n ) .
Theorem 4.3 Suppose f and f^ satisfy (2) I f(.' k)l 5 C(1 -k IzI)NPs(z), (z,k) E M ( n ) (ii) ~ l f ^ ( ~ O>II , 5 c ( l + x)N e-'", (A, 01 E R+ x 2. Then f = 0 if s < t and when equality holds f can be written as a finite linear combination of functions of the form
A Survey of Hardy Type Theorems
47
where Pmjare solid harmonics of degree m and gmj are bounded functions on
K. This theorem was recently proved in [34]. As we can estimate (-A,)"lp?(z) precisely, it is possible to determine for what values of m the functions f m j occurs in f . What happens when the polynomial factors (l+lxl)Nand ( l + X ) N are replaced by functions of more general growth is still unknown.
5 Beurling's Theorem In 1991, Lars Hormander published a proof of the following theorem attributed to Arne Beurling.
Theorem 5.1 There is n o nontrivial f in L1(!R)for which j+z), --w
ImIelZYl& dY < 00.
--w
The case a6 > $ of Hardy's theorem is an immediate consequence of Theorem 5.1. This is again a theorem on entire functions. Beurling's theorem was extended to higher dimensions by Bagchi and Ray [ll]but a far reaching generalisation was obtained by Bonami, Demange and Jaming [30] recently.
Theorem 5.2 Let f E L2(Rn)be such that
for some N 2 0 . Then f = 0 whenever N 5 n and when N > n, the above holds i f and only i f f can be written as f(x) = P ( x ) e-$(Axyx) where A i s a real positive definite symmetric matrix and P is a polynomial of degree < + ( N- n).
As in the one dimensional case, the proof of this theorem also uses heavy doses of complex analysis. Theorem 2.1 is a corollary of this result; so are the uncertainty principles of Cowling-Price and Gelfand-Shilov. The result of Cowling and Price [3] is an LP - Lq version of Hardy's theorem. They replaced the L" estimates for the functions f(x) ealXl2and f(<)eblcI2 by LP and Lq estimates. Here is their result.
Theorem 5.3 Let N 2 0 , l 5 p , q 5 00. Assume that f E L2(Rn)is such that thefunctions f(z) ealz12(l+JzJ)-Nand eblc12(1+JeI)-N are in LP and Lq respectively. Then f = 0 when a6 > and when a6 = $, f (x)= P ( x ) e--als12 for some polynomial P.
a
f(c)
48
S. Thangavelu
Another result that can be deduced as a corollary of Theorem 5.2 is the following uncertainty principle of Gelfand-Shilov type.
Theorem 5.4 Let N 2 0 , 1
<
p ,q
<
00,
f
+f
= 1 and ab
f E L2(Rn) be such that both the functions If(.)[
2
$. Let
+
e:("I"I)P(l and eb(b15i)'(l I x ~ ) - ~ are integrable. Then f = 0 unless p = q = 2 and ab = in which case f ( x ) = P ( x ) e-ia21"I2 for some polynomial P of degree less than ( N - n ).
If^(r)l
+
For the Euclidean motion group the following version of Beurling's theorem has been proved in [34].
Theorem 5.5 Let f E L1 n L 2 ( M ( n ) )be such that
for every u E
h;r
with 0 5 N
5 n. Then f
= 0.
As opposed to the proof of Theorem 5.2 which mainly uses complex analysis, the proof of the above result relies on detailed analysis of the representations q,. The case N > n of Theorem 5.5 is still unsolved except when n = 2 and N = 3. In this particular case the conclusion is that (with z E @. E R2) f(Z,
= g(eip)e--alzlz
for some function g E L2(T)(2' E SO(2)).We conjecture that a similar result is true in general but probably new ideas are required to solve this problem.
Part 11: Semisimple Lie Groups
6 Introduction As we have seen in Part I of this survey, Hardy's theorem for Fourier transform pairs, proved in 1933, remained as an isolated piece of work until very recently except for two generalisations: (i) the LP-Lq version due to Cowling and Price; (ii) the most optimal version due to Pfannschmidt. We have discussed these two results in Part I. However, both these generalisations dealt only with the Euclidean Fourier transform. In 1995 Sitaram and Sundari established analogues of Hardy's theorem for the Heisenberg group and the Euclidean motion group. But it was only in 1997 an analogue of Hardy's theorem for a semisimple Lie group was considered. This paper of Sitaram and Sundari [8] triggered a lot of activity as can be seen from the number of papers written on this topic.
A Survey of Hardy Type Theorems
49
In this brief survey we shall describe some of the important Hardy type theorems proved for semisimple Lie groups and the associated Riemannian symmetric spaces. The results will be described under three headings: (i) Group Fourier transform on semisimple Lie groups, (ii) Helgason Fourier transform on Riemannian symmetric spaces and (iii) Fourier transform on NA groups. There is a fourth section dealing with the special case of SL(2,R) in which a complete analogue of Hardy's theorem will be stated. In order to keep the survey short and crisp, we follow the strategy of describing only the first and latest results under each section. We provide the reader with a more or less complete bibliography so that he/she can browse through the papers to learn more about the known results. For the group Fourier transform on a semisimple Lie group the case ab > 1 of Hardy's theorem has been completely settled. However, the case ab = a4 has been resolved only for functions on symmetric spaces. The case ab > is viewed as an instance of uncertainty principle whereas the equality case is a characterisation of the heat kernel. Even on the Euclidean motion group the Hardy conditions on a function f and its Fourier transform need not imply that f is a constant multiple of the heat kernel. But it has been proved that f factors as f(z,Ic) = g(Ic)pr(z) where p y is the heat kernel on R". Writing $ ( T ) in place of p r ( z ) with 1x1 = T we note that (47Tt)"p;+"(T)
= p?(r).
By expanding g ( k ) in terms of spherical harmonics and using Hecke-Bochner formula we get an expansion
m = O j=1
where cm,j are the spherical harmonic coefficients of the function g. Thus f is an infinite superposition of various heat kernels. We conjecture that a similar result is true for at least all semisimple Lie groups of real rank one. At the time of writing this survey, we know how to prove this only for the group SL(2,R). The various building blocks of f are exactly the heat kernels associated to various Jacobi operators. Unlike the Euclidean case where all the heat kernels p;+2m are given by the same function, namely the Gaussian, the situation of SL(2,R)is different. The various heat kernels that appear in an analogous expansion of f on SL(2,R) are all different. This makes the equality case ab = of Hardy's theorem for semisimple Lie groups more subtle and difficult. And at least for the author, the difficulty makes it more interesting and worth pursuing.
7 Group Fourier Transforms on Semisimple Lie groups We begin by setting up the notation. Let G be a connected noncompact semisimple Lie group with finite centre. Let G = K A N be an Iwasawa de-
50
S. Thangavelu
composition of G and let P = M A N be the associated minimal parabolic subgroup of G. As usual we denote the Lie algebras of A and G be a and g respectively. The norms on a and a* induced by the Killing form are denoted by 1.1. For a E h;r and X E $ let T A , ~be the principal series representations which are realised on certain Hilbert spaces H ( K ,a). If x = kak’ is the polar decomposition of x E G then we define 1x1 = I logal. For suitable functions f on G we define the group Fourier transform by
s,
f(X, a)= f (xc)%4~c)dx.
(7.1)
Note that !(A, a ) is a bounded linear operator on H ( K ,a). For f E L2(G)it is known that f(X, a ) is a Hilbert-Schmidt operator. We denote the operator and Hilbert-Schmidt norms of f(X, a ) by a ) [ and / Ilf(X, a ) l I ~respectively. s With these notations Sitaram and Sundari [8]proved the following Hardy’s theorem in 1997. Theorem 7.1 Assume that G has only one conjugacy class of Cartan subgroups. Let f € L1(G) satisfy the following two conditions. (i) c e-+IZ, x E G, (22) IIf(X, .)I1 5 c e- Pl’Iz, a E A?, x E a* where a , ,8 > 0. T h e n f = 0 whenever a,8 > +.
If(.)[
<
The proof of this theorem is rather straightforward. For cp,$ E H ( K , a ) one looks at the function F(X) = (f”(X,a)cp,$)and shows that this can be extended to (C’, 1 being the rank of G, as an entire function of order two satisfying the estimate
IF(X> 5 ~ce+Iz,
x E C’
(74
i.
for some y > 0 satisfying ,By> Then by appealing to the multi-dimensional version of the complex analytic lemma, Lemma 2.2 of Part I, one concludes that f(X,a) = 0 for all X E a* and a E dl.As the group is assumed to have only one conjugacy class of Cartan subgroups, the Plancherel measure is supported on the principal series representations and consequently f = 0. The estimate (7.2) is proved by using the hypothesis (i) on f (z) and an elementary estimate on the matrix coefficients (~~,~(x)cp, $). Indeed, if cp and $ are K-finite and smooth, then
I(~A,,,(x)cp, 5 CP,$ (1
+ Izl)melzm(X)IIzle-p(loga)
(7.3)
which follows from an estimate on elementary spherical functions. Here the constant CP,+depends on L” norms of cp and $. The extra condition in Theorem 7.1 was removed in two papers: Cowling et a1 [13] used the subquotient theorem of Harish-Chandra to prove f = 0 whenever f(X, a ) = 0 for all X E a* and a E h;r. On the other hand Sengupta
A Survey of Hardy Type Theorems
51
[15] used results of Casselman and Milicic on the asymptotic behaviour of the matrix coefficients of admissible representations. The equality case in Theorem 7.1 was not considered until recently when Narayanan and Ray treated it for right K-invariant functions on G. We will say more about this theorem later in this article. Let us also mention that Sarkar [28],[29] has considered the equality case for functions on the group and obtained some results at the level of matrix coefficients of the Fourier transform. We now proceed to state a general result for the case ap = The estimate (7.3) can be slightly improved. For every X E $ and 'p,$ E H ( K , a ) we have the estimate
i.
1(%7(4(P,
+)I I ll'pll 11+ll
eIzm(x)lIz'-
Note that we have lost the exponential factor e-P(loga) appearing in (7.3) but this is a small price we pay for the above estimate which involves L2 norms of 'p and $. This allows us to use operator theoretic arguments in establishing the following result.
Theorem '7.2 Let f E L1(G)satisfy the following two conditions f o r some s > 0 , t > 0 with st 2 (i) lf(z)I 5 c ( l + 1zl)N e-+12 e-P(loga), z = kak' E G, (ii) ~ ~ j ~5 c~( l +, IaX ~) >e--tl'I2, ~ N ~ aE E a*. Then f (A, a ) = 0 unless st = in which case
a.
a
M,x
for all X E $ and a E M where A, are certain operators acting o n H ( K ,a ) . Actually, it turns out that A,'s are the restrictions to H ( K , a ) of certain fixed operators which are given in terms of the function f . It is interesting to note that the conditions (i) and (ii) allow us to factorise f(X, a) as above with A, independent of X and cr. It is easily seen that all the earlier versions of Hardy's theorem follow from the above result. However, we cannot be too happy with this theorem though it characterises functions satisfying (i) and (ii) on the Fourier transform side. From the equation
it is not easy to draw any conclusion on f without some extra assumptions such as right K-invariance. This will take us to the realm of Riemannian symmetric spaces which we treat in the next section. We return to Theorem 7.3 in Section 10 where we deal with the special case SL(2,R).
52
S. Thangavelu
8 Helgason Fourier transforms on Riemannian Symmetric Spaces In this section we specialise to the case of right K-invariant functions on G. Here K is the maximal compact subgroup appearing in the Iwasawa decomposition. Any such function f can be thought of as a function on the Riemannian symmetric space G / K . It is well known that if f is right K-invariant then f^(A,a)= 0 unless a = 1 is the trivial representation of M . In this case all the representations T X , J are realised on the Hilbert space L 2 ( K / M ) .There is an orthonormal basis {vj : j = O , l , ....} of L 2 ( K / M )with vo = 1 such that !(A, 1)vj = 0 for all j 2 1. Therefore, we can identify the group Fourier transform of a right K-invariant function f with the function ?(A, k) = !(A,
l)vo(k)
on a* x K/M. This is known as the Helgason Fourier transform of f and we have
We also have the expression
where H ( z ) is the unique element in a given by the Iwasawa decomposition x = k(x)eH(z)n(z). Returning to the equation (7.4) we see that for a right K-invariant function
f,
and f^(X, a) = 0 for all other a.Taking N = 0 we are led to the equation f(A, 1) = e-tlX12A
(8.3)
where A : L 2 ( K / M ) L2(K/M) is such that Auj = 0 for all j >_ 1. But something more is true: from (8.2) and (8.3) we infer that --$
As the right hand side vanishes for X = i p for all j 2 1we see that for all j 2 1. Hence ?(A, k) = co e-tlA12.
(Avo, wj) = 0
A Survey of Hardy Type Theorems
53
This simply means that f is a constant multiple of the heat kernel p t ( z ) associated to the Laplace-Beltrami operator A on G / K . There is a conjecture due to Anker and Ji which says that
/pt(z>l5
(8.4)
~t~(j~l)e-ttPtZe-~t"tze-P(log~)
where P is an explicit polynomial. This estimate has been proved for all complex semisimple groups and also for groups of real rank one. For such groups Hardy's theorem can be viewed as a characterisation of the heat kernel in terms of the Helgason Fourier transform.
Theorem 8.1 Let f be a function o n G / K which satisfies the estimates (i) If(.)l I C(1+ Izl)"pt(4, 2 E G/K'
(ii)
(S,,, If(X,
k)12dk)' 5 Ce-tlX12
for some N1 2 0. Then f is a constant multiple of the heat kernel pt.
This result was proved by Narayanan and Ray [25] a couple of years ago. Note that in the hypothesis (i) we have allowed an extra factor of (1 1 ~ 1 ) ~ but in (ii) we have not assumed any such growth. It is therefore natural to see what happens if we replace (ii) by the estimate
+
'
This condition arises naturally when we try to prove Hardy's theorem for N A groups as will be shown in the next section. But now things get more complicated: the upper estimate (8.4) for the heat kernel which we used in the proof of Theorem 8.1 no longer suffices. We also need a lower bound for the heat kernel. At present an analogue of Theorem 8.1 with (ii) replaced by (8.5) is known only when the rank of G / K is one though it is reasonable to believe that this restriction is really not needed. For the rest of the section we assume that G / K is a Riemannian symmetric space of rank one. In this case KIM is identified with the unit sphere in a Euclidean space. Let K, denote the set of all class-1 representations of K. Then for each 6 E K M there is a unique vector w6 in the Hilbert space V6 on which 6 is realised such that G(m)wJ= w6 for all m E M . Using these representations we can obtain an orthonormal basis (y6,j : 1 5 6 5 d6,G E i ? ~ } for L 2 ( K / M ) .These functions can be identified with spherical harmonics. When f is a function on G / K writing f(X) = f(X, 1) where 1 stands for the trivial representation of M we have (with YO= 1)
The integrals
54
S. Thangavelu
known as Eisenstein integrals can be explicitly calculated. It is known that there exist polynomials Q a ( i X p ) , called Kostant polynomials and Kbiinvariant functions ( P ~ Jsuch that
+
@x,a(ka) = Qa(iX
+ P)Ya,l(k)cpA,a(a).
(8.8)
Moreover, we have the interesting formula e-(iA+p)H(s-’
‘)fi,j( ~ c ) d = k @ x , s ( a ) f i , j(k’)
(8.9)
SKIM
where x = k’a. In view of these formulas we have
where J ( a ) is the density appearing in the formula dg = J(a)dadkdk’ of the Haar measure dg on G in terms of the polar decomposition. Defining
using (8.8) and recalling the definition of ?(A, k) we obtain
It can be shown that
are expressible in terms of certain Jacobi functions which are eigenfunctions of an elliptic operator As related to A . Let pf be the heat kernel associated to this As characterised by the equation (Px,,5(U)
LP;(a)vA,s(a)J(a)da= e-t (AZ +P i ) where pa are certain constants associated to 6. Very precise upper and lower bounds on the heat kernels p f ( a ) are known, thanks to the works of Anker, Damek and Yacoub. Using all the above ingredients we are able to prove the following result ~71.
Theorem 8.2 Assume that G / K is of rank one. Let f be a function o n G I K which satisfies the estimates 6) If >I.( 5 C(1+ I 4 ) ” P t ( X h x E G I K , (ii)
(SKIMI ~ ( xk)12dk) ,
‘
5 ~ ( 1 I+ XI>N2e-tlAI2,
x E a*
for some N l , N2 2 0. Then f is afinite linear combination of terms of the f o r m fa( k a ) = Y a , j (k)Pa(Aa)pf( a ) where Ps are some polynomials. In particular, when one of Nj is zero f is a constant multiple of the heat kernel pt.
A Survey of Hardy Type Theorems
55
The proof of the above theorem actually shows that the following refined version is true. Let us write
Theorem 8.3 Let G / K be as in the previous theorem. Let f be a function o n G / K which satisfies, for each 6 E K M and j the following estimates: (2) lf(4I I C(1+ 1 4 ) N P t ( 4 z E G / K , ( i i ) l ~ a , j ( ~I ) ~~ a , j e - t l ' 1 ~ , x E a* Then f = 0 when s < t ; f is a constant multiple of the heat kernel pt when s = t and there are infinitely many linearly independent functions satisfying both conditions when s > t. It is still an open problem whether Theorems 8.2 and 8.2 are valid for all symmetric spaces irrespective of their ranks.
9 Hardy's Theorem for NA Groups Harmonic N A groups form a class of solvable Lie groups equipped with a left invariant metric and they include all Riemannian symmetric spaces of noncompact type and rank one. Indeed, writing the Iwasawa decomposition of a semisimple Lie group G as G = N A K , the symmetric space G / K can be realised as the group S = N A with A = R+. For the real hyperbolic spaces Hn(R) the group N is abelian. For the complex hyperbolic space H"(C), N is the Heisenberg group H"-'. For other cases N is a nilpotent group belonging t o the class of H-type groups. We briefly recall the definition of a H-type group. Let n be a two step nilpotent Lie algebra equipped with an inner product (, ). Let z be the center of n and v its orthogonal complement so that n = v @ z. Following Kaplan we say that n is a H-type algebra if for every 2 E z the map Jz : v + v defined by ( J Z X Y ) = ([X, YI, y Ev (9.1)
a, x,
satisfies the condition Jg = -12I2I, I being the identity on v. A connected and simply connected Lie group N is called a H-type group if its Lie algebra is a H-type algebra. Since n is nilpotent, the exponential map is surjective and hence we can parametrise elements of N = exp n by ( X ,2 ) where X E v, 2 E z. By Campbell-Hausdorff formula, the group law is given by
( X ,Z ) ( X ' , 2')=
(x + X ' ,
2
+ 2' +
(9.2)
The Haar measure on N is given by dX dZ where dX and dZ are Lebesgue measures on v and z. The best known example of a H-type group is the Heisenberg group about which we say more in Part I11 of this survey.
56
S. Thangavelu
Given a H-type group N let S = N A be the semidirect product of N with A = R+ with respect to the action of A on N given by the dilation ( X ,2)+ ( a i X ,aZ),a E A. We write ( X ,2,a) to denote the element exp(X + Z)a. The product law on N A is given by
(X, 2,U ) ( X ’ , Z’, a‘) =
1 z+ aZ‘ + -a+ [X, X’], aa’ 2
(x +
).
(9.3)
For any 2 E z with 1 2 1 = 1, JZ = -I and hence Jz defines a complex structure on v. Consequently v is even dimensional. Let 2m be the dimension of v and Ic the dimension of z. Then Q = m Ic is called the homogeneous dimension of S. The left Haar measure on S is given by a-Q-ldudXdZ. The Lie algebra s of S is simply n @ R equipped with the inner product
+
((X, 2,t ) ,(XI, Z’, t’)) = (X, X’)+ ( Z , Z ’ )+ tt’. This makes S into a Riemannian manifold which is a harmonic space. Rank one symmetric spaces of noncompact type constitute a very small subclass of N A harmonic spaces. Analysis on such groups has drawn considerable attention during the last decade. See the works of Damek, Ricci, Anker and others given in the references of [16],[33].Despite the fact that on non-symmetric N A groups there is no analogue of K acting transitively on the spheres in N A , we can define a notion of radiality and a whole lot of spherical analysis can be done. In particular, an analogue of the Helgason Fourier transform on S has been introduced and studied by Astengo, Camporesi and Di Blasio. This was done in terms of the Poisson kernel associated to the Laplace-Beltrami operator A on S . If f is a bounded harmonic function on S , then as proved by Damek, f can be represented as f(2) =
S P ( 2 ,n)F(n)dn,
2
E
s
N
where F is the restriction of f to N and P(2,n) is the Poisson kernel defined as follows. For a E R+ and n = (X, 2 ) define
where cmk = Pl(0,O). Then P ( 2 , n ) = Pa(nlln) if complex number X define
P+n)
1
ix
2
=
1
iX
n10
E S. For a
= ( P ( 2 , n ) ) T - v = (P,(n;ln))T-v.
(9.5)
Given a C r function f on S , its Helgason Fourier transform on C x N given by
f” is the function
A Survey of Hardy Type Theorems
f(X, n) =
1f(x)P,(x,
+x.
57
(9.6)
S
Plancherel and inversion formulas are known for this Helgason Fourier transform. We denote by h,(z) the heat kernel associated to A which is a radial function. This kernel is characterised by the requirement that i L t ( ~ n) , = PA(e,n)e-t(A2++Q2)
(9.7)
where e = (O,O, 1). In 2000, Astengo, Cowling, Di Blasio and Sundari I161 proved the following version of Hardy’s theorem for N A groups.
Theorem 9.1 Let f be a function on the N A group which satisfies the estimates (2) If (4I Chs(4, (ii) J [?(A, Y,Z ) I ~ dz ~ Y5 c e-2t’2 N
for all X E R. Then f = 0 whenever s
< t.
Actually, they proved that the above theorem is valid even if the L2 norm is replaced by LP norm for any 1 5 p 5 00. They used the horocyclic Radon transform (Abel transform) to reduce matters to the Euclidean case. In the above result the case s = t , which we have been able to settle recently [33], was left open. Recall that the heat kernel is characterised by (9.7) and hence in view of (9.4) and (9.5) it is easy to see that
N
9.
for any 0 < y < Thus by slightly strengthening the hypothesis on we can treat the equality case and prove
Theorem 9.2 Let f be a function on the N A group which satisfies c ht (x). Further assume that
/
+
If”(X,Y, Z)I2(1 IZI2)%Y dZ
I f I).(
f^ 5
I c (1+ 1 XI )fle-2tA2
N
for all X E R and some y >
9. Then f (x)= cht(x).
The proof of this theorem also uses Radon transform, not the horocyclic one used by Astengo et a1 but the ordinary Euclidean Radon transform in the Z variable. Given a function f (X, 2,a) on S define its partial Radon transform f w ( X , t , a ) , t E .IR, w E S“’ by
58
S. Thangavelu
Ricci has already introduced this transform and used it to show that the subalgebra L&d(S) of L1(S) consisting of radial functions is commutative. He also used it to deduce the inversion formula for the spherical Fourier transform on S from the corresponding result for rank one symmetric spaces. The usefulness of this transform stems from the following fact. Given w E Sk-llet 2, = exp w' which is a subgroup of S . Then the quotient group S , = S/Z, is a symmetric space which can be identified with the complex hyperbolic space. Moreover, for functions f and g on S we have (f * g ) , = fu *, g, where the convolution on the right hand side is in S,. Therefore, the partial Radon transform allows us to reduce matters to the complex hyperbolic space. I f f is as in Theorem 9.2 then it can be shown that the function g w ( K Y,4 = .--f,(x, 97.) satisfies the conditions
and
where p t is the heat kernel on the symmetric space S, and ij, is the Helgason Fourier transform on S,. Therefore, we can appeal to Theorem 8.2 to complete the proof of Theorem 9.2, In fact, an analogue of Theorem 8.3 is also true. We refer to [33] for the formulation and proof.
10 Hardy's Theorem for S L ( 2 , R) In this section we specialise to the case of SL(2,R) for which a complete analogue of Hardy's theorem is known. The Iwasawa decomposition G = K A N of SL(2,R) is given by
K
= (Ice = diag(e:',e
- "0 2
) : 0 5 8 < 47r}
and
A = {a, =
coshr sinhr sinhr coshr
:rER}.
Then M = {Ico, &}, I? consists of all characters xn,n E iZ given by xn(Ice) = einO and &iconsists f of xo and x4 restricted to M. Let us simply denote xo by 0 and xi by so that A? = (0, ?j} and the principal series representations of G associated to the above Iwasawa decomposition can be simply
A Survey of Hardy Type Theorems
59
written as 7 r ~ , oand 7rA,+. These representations are realised on the Hilbert spaces H(K,O) and H ( K , $). For each IS E h;r there exists an orthonormal basis {ej : j E Z + o} with the property that 7rA,u(k6)ej= Xj(k6)ej = eijeej.
(10.1)
With these preliminaries let us look at the conclusion of Theorem 2.4 with G = SL(2,R) and N = 0. We have ..
q U ( k e ) e j = xj(k6)ej = ev6ej,
where A, is a bounded linear operator on H ( K ,IS).If {em : m E the basis described above, e-tA2 (Auej,em) =
S,
f(x)(nA,u(x)ej,em)dx.
(10.2)
Z+ IS}is (10.3)
Now the matrix coefficients are expressible in terms of Jacobi functions. Let us recall the definition of the Jacobi functions cpr’”(r). For each a ,p, X E @. with a # -1, -2, -3,. . ., these functions are defined by 1 + p + 1 - i X ) , -(a + ,O + 1 + iX),a + 1,- sinh2r 2
‘PA
where F is the Gaussian hypergeometric function
Here ( u ) = ~ 1 and (u)k = u(u and m,j E Z o
+
+ 1)....(u + k - 1) for k 2 1. Then for IS E A?
where the functions c ~ , ~ ( m are , j )given by (10.5)
if rn 2 j and (10.6)
+
, 1 for all m E Z IS. if m 5 j . Note that c ~ , ~ (mm) = Since the Haar measure on G is a constant multiple of (sinh 2r)drdBdq the equation (10.3) becomes
60
S. Thangavelu
Using (10.4) and observing that cx,a(m,j)vanishes for certain values of X when m # j which forces (A,ej, em) = 0 whenever m # j . And when m = j e - t X 2 (A,em, em) = c
i
fm(r)(pf'2m'(r)(2 sinhr)(2 c ~ s h r ) ~ ~ + ' d(10.7) r
0
where we have written (10.8)
It turns out that we cannot eliminate any of the functions f m ( r )as all of them satisfy the same estimates as the heat kernel pt(x). Theorem 10.1 Let f be a function on SL(2,R) satisfying the following two conditions: (i) If(x)I 2 Cpt(x) (ii) I l j ( X , a ) l l ~ 5 s Ce-tX2. Then f can be written as in (10.9) and hence there are infinitely many linearly independent functions satisfying the hypotheses of the theorem. In order to prove this theorem we have to justify our claim that all the functions f m ( r )satisfy the same estimates as pt(x).To this end, let us recall the definition of the Jacobi transform Ja,p of type (a,p): J,,pg(X) =
lw
g ( r ) y ~ ~ ' ~ ) ( r ) ( 2 s i n h 2 r ) ~ " + ~ ( 2 c o s h 2 r ) ~ ~ + ~(10.10) dr.
Let hi"") be the heat kernel associated to the elliptic operator d2
Aa,p = - + ((2a + 1)cothr + (2p dr2
+ 1)tanhr)-.drd
Then hf"") is characterised by the equation Ja,ph!"'"(X) = e-t(X2+p2)where p =a p 1. Moreover, pt(a,) is a constant multiple of hL0")(r).Anker, Damek and Yacoub have obtained precise upper and lower bounds for the heat kernels hi"") under some conditions on a and p.
+ +
A Survey of Hardy Type Theorems For each n E N and
0
61
E (0, % }we can show that
In terms of Weyl fractional integral operators we can express fnCu as
l m ( s i n h s)(cosh 2s - c o s h 2 ~ - ) ~ ~ ~ ~ ~ ~ ~ h (10.12) ~ ~ ~ ~ ~ Thus we see that f m ( r ) are given by fractional integrals of the heat kernels h!2n’0)(r). Using this expression we can show that f m ( r )and h?”)(r) satisfy the same estimates proving our claim. Therefore, the most general f satisfying the hypotheses of the theorem is given by (10.9) where fm are explicitly given by (10.12). Unlike the Euclidean case these heat kernels hi2“’O)are linearly independent.
Part I11 : Nilpotent Lie Groups
11 Introduction In this part of the survey on Hardy type theorems, we deal with the realm of nilpotent Lie groups. Though nilpotent Lie groups are closer to Euclidean spaces than any other class of non-abelian groups, formulating and proving an analogue of Hardy’s theorem poses certain problems. For one thing, the Fourier transform on a nilpotent Lie group is operator valued. Though the same remark applies to semisimple Lie groups, there is a remarkable difference between these two. As we have seen in part 11, the group Fourier transform associated to the principal series representations of a semisimple Lie group is parametrised by X which can be complexified and under certain decay conditions on the function f,its Fourier transform can be holomorphically extended. And this fact is used to reduce matters to the basic complex analytic lemma on entire functions. The representation theory of nilpotent Lie groups has been well understood and is given by Kirillov theory. The relevant representations are parametrised by the co-adjoint orbits and this parameter domain has certain inherent discontinuities. Owing to this fact, it is not possible to extend the group Fourier transform as a holomorphic function on the parameter domain. This problem manifests itself even in the simplest case of Heisenberg groups. For the group Fourier transform on a nilpotent Lie group, it is possible to formulate several versions of Hardy’s theorem. One version which is the direct analogue of original Hardy’s theorem turns out to be a theorem for
62
S. Thangavelu
the central variable. Another version, namely the heat kernel version, seems to be the right analogue at least for the class of stratified groups. There is yet another version which assumes no condition on the function but two sets of conditions on the Fourier transform side. This survey has two sections. In section 12 we review results on Heisenberg groups and in section 13 we deal with general nilpotent Lie groups.
12 Heisenberg Groups The most well known example from the realm of nilpotent Lie groups is the Heisenberg group H" which is simply C" x R equipped with the group law (Z,t)(W,S)
=
1 2
(
z+w,t+s+-Im(z.w)
).
The irreducible unitary representations of H" that are relevant for the Plancherel theorem are parametrised by non-zero reals X and are realised on the same Hilbert space L2(Rn) in the Schrodinger picture. For each X E R, X # 0 the representation 7rx is given by
where 'p E L2(R"). The Haar measure on H" is simply the Lebesgue measure dz dt on C" x R and we form the L* spaces using this measure. For f E L 1 ( H n ) its group Fourier transform is the operator valued function
A X ) = x x ( f )=
J
H"
f x ( z , t ) .rrx(z,t) dz dt.
It is known that for f E L1 n L2(H"),f^(X)is Hilbert-Schmidt and the Plancherel formula holds (see [ 5 ] ) :
H"
-W
In the above formula I l f ^ ( X ) l l ~stands ~ for the Hilbert-Schmidt operator norm of f^(X). Measuring the decay of the Fourier transform in terms of the HilbertSchmidt norm, the following theorem was proved in [5].
Theorem 12.1 Let f E L1(H") be such that If(z,t)l 5 g ( z ) e-at2 and Ilf"(X)ll~s5 c cbX2 f o r all ( z , t ) E H" and X E R,X # 0 where g E L1 n L2(C"). Then f = 0 whenever ab >
a.
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63
We see that the function g in the hypothesis is completely arbitrary and a glance at the proof shows that the above is actually a theorem for the central variable. The most natural analogue of Hardy's theorem for H" is the heat kernel version which we describe in a moment. On the Heisenberg group there is a subelliptic operator L called sublaplacian which plays the role of Laplacian on R". This operator is explicitly given by
where A, is the Laplacian on C".It is well known that L generates a heat diffusion semigroup whose kernel qa(z, t ) is explicitly given by
Here, f x ( z ) stands for the partial Fourier transform 00
-00
The (group) Fourier transform of the heat kernel qa is given by &(A) = e - a H ( X )where H ( X ) = - A X21zJ2is the scaled Hermite operator on R". An easy calculation reveals that the Hilbert-Schmidt operator norm of &(A) does not have any Gaussian decay. Therefore, instead of measuring the decay of !(A) in terms of the Hilbert-Schmidt norm we compare the non-negative operator f^(X)*.f(X) with &,,(A). The following is the heat kernel version of Hardy's theorem for the Heisenberg group.
+
Theorem 12.2 Let f E L'(H") be such that If(z,t)l 5 c p,(z,t) and f(X)*f(X) 5 c &?b(X) f o r all ( z , t ) E H" and X E R,X # 0. Then f = 0 whenever a < b. Compare this theorem with Theorem 2.1 in part I of this survey. As we have mentioned in the introduction, .f(X) cannot be defined as a holomorphic function of A. Therefore, the proof of the above theorem given in [26] is not straightforward. Using pointwise estimates for the heat kernel q,(z, t ) we first show that for fixed z E C" the function f X ( z ) extends to a holomorphic function of X E C in the strip IIm(X)I < A for some A > 0. Therefore, it is enough to show that f X = 0 for 0 < X < b for a suitably chosen 6. In order to prove this we observe that
/
f(X> = fX(4n ( z ,O)dz @"
64
S. Thangavelu
and operators of the form W ( 9 )=
J
Cn
g ( z ) d z , O ) dz
are called Weyl transforms. The theorem reduces to proving that the conditions
5 c d ( z ) , wX(g)*wA(g) 5 c 6 2 b ( X ) imply g = 0 whenever a < b. This is achieved by a longwinded process which /!?(z)1
uses several properties of special Hermite functions, eventually reducing matters to the Euclidean situation. We refer to [26] for details. As in the Euclidean case we also have a refined version of Theorem 12.2. There are operator analogues of spherical harmonics using which we can formulate and prove a theorem which assumes conditions on the decay of the spherical harmonic coefficients of !(A). We refer to [26] for the formulation and proof. Let us now turn our attention to the equality case in Theorems 12.1 and 12.2. In Theorem 12.1 if we replace the second condition by
1x14 I I ~ ( X11H.s ) L c ePbx2
i,
then the conclusion reads: f = 0 whenever ab > and when ab = f ( z ,t ) = g ( z ) e - a t 2 . This has been observed only recently. In Theorem 12.2 we expect that f ( z ,t ) = c q a ( z ,t ) when a = b. That is the conditions
f ( z , t ) = g ( z , t> q d z , t ) , f(X)*f(X) L c 6 2 a ( W where g E L"(H") imply that f is a constant multiple of the heat kernel. However, this has been proved only when g is independent o f t .
Theorem 12.3 Let f E L1(H") be such that f(z,t) = g ( z ) q a ( z , t ) where g E L"(@") and f^(X)*f(X) 5 c &(A). Then f ( z , t ) = c q a ( z , t ) . The method of the proof (given in [26])requires that we have the estimate Ifx(z)l 5 c q?(z) for all X E R, X # 0 and this can be proved when g is independent o f t . Otherwise, we have a problem in estimating f A ( z ) and the equality case in Theorem 12.2 remains open. We remark that the condition o f f can be replaced by the condition I f x ( z ) l 5 c q;(z) leading to the same conclusion.
13 General Nilpotent Lie Groups In the previous section we have stated several versions of Hardy's theorem for the Fourier transform on the Heisenberg group which is the most well known
A Survey of Hardy Type Theorems
65
example from the realm of nilpotent Lie groups. Therefore, it is natural to ask to what extent the results of the previous section generalise to the case of general nilpotent Lie groups. A partial analogue of Theorem 12.1 for general nilpotent Lie groups has been proved by Kaniuth and Kumar [18]. Let G be a connected, simply connected nilpotent Lie group with Lie algebra g . Fixing a strong Malcev basis for g we can identify g with Rd and as the exponential map exp : g + G is a diffeomorphism, G can also be identified with Rd. With this identification we can define 1x1 for z E G. Let A be a suitable cross section for the generic co-adjoint orbits in g', the vector space dual of g and define (XI as the Euclidean norm of X E A . A parametrises all the irreducible unitary representations of G that are relevant for the Plancherel theorem. Let 7rx stand for the representation associated to X E A and define q ( f = ) JG f(z)7rx(z)dz.We have the following result (see [W. Theorem 13.1 Suppose f is an L2 function o n G which satisfies the estimate I f I).( 5 Ce-alz12 f o r all z E G. Further assume that f o r every X E A we have I l 7 r x ( f ) l l ~ s 5 Ce-blX12. Then f = 0 whenever ab >
i.
A different version of this theorem where the pointwise estimates are replaced by integral conditions have been proved by Ray [21] and Astengo et a1 [16] for all step two nilpotent Lie groups. We remark that in the above theorem as well as the results proved by Ray and others the equality case was not treated. What happens when ab = still remains open though it is reasonable to believe that the same conclusion as in the case of H" can be proved at least for all step two nilpotent Lie groups. We further remark that the strong Gaussian decay of f in all the variables is not really required for the conclusion of the theorem. It is enough to assume such a decay only in the central variable. For this reason the above theorem is a result for the central variable. An exact analogue of Theorem 12.2 can be formulated when the group G is stratified. This means that the Lie algebra g admits a vector space decomposition
a
m
j=1
where V, are vector subspaces of g and V1 generates g as a Lie algebra. The Lie algebra is then equipped with a natural dilation structure S, such that I
\
+Xj,X j
E
V, and r > 0. Using exponential map which is
a global diffeomorphism we can define a dilation structure exp o 6, o exp-l on G. By abuse of notation we denote this also by 6,. On every stratified group G there exists a homogeneous norm such that l6,zl = rlzl, z E G, r > 0. Fixing an orthonormal basis XI, Xa,. . .Xe of V1 we define the sublaplacian L
66
S. Thangavelu
e on G by L: = - C Xj". The heat operator associated to L is the differential j=1 operator 8,+ L on G x (0, m). There exists a function p t ( x ) on G x (0, m) called the heat kernel such that f * p t ( z ) solves the heat equation with initial condition f . The heat kernel has several interesting properties for which we refer to the monograph 'Hardy spaces on homogeneous groups' by Folland and Stein. We are now ready to formulate the following heat kernel version of Hardy's theorem. Conjecture: Let G be a stratified group and let A and pt be as above. Suppose f E L1(G) satisfieslf(z)l I c p , ( x ) a n d n x ( f ) * n x ( f )I c q ( p 2 t ) f o r a l Z x E G and X E A . Then f = 0 whenever s < t and f = c pt when s = t. As we have seen in the previous section, the equality case s = t is open even for the Heisenberg group. The case s < t of the above conjecture has been recently proved (see [35]) for all nonisotropic Heisenberg groups and for a class of step two groups which includes all H-type groups. The proof given for the isotropic Heisenberg group doesn't work for the non-isotropic case, so a different proof was found in [35]. The general case of the above conjecture is still open. A close examination of the proof of Hardy's theorem for the Heisenberg group reveals that the proof depends on the following facts: (i) the heat kernel qa(z,t ) is known to satisfy a good estimate of the form
Iqa(Z,t)l
I
e-$(~44+t2)+;
(ii) the kernel q;(z) is explicitly known and (iii) (H", U ( n ) ) is a Gelfand pair where U ( n ) is the unitary group acting on H" as automorphisms. For a general stratified group G, though a good estimate is available, from the work of Dziubanski et al, for the heat kernel, we do not have an explicit formula for the partial Fourier transform of p t in the central variable. Such a formula is available in some special cases of step two groups. Moreover, we do not have a Gelfand pair at our disposal unless G is a direct product of Heisenberg groups. So, there are many technical problems in extending from H" to general stratified groups even the inequality case of Hardy's theorem. Therefore, we look for an alternate description of Hardy's theorem. To motivate our result, let us consider the conditions f(X)*f(x) 5 c &(,(A) and Ifx(z)l 5 c q;(z) on H". Since H(X) is unitarily equivalent to lXlH it is clear that the condition on !(A) can be replaced by f ( X ) * f ( X ) 5 c e-2blXlH.We also note that
as can be easily verified from the explicit formula for qa. Thus Hardy's theorem for H" can be stated in terms of the fixed semigroup e-2bH and the fixed kernel q:(z). In the case of general nilpotent groups when the relation between different n(f)are not explicitly known, such a reduction is of great importance. Thus we may consider a condition of the form .rr(f)*.rr( f ) 5 c e-2bH.
A Survey of Hardy Type Theorems
67
Since we do not have an explicit formula for the heat kernel we would like to replace the condition lf(z)I 5 c ps(z)by another condition on the Fourier transform. For the Euclidean Fourier transform we can prove the following version of Hardy's theorem by assuming estimates only on f and its derivatives but not on f .
Theorem 13.2 Let f which satisfies
E
L1(Rn)be such that la"lf"(<)12 5
for all
Q
E
N" and E E R".
f(E)
is a C" function o n R"
c a! ulaI
Further assume that
~ f ^ ( ~ > l 5 c ( 1 + Itl)me-blEla. Then f = 0 whenever a < 2b. We briefly indicate a proof of this theorem which is proved in [35].Consider the Taylor expansion
of f". Let a < bl < 2b and apply Cauchy-Schwarz inequality to get
By the hypothesis on a"f and by the choice of bl we see that the first sum on the right hand side of the above is finite. This shows that f" is real analytic and l f ( ~ q)12 5 c eb11q12.
+
The same argument shows that function satisfying the estimate
f"(0can
be extended to @" as an entire
We can now appeal to the complex analytic lemma to complete the proof. We also have an analogue of Theorem 13.2 for nilpotent Lie groups. The formulation requires non-commutative analogues of the derivatives a". For a bounded operator acting on L2(Rn) we define the noncommutative derivatives S j T = [ A j , T ] , Sj = [T,Aj*] for j = 1 ' 2 , ...'n where [T,S] = T S - ST and A j , A; are the annihilation and creation operators of quantum mechanics. For multi-indices a ,,B define ba and in the usual way. The following theorem has been proved in [35].
r'
68
S. Thangavelu
Theorem 13.3 Let G be a connected, simply connected nilpotent Lie group and let A be a cross section f o r the generic coadjoint orbits parametrising elements of G relevant to the Plancherel measure. Let f E L1 n L2(G) satisfy the conditions (i) m(f)*m(f)I c e-2bH, (ii) IIa"ap(71-x(f)*71-x(f))II~S 5 c ( a + p)! ulal+lPI f o r all X E A and a,/?E N". Then f = 0 whenever a < 2(tanh2b). This result is actually a theorem for Hilbert-Schmidt operators and the proof uses the Fourier-Weyl transform introduced in connection with the Paley-Wiener theorem. If xx(f) is the Weyl transform of a function, say fx on @" then the estimates on the noncommutative derivatives of nx(f)*rx(f) can be transformed into estimates on the ordinary derivatives of the FourierWeyl transform of the function fx. Using explicit formulas for special Hermite functions and generating function identities for Laguerre polynomials we can estimate fx and its Euclidean Fourier transform. Finally, appealing to Hardy's theorem on @" we conclude the proof. The equality case in Theorem 13.3 is still open.
References 1. G. H. Hardy, A theorem concerning Fourier transforms, J. London Math. SOC. 8 (1933), 227-231. 2. I. M. Gelfand and G. E. Shilov, Fourier transforms of rapidly increasing functions
and questions of uniqueness of the solution of Cauchy problem, Uspekhi Mat. Nauk. 8 (1953), 3-54. 3. M. Cowling and J. Price, Generalisationsof Heisenberg's inequality, in Harmonic analysis (G. Mauceri, F. Ricci and G. Weiss, eds), Lecture notes in Math. 992, Springer, Berlin. 4. L. Hormander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Math. 29 (1991), 237-240. 5. A. Sitaram, M. Sundari and S. Thangavelu, Uncertainty principles on certain Lie groups, Proc. Indian Acad. Sci. 105 (1995), 135-151. 6 . C. Pfannschmidt, A generalisation of the theorem of Hardy: A most general version of the uncertainty principle for Fourier integrals, Math. Nachr. 182 (1996), 317-327. 7. A. Miyachi, A generalisation of a theorem of Hardy, Harmonic Analysis seminar, Izunagaoka, Shizuoka-ken, Japan, 1997. 8. A. Sitaram and M. Sundari, An analogue of Hardy's theorem for very rapidly decreasing functions on semisimple groups, Pacific J. Math. 177 (1997), 187-200. 9. M. Sundari, Hardy's theorem for the n-dimensional Euclidean motion group, Proc. Amer. Math. SOC.126 (1998), 1199-1204. 10. M. Eguchi, S. Koizumi and K. Kumahara, An analogue of the Hardy theorem for the Cartan motion group, Proc. Japan Acad. 74 (1998), 149-151. 11. S. C. Bagchi and S. K. Ray, Some uncertainty principles like Hardy's theorem on some Lie groups, J. Aust. Math. SOC.65 (1998), 289-302.
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12. M. Ebata, M. Eguchi, S. Koizumi and K. Kumahara, A generalisation of the Hardy theorem to semisimple Lie groups, Proc. Japan Acad. Sci. 75 (1999), 113-114. 13. M. Cowling, A. Sitaram and M. Sundari, Hardy’s uncertainty principles on semisimple groups, Pacific J. Math. 192 (2000), 293-296. 14. N. Shimeno, An analogue of Hardy’s theorem on the Poincare disk, The Bulletin of Okayama University of Science, 36 (2000), 7-10. 15. J. Sengupta, An analogue of Hardy’s theorem for semisimple Lie groups, Proc. Amer. Math. SOC.,128 (2000), 2493-2499. 16. F. Astengo, M. Cowling, B. Di Blasio and M. Sundari, Hardy’s uncertainty principle on some Lie groups, J. London Math. SOC.62 (2000), 461-472. 17. M. Eguchi, S. Koizumi and K. Kumahara, An Lp version of the Hardy theorem for motion groups, J. Aust. Math. SOC.68 (2000), 55-67. 18. E. Kaniuth and A. Kumar, Hardy’s theorem for simply connected nilpotent Lie groups, Proc. Cambridge Philos. SOC.131, 487-494. 19. K. Grochenig and G. Zimmerman [2001], Hardy’s theorem and the short-time Fourier transform of Schwartz functions, J. London Math. SOC.63 (2001), 205214. 20. N. Shimeno, A note on the uncertainty principle for the Dunk1 transform, J. Math. SOC.Univ. Tokyo 8 (2001), 33-42. 21. S. K. Ray, Uncertainty principles on two step nilpotent Lie groups, Proc. Indian Acad. Sci. 111 (2001), 1-26. 22. S. Thangavelu, An analogue of Hardy’s theorem for the Heisenberg group, Colloq. Math. 87 (2001), 137-145. 23. J. Sengupta, The uncertainty principle on Riemannian symmetric spaces of the noncompact type, Proc. Amer. Math. SOC.130 (2002), 1009-1017. 24. E. K. Narayanan and S. K. Ray, L p version of Hardy’s theorem for semisimple Lie groups, Proc. Amer. Math. SOC.130 (2002), 1859-1866. 25. E. K. Narayanan and S. K. Ray, The heat kernel and Hardy’s theorem on symmetric spaces of noncompact type, Proc. Ind. Acad. Sci. 112 (2002), 321330. 26. S. Thangavelu, Hardy’s theorem on the Heisenberg group revisited, Math. Z. 242 (2002), 761-779. 27. S. Thangavelu, Hardy’s theorem for the Helgason Fourier transform on rank one symmetric spaces, Colloq. Math. 94 (2002),263-280. 28. R. Sarkar, Revisiting Hardy’s theorem on semisimple Lie groups, Colloq. Math. 93 (2002), 27-40. 29. R. Sarkar, A complete analogue of Hardy’s theorem on S L 2 ( R ) and characterisation of the heat kernel, Proc. Ind. Acad. Sci. 112 (2002), 579-594. 30. A. Bonami, D. Demange and P. Jaming, New uncertainty principles for the Fourier and windowed Fourier transforms, Revista Math. Ibero. 19 (2003), 2355. 31. S. Thangavelu, An introduction t o the uncertainty principle: Hardy’s theorem on Lie groups, Prog. Math. 217, Birkhauser, Boston (2003). 32. S. Thangavelu, On theorems of Hardy, Gelfand-Shilov and Beurling on semisimple Lie groups, Publi. RIMS 40 (2004), 311-344. 33. S. Thangavelu, On Paley-Wiener and Hardy theorems for NA groups, Math. Z. 245 (2003), 483-502. 34. R. P. Sarkar and S. Thangavelu, On theorems of Beurling and Hardy for the Euclidean motion group, Tohoku Math. J. (to appear).
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35. S. Thangavelu, An uncertainty principle for operators with applications to nilpotent Lie groups, (preprint) 36. S. K. Ray and R. Sarkar, L P version of Hardy’s theorem and characterisation of heat kernels on symmetric spaces, (preprint) 37. C. Smitha and S. Thangavelu, An optimal version of Hardy’s theorem for the Dunk1 transform (under preparation).
Optimal sk-Spline Approximation of Sobolev’s Classes on the 2-Sphere C. Grandison and A. Kushpel Department of Mathematics, Physics and Computer Science Ryerson University 350 Victoria Street Toronto Ontario M5B 2K3 Canada [email protected], [email protected]
Summary. The space of sk-splines is the linear span of shifts of a single kernel K . In this article we introduce sk-splines on S. It is shown that, with suitably chosen kernel K , the subspace of sk-splines realizes sharp orders of Kolmogorov’s n-widths in different important situations.
1 Introduction In the case of S2, it is not possible to construct, in general, an equidistributed set of points since there are only finitely many polyhedral groups. Extensive computations for optimal configurations have been reported in a number of articles (see, e.g., [S]). However, attempts to find sets of points on the sphere which imitate the role of the roots of unity on the unit circle have usually led to very deep problems in the Geometry of Numbers, Theory of Potential, etc., and usually these approaches give us just a measure of the uniformity of the distribution of points (like cup discrepancy [9] or the minimum possible energy of a configuration, [4], [24]) rather than explicit constructions. In the univariate case (on the circle Sl) the rate of convergence of the best (in L, sense) piecewise polynomial spline of degree r - 1 with n fixed equi-spaced knots on the Sobolev class WL(S1), 1 5 p , q 5 00, has the order n--T+(11P-1/4)+as n + m, for r E Z+, where (u)+ := max{u, 0); see e.g. [23]. The rate of best approximation to Wi((sl) from Tn & Wi(S1), the subspace of trigonometric polynomials of degree 5 n, has the same order of convergence. A fundamental question for the spline theory o n S2 is whether with n suitably chosen knots (and kernel) the subspace of sk-splines with these knots can give the same order of L1 approximation o n Sobolev classes W [ ( S 2 )as Kolmogorov’s n-widths dn(W[(S2), L1(S2))as n + oo? If yes, then how do we construct optimal approximants?
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C. Grandison and A. Kushpel
The purpose of the current article is to construct a subspace of sk-splines which gives the same order of convergence as the subspace of polynomials of the same dimension and the respective n-widths. sk-Splines were introduced and their basic theory developed by Kushpel [7]-[8], [lo]-[17] and [23]. Suppose that A is a convex, compact, centrally symmetric subset of a Banach space X with unit ball B. The Kolmogorov n-width of A in X is defined by
&(A, X)= &(A, B ) = inf
sup
inf
XnLX f E A SEX,
Ilf
-g((x,
where X, runs over all subspaces of X of dimension n. Since it is known that W,.l(Sd)is a compact subset of L q ( S d )when a > d - - - the function of
(1:
3+
interest to us, d, (W:(Sd),L q ( S d ) is ) well defined in these cases. In many situations sk-splines give new examples of optimal subspaces in sense of Kolmogorov's n-widths (see e.g. [lo]-[17]). It will be shown that with suitable kernel the subspace of sk-splines with n knots on the equiangular grid realizes Kolmogorov's n-width d,(W[(S2),Ll(S2)) in the sense of order.
2 Elements of Real Harmonic Analysis on
S2
Let ( X , Y > = 5 1 ~ 1 + 1 c 2 ~ 2 + ~ 3 ~ 3 , w h e r e x = ( z l , 5 2 ,Ez 3R)~ , Y = ( Y I , Y ~E, Y ~ ) R3 be the usual scalar product in R3, and S 2 be the 2-dimensional unit sphere in R3, S2 = {zlz E R3, (z,z) = 1). For each k = 0,1,2,. . ., let {Y;(Z)}~=-~ be an orthonormal basis' for H k , the eigenspace corresponding to the eigenvalue ~k = -k(k 1) of the Laplace-Beltrami operator A for the sphere, i.e., AYL = rkYL, -k 5 I 5 k. Denote by T, := @2=oHk, the space of spherical polynomials of degree 5 n, then dimT, = (n 1)2. In this paper such polynomials will usually be represented as functions of 9 and p. Let p be the normalized rotation invariant measure on the sphere, so that
+
+
For any
4 : S2 + R define
'We view basis is:
Hk
as a subspace of
Lz(S2)here. One choice for this real orthonormal
where -Ic 5 1 5 Ic and Ic = 0 , 1 , 2 , . . . , and P r ( t ) are the associated Legendre functions.
Optimal sk-Spline Approximation of Sobolev's Classes on the 2-Sphere
73
if 1 5 p < 00, l$I"dP),"" if p = 00. esssupZEs2 Iq5(z)(, (JSZ
IldllP =
The space L2(S2) = {$ : S2 --t R orthogonal decomposition
I 4 measurable and IIq511p <
00)
has the
M
k=O
A function K , on S2 is said to be zonal with respect to a pole q E S2 if it is invariant under the action of all rotations IS of S2 which fix q, i.e. K,(z) = K , ( I s ~for ) all z E S2,and IS E SO(3) with aq = q. Then K, = K ( ( x ,q ) ) for some defined on [ - 1 , 1 ] . For any integrable function with domain [ - 1 , 1 ] we define its spherical convolution with any integrable function 4 on S2 by
We shall need the Legendre polynomials, p k , k = 0 , 1 , 2 . . . which can be defined in terms of the generating function M
k=O
where 0 _< JpI < 1 and It1 5 1 . It can be shown that the kernel for orthogonal projection on
Hk
k
yL(zc)yL(q) = ( 2 k -t- l ) P k ( ( z , q ) ) . I=-k
This is clearly real valued, zonal with pole q and is a member of H k for each q E S 2 . Let cose = ( x , q ) . Hereafter we shall use Z(')(t) to denote ( 2 k 4-1 ) P k ( t ) , SO that z ( k ) ( ( z , q ) )= zF)(x)= ( 2 k
Let $ E L l ( S 2 ) ;then monics
+ l ) P k ( ( z , v ) ) = (2k + 1)&(cose).
4 has a formal Fourier expansion in spherical harm
k
k=l
I=-k
For this d consider 00
k
74
C. Grandison and A. Kushpel
where a > 0. The function 4a(z)is called the athfractional integral of @(z). Let Aa4(z) = be the function in Ll(S2) defined by
k= 1
l=-k
The function is called ath fractional derivative of 4. It is known (see [l])that if a > 0 then the function &(x) is well-defined and the spherical fractional integral is essentially the spherical convolution ja * 4 where ija : [-1,1] -+ R E Ll([-l, l]),is defined by
Using this definition, for a , ,Ll> 0 we have A* o Ap = AaSp. It is shown in [l]that &((*,q)) E L p / ( S 2 for ) cr
> 2 / p and 1 I p 500.
(3)
The classical Sobolev classes on the sphere can alternatively be defined thus
It is clear that the function sets W,.l(S2)are convex and centrally symmetric, they are also shift-invariant, i.e., for any f E W;(S2) and any cr E S0(3),we have f&) = f(a-lz) E W,.(S2). References to the previously mentioned results from harmonic analysis can be found in [l],[5], and [25]-[27]. Let E(z)be a fixed continuous function on [-1,1] and A N = {zj}y=l be a fixed set of points on S2,referred to as the knots. An associated sk-spline on S2 means any function of the form N
sk(z) = a0
+ C a j E ( (z,
zj)),
(4)
j=l
where aj E Iw, 0 5 j 5 N . For any fixed k and AN the space of all functions of the form (4)will be denoted SK;".
3 Estimates of the Best Approximation by sk-Splines on S2 We parameterize the points z on S2 by their spherical coordinates z = (0, 'p) E [ 0 , 4 x [0,27r). Let us fix some b = 2n, n E Z+, and consider the grid points
Optimal sk-Spline Approximation of Sobolev's Classes on the 2-Sphere
75
x,,~= ( B T , c p j ) , where 0, = 7 ~ r / ( 2 b )1, 5 r 5 2b - 1 and cpj = ?rj/b, 0 5 j 5 2b - 1. This set of 2b(2b - 1) equiangular points xT,jE S2 (excluding north and south poles) will be called the "atlas points" and denoted A 2 b x 2 b . Let S K A Z b x Z b be the space of sk-splines on the sphere associated with : [-1,1] + R. Let ck,l(h) be the Fourier knots A 2 b x 2 b and the kernel coefficients of h associated with the orthonormal basis ( 1 ) for L2(S2)as in ( 2 ) and define sk$)(x) E S K A Z b x Z b for each k = 0 , 1 , . . . ,b / 2 - 1 by
aa
2b-1
where
2b-1
b-1
b
1
a, = -sin 2b2
(g)
m=O
1 2m+1 -sin
(
+
(2m 1 ) r r 2b
)
and xjPl2-1)
=
{
if 0 5 k 5 l ( b / 2 - 1 ) / 2 J ; 1 1 2 - 2 k / ( b / 2 - l), if [ ( b / 2 - 1 ) / 2 J 1 5 k 5 b / 2 - 1.
+
The main statement of this section is > 0 then for all b E 2Zt
Theorem 3.1 Let a
We also prove Theorem 3.2 Let a > 0. Let A = A N = {xj}jN_1 be any fixed set of N distinct points on S2. Let S K 2 be the associated space of sk splines induced by the kernel ija and knots A N . Then
inf AN
sup
inf
[If
- skllLl(sz) x N;"l2 as N
00.
f E W r ( S 2 ) skESKf/
It is known (see [2]) that
So from the Theorem 3.2 follows Corollary 3.2 If n and N are such that dimTn = d i m S K 2 then the space of sk-splines S K C gives the same order of L-1 approximation t o W,.l(S2)as the space of spheracal polynomials 'Jn.
76
C.Grandison and A. Kushpel
Proof of Theorem 3.1 Our estimates of the best approximation by sk-splines are based on the following result by Kushpel [ 2 1 ] , [22]. Let
we will use the following optimal reconstruction o f f from its values at by a member of ( J b / 2 - 1 studied in I211
A2bx2b
26- 1 2b- 1 r=l j = O b/2-1
where Xerrvj(x)is a zonal member of
Ybp-1
with pole at ( O r , cpj) defined by
bl2-1
k=O
with uf and Af'/2-1) as defined in the prelude to the theorem and
zif!vj(x)= ( 2 k
+ l)pk((x,Xr,j))*
Here Pk(t),t E [-1,1],k = 0,1,. . . are the Legendre polynomials normalized so that pk(1)= 1. Note that b'
dimYb~b/~-~ = - x 2b(2b - 1) = d i m S K t 2 b x 2 bas b 4 00.
4
(5)
It has been proved in [21] and [22] that as b 4 00
Using this integral representation, (5) and ( 6 ) we obtain that as b --t
00:
Optimal sk-Spline Approximation of Sobolev's Classes on the 2-Sphere
.
.
<
.
. .. . . ... .. .
.
..
-0.5
.. .
..' .
,,.:.:
.
,,...
' .. . .. . . .. . ...'
.
'::, ." ...'
..a;
0.5
'.
-0.5
Fig. 2. Graph of r = 1 4 -
Fig. 3. Graph of r = 1+
..
(for CY = 2.1). skg12 (6%)
ll.k~,~zIlm
(for CY = 2.1)
77
78
C . Grandison and A. Kushpel
where
Let us put
r=l
by ( 3 ) sk:;) E Ll(S2) for any a
j=o
> 0 and we have finally
Optimal sk-Spline Approximation of Sobolev’s Classes on the 2-Sphere
79
Proof of Theorem 3.2 The upper bounds in Theorem 3.2 follow from Theorem 3.1. To get lower bounds we have used known asymptotics for Kolmogorov’s n-widths (see [2], [3] [18] - [20]) which ensure in particular t h a t
d n ( w ~ ( sL 2 )~, ( sx~nTffI2 > > as n
--f
00.
References 1. Askey, R., Wainger, S., On the behavior of special classes of ultraspherical expansions-I, 11, J. Analyse Math. 15, 1965, 193-244. 2. Bordin, B., Kushpel, A. K., Levesley, J., Tozoni, S. A., n-Widths of multiplier operators on two-point homogeneous spaces, In: Approximation Theory IX, Vol. I, C. K. Chui and L. L. Schumaker eds., Vanderbilt Univ. Press, 1998, 23-30. 3. Bordin, B., Kushpel, A. K., Levesley, J., Tozoni, S. A., Estimates of 7%-Widthsof Sobolev’s Classes on Compact Globally Symmetric Spaces of Rank One, Journal of Functional Analysis, 202, 2003, 307-326. 4. Depczinski, U., Stokler, J., A differential geometric approach to equidistributed knots on Riemannian manifolds In Approximation Theory IX, Vol. I, C. K. Chui and L. L. Schumaker eds., Vanderbilt Univ. Press, 1998, 97-104. 5. Erdklyi, A., Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, (1953). 6. Glasser, L. and Every, A. G., Energies and spacings of point charges on a sphere, J . Phys A: Math. Gen., 25, 1992, 2473-2482. 7. Gomes, S. M., Kushpel, A. K., Levesley, J., Ragozin, D. L., Interpolation on the torus using number theoretic knots, in Wavelets, Images and Surface Fitting, L. L. Schumaker (eds), World Scientific Publishing, Nashville, TN, 1997, 143-150. 8. Gomes, S. M., Kushpel, A. K., Levesley, J., Ragozin, D. L., Interpolation on the torus using number theoretic knots, Journal of Approximation Theory, 98, 1999, p. 56-71. 9. Grabner, P., Tichy, R. F., Spherical designs, discrepancy and numerical integration, Math. of Comp., 60, 201, 1993, 327-336. 10. Kushpel, A. K., Extremal Properties of Splines and n-Widths in space CZ,, Preprint 84.25, Inst. Math. Acad. Sci. of Ukrainian SSR, 1984. 11. Kushpel, A. K., sk-Splines and exact estimates of n-widths of functional classes in Cz,, Preprint 85.51, Inst. Math. Acad. Sci. of Ukrainian SSR, 1985. 12. Kushpel, A. K., Rate of convergence of sk-spline interpolants on classes of convolutions, In Investigations in Approximation Theory, Inst. Math. Acad. Sci. of Ukrainian SSR, 1987, 50-58. 13. Kushpel, A. K., A Family of Extremal Subspaces, Ukrain. Math. Zh. 39, 6, 1987, 786-788. 14. Kushpel, A. K., Sharp estimates of the widths of convolution classes, Izvestia Acad. Nauk SSSR, 52, 1988, 1315-1332. 15. Kushpel, A. K., Estimates of the Diameters of Convolution Classes in the Spaces C and L, Ukrain. Math. Zh. 41, 8, 1989, 1070-1076. 16. Kushpel, A. K., Levesley, J., Interpolation on Compact Abelian Groups using Generalized sk-splines, In Approximation Theory VIII, v.1: Approximation and Interpolation, Charles K. Chui and Larry L. Schumaker (eds.), World Scientific Publishing, 1995, 317-324.
80
C. Grandison and A. Kushpel
17. Kushpel, A. K., Levesley, J., Light, W., Approximation of smooth functions by sk-splines, In Advanced Topics in Multivariate Approximation, F. Fontanella, K. Jetter and P.-J. Laurant (eds.), World Scientific Publishing, 1996, 155-180. 18. Kushpel, A. K., Levy Means associated with Two-Point Homogeneous Spaces and Applications, 49 Seminkio Brasileiro de Anklise, 1999, 807-823. 19. Kushpel, A. K., Estimates of n-Widths and €-Entropy of Sobolev’s Sets on Compact Globally Symmetric Spaces of Rank 1, 50 Seminbio Brasileiro de Analise, 1999, 53-66. 20. Kushpel, A. K., n-Widths of Sobolev’s Classes on Compact Globally Symmetric Spaces of Rank 1, In Trends in Approximation Theory, K. Kopotun, T. Lyche, M. Neamtu (eds.), Vanderbilt University Press, Nashville, TN, 2001, 201-210. 21. Kushpel, A. K., Optimal Reconstruction of Sobolev’s Classes on 2-Sphere, 53 Seminario Brasileiro de AnBlise, 2001 , 233-244. 22. Kushpel, A. K., Optimal Distribution of Data Points on S2 and Approximation of Sobolev’s Classes in L,(S2), Journal of Concrete and Applicable Mathematics, (to appear). 23. Levesley, J., Kushpel, A. K., Generalized sk-spline Interpolation on Compact Abelian Groups, Journal of Approximation Theory, 97, 1999, 311-333. 24. SafF, E. B., Kuijlaars, A. B. J., Distributing many points on a sphere, The Math. Intelligencer, 19, 1, 1997, 5-11. 25. Stein, E. M., Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Pres, (1971). 26. Szego, G., Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I. (1959). 27. Vilenkin, N. J., Special functions and theory of representation of groups, Nauka, Moscow, (1965).
&Spline Approximation on the Torus A. Kushpel Department of Mathematics, Physics and Computer Science Ryerson University 350 Victoria Street Toronto, Ontario M5B 2K3 Canada akushpel0ryerson.ca
Summary. Almost optimal error bounds for sk-spline approximants of the same order, in power scale, as best trigonometric approximation on Sobolev's classes W T ( T d )in L l ( T d )are established.
Introduction Let the &dimensional torus T d= IRd/27rZd and 6, = {wl, ..., wn}c T d , be a fixed set of nodes. Then, if K E C(Td)has mean value 0, an sk-spline on 'ITd is a function of the form
k=l
Such functions are natural generalization of periodic polynomial splines, realized when K is a Bernoulli monospline of appropriate order. The sksplines were introduced, and their basic theory developed by Kushpel [9,7].In this paper we continue with the development of error estimates for sk-spline approximation begun in the univariate case in [S, 71, and extended in the case of multidimensional torus in [4, 5, 11, 121. For an overview of approximation by sk-splines see [lo]. In the univariate case the rate of convergence of spline approximants with n equidistant fixed knots on Sobolev's classes Wi('JI'')in Lq(T1), 1 5 p,q 5 a, has order n-T+(l/P-l/q)+ as n + 00, for T E IN, where (a)+ = max{a, O}; see e. g. [S]. The rate of best approximation from In, the subspace of trigonometric polynomials of degree 5 n,has the same order of convergence. A fundamental question for multidimensional spline theory is whether the subspace of multidimensional splines will be as good as the subspace of trigonometric polynomials of the same dimension in the sense that they have the same rate of convergence on Sobolev's classes?
82
A. Kushpel
The purpose of the current article is to construct subspace of sk-splines which gives the same up to some logarithmic factor rate of convergence in L l ( T d ) as the subspace of trigonometric polynomials 7(r;)of the same dimension on Sobolev's classes Wr('ITd). For any f E Ll('lrd)define the Fourier coefficients
c(z) = z = (z1,...,z d )
E
s
f(x) eizxdx,
22d , x = (21, ..., z d )
E
Td.
where zx denotes the scalar product of the vectors z and x, and dx is the normalized Haar measure on the torus. Let K E Ll('lrd),4 E L 1 ( T d ) ,then
( K * 4>(x>=
J K(x
- Y)4(Y)dY
In the multidimensional setting, on W d ,we consider the usual anisotropic Sobolev class W i ( T d )with smoothness r = ( q ,. . . , r d ) , 0 < r = r1 = r, < r,+1 5 ... 5 T d . w e will use the fact that
= {c+Kt; *4, c E
IR, 4 E U p } ,
Up being the unit ball of L p ( T d ) , d
K'(x) = k= 1 and
1 kE
IZkI-Tke-k
=
Z\{0)
IzI~-"...Iz~~-~~,
1k, eizx ZEZd
if z, # 0 for any 1 5 m 5 d otherwise.
kz={O For any s = (s1,..., S d ) E INd define
r; :=
U
qs).
r-lrssm
It is known that n = Card(T6) x 2mmv-1. The order of approximation of functions from W i ( T d )in L q ( T d ) by , the trigonometric polynomials
T(x) E T(r6)= lin{eizxI z E
r;}
sk-Spline Approximation on the Torus
83
from the optimal hyperbolic cross TG containing n harmonics, is (see [3]) (log n ) ( v - l ) ( T - ( l / P - l / q ) + ) ,
n-T+(llP-'/q)+
1 < p , q < 00, r > (l/P - l/q)+. Approximation by splines with gridded n knots gives the order of convergence n-T/d+(l/P-l/q)ld
1 I p I 2 I q I m, l / p - l / q 1 1/2, r > 1 (see [ll, 12, 21) which is much slower than the order of best trigonometric approximation. In [4] we established the upper bound n-T+llP-
1I p 5 2
l / q (log n)Tv+(d-l)(l/P-l / q ) 1
I q 5 00,
l / p - l / q 2 1/2, r
> 1.
Using more sophisticated techniques we improved this result in [5]by decreasing the order of the logarithmic term n-T+l/P--llq
1I p
(log n)Tv-(l/P-l/q)
I 2 5 q I 00,
l / p - l / q 2 1/2, r
> 1.
An important component of our method is that the nodes of splines are generated using prime numbers and associated number theoretic ideas. For a fixed prime number P let Zd 3 G p = { g = (91, ...,g d ) : 1 5 gi 5 P - 1, i = 1,..., d } . Then for any fixed g E G p , let A$ = {wj = 2 n j g / P : j = 0, ..., P 1) E W d . Theorem 1 Let r = r1 = ... = T d then there is such g* E Gp that for 6 p = A%* we have sup
Ilf
- s-ksp(f)ll1<< P-T(logP)T2d+2d-1
7
P-tw,
f Ew;,o
where
Skdf) =
c w j€A%'
Moreover,
vwj (f)Kr;(x- Wj),
84
A. Kushpel
1 Error Estimates In this article we shall be concerned just with the case r = r1 = ... = r d . Let
It is known that @m(t)2 0 for any t
E 'It". Let
us define
Al(t) = V~I-1 ( t )- V~I-2( t ) , I 2 2,
n d
AS(x) =
Asj (
~ j ) ,s
=(
~ 1 ..., , sd)
E Zd,
j=1
and
~ ( 2 ~36K,(x) )
=
C A,(x),
e := (1,..., 1) E
zd.
selm
It is easy to check that for any s E Z d we have llAs(x)lll << 1,
IIKmII1 <<
C
Ossesm
IIAs(x)IIl << m d ,
and T(x) = (T * Km)(X) for any polynomial T E 7(T6), Using this fact we get for any T E 7 ( T 6 )
sk-Spline Approximation on the Torus
85
Remark that for any z E T dwe have 0 5 E ( z ) E T(T6).It is known (see [ 6 ] ) that there is such g* E G p that for any polynomial T ( x ) E I(T6)with
Q =: P(1og P)-rd, where
1: = (TI,
..., T d ) , T = T I
(2)
= ... = ~d we have
Let w p be the discrete normalized measure on A$. Using (1) and (3) we obtain 1
1
I ~ ( X ) l d W P ( X )=
p
c
IT(Wj)l
wj € A%*
5 CmdJ IT(Y)IdY. Comparing (3) and (4) we get that for any T(x) E
K:
g*
(T,x) :=
A,
and
c
T ( W j ) ($K,(X
(4)
T(r6)
1
P
c
)
- Wj) = T(x)
WjEAgd
IKm(X-Wj)l i C m d .
wj €A:*
From the last estimate and the Lebesgue inequality we find
It it known that (see [l, 131)
86
From
A. Kushpel
(a), (5) and
(6) we conclude
5 CP-'(log
P)'*d+2%
(7)
Using duality arguments we come to the estimate Cp-r(log
where vwj ( h )=
p)r2d+2d-1
1
K k q x - Wj)h(X)dX
c
and
Ip)(X)
=
(kz)-Vzx.
Zcr;
Applying Bernstein's inequality (see [l, 131) we find that for any T E 7 ( T ; )
II 1 K &'(log QId-' IIT II 1, where
T(-r) = T *
(k,)-'eizx
za-6 and d n ( y - , o ,L1)
L bn(w;,o, L1) 2
n-'(log n)- ('-
l)(d--l)
where d, and b, are Kolmogorov and Bernstein n-widths respectively. This means that
&-Spline Approximation on the Torus
87
References 1. K. I. Babenko, Approximation of a set of periodic functions of m a n y variables by trigonometric polynomials, Doklady AN SSR, 132(5):982-985, 1960. 2. N. Dyn, J. Narcowich and J. D. Ward, Variational principles and Sobolev-type estimates f o r generalized interpolation on Riemannian manifold, Constr. Approx., 15:175-208, 1999. 3. E. M. Galeev, Approximation by Fourier sums of sets of functions with bounded derivative, Mat. Zametki, 22:197-211, 1978. 4. S. M. Gomes, A. K. Kushpel, J. Levesley and D. L. Ragozin, sk-Spline interpolation on the torus using number theoretic knots. In Curves and Surfaces with Application in CAGD (A. Le Mehaute, C. Rabut and L. L. Schumaker, eds), 143-150, 1997, Vanderbilt Univ. Press, Atlanta, GA. 5. S. M. Gomes, A. K. Kushpel, J. Levesley and D. L. Ragozin, Interpolation o n the torus using sk-splines with number theoretic knots, J. Approx. Theory, 98:56-71, 1999. 6. N. M. Korobov, Trigonometric Sums and its Applications, Nauka, Moscow, 1989. 7. A. K. Kushpel, sk-Splines and exact estimates of n-widths of functional classes in the space Cz,, Preprint 85.51, Inst. Math. Akad. Nauk Ukrain. SSR, Kiev, 1-47, 1985. 8. A. K. Kushpel, Rate of convergence of sk-spline interpolants o n classes of convolutions, Inst. Math. Acad. Nauk. Ukrain. SSR, 50-58, 1987. 9. A. K. Kushpel, Sharp estimates of widths of convolution classes, Math. USSR Izvestia, AMS, 33(3), 631-649, 1989. 10. A. K. Kushpel, J. Levesley and W. A. Light, Approximation of smooth functions by sk-splines, In Advanced Topics in Multivariate Approximation (F. Fontanella, K. Jetter and P.-J. Laurent, eds), 155-180, 1996, World Scientific, Singapore. 11. J. Levesley and A. K. Kushpel, Interpolation o n compact abelian groups using generalized sk-splines, In Approximation Theory VIII (C. K. Chui and L. L. Schumaker, eds), vol. I, 317-325, 1996, World Scientific, Singapore, 1996. 12. J. Levesley and A. K. Kushpel, Generalised sk-spline interpolation on compact abelian groups, J. Approx. Theory, 97:311-333, 1999. 13. S. A. Teliakovsky (Teljakovskig, Some estimates for trigonometric series with quasi convex coeficients, Math. Sbornik, 63(105), 426-444, 1964.
Entropy Numbers of Sobolev and Besov Classes on Homogeneous Spaces A. Kushpell and S. Tozoni2 Department of Mathematics, Physics and Computer Science Ryerson University 350 Victoria Street Toronto, Ontario M5B 2K3 Canada [email protected]
IMECC-UNICAMP CAIXA Postal 6065 13081-970 Campinas SP Brazil tozoniQime.unicamp.br Summary. Sharp orders of entropy numbers en(Wp,L,) and en@&, L,) of Sobolev and Besov classes Bi,Ton compact globally symmetric spaces of rank 1 classes
arefoundfor l < p , q < m a n d l _ < r < c a .
Introduction Let A be a compact subset of a Banach space X. Let us denote by en(A,X), H,(A, X), dn(A,X), and d"(A, X) entropy numbers, €-entropy, Kolmogorov's and Gelfand's n-widths respectively. In the present paper we investigate the asymptotic behavior of the entropy numbers of Sobolev classes and Besov classes pi,Tin L , on a compact globally symmetric space of rank 1 or twopoint homogeneous space M d , of dimension d. On any such manifold there is an invariant Riemannian metric d(., .), and a measure du which is induced by the normalised left Haar measure on (2 and is invariant under the action of Q. Two point homogeneous spaces admit essentially only one invariant second order differential operator, the LaplaceBeltrami operator A. A function 2 : M d + R is called zonal if Z(h-l.) = Z(.) for any h E 'Ft. A complete classification of the two-point homogeneous spaces was given by Wang [20]. They are the spheres Sd, d=1,2,3, ...; the real projective spaces P d ( R ) ,d=2,3,4 ...; the complex projective spaces Pd(C), d=4,6,8, ...; the quaternionic projective spaces Pd(lH),d=8,12, ... and the Cayley elliptic plane P16(Cay).
mi
90
A. Kushpel and S. Tozoni
For each zonal function z on M d ,we have an univariate function 2, defined on [-1,1], z(z) = .Z(cos(2Xd(z, o ) ) ) ,2 E M d , where X is either 7r/2L or 7r/4L, depending on the homogeneous space M d and L is the diameter of G / N . Let L, be the set of all complex measurable functions f on M d of finite norm, given by
II f I,=
{
if 1 I P < 00, ( J M d If (.)l"d.(.))l/", ess sup{lf(z)l : z E M d } ,i f p = 00.
+
Let z" E L1([-l, 11,(1- z)"(l z) pdz) . Then, for any integrable function g we define the convolution z * g on M d by
For each k E IN, let H k be the eigenspace of the Laplace-Beltrami operator corresponding to the eigenvalue - k ( k a p l),where a and p are numbers associated with the particular homogeneous space M d . We denote IN = f3$=oHk and we have dimIN =: N d and H k IH l , k # 1. The Hilbert space L2 with the usual scalar product
+ + +
. is a unique real zonal eigenfunchas the decomposition L2 = @ ~ = o H kThere tion (up to scalar multiplication) z k E Hk such that, the orthogonal projection from L2 onto H k is given by the convolution operator f H z k * f . For any s > 0, the function
is integrable on M d . The Sobolev space W;, s > 0, is defined by W; := {f E L , : g-s * f E L,}, with norm llfll; := 11g.+ * f I l p . Here we have identified functions which differ by a constant, i.e., if f - g = c (constant) a.e., then f = g in W;. The Sobolev class is the closed unit ball of W; and is given
+
wi
up}.
by = { c gs * f : c E IR,f E Let cpk := K 2 k - K 2 k - 1 , where K2n is a natural generalization of the de la Vallhe Poussin polynomial Vn,znon S 1 . The function f E L, belongs to the Besov space BG,T,s,p, T E R,s > 0, 1 I p , r 5 00, if
We will identify two functions in B$ which differ by a constant. It is easy to see that Bd,Tis a normed vector space with norm 11 . The Besov class
Entropy Numbers of Sobolev and Besov Classes on Homogeneous Spaces
91
--s
Bp,r is the closed unit ball of B&. References to the previously-mentioned results from harmonic analysis can be found in [2, 5 , 61. Consider s, p , r, q E R with 1 5 p , q, r 5 00 and s > d ( l / p - l / q ) + . It is known that the convolution operator f H gs * f is a compact operator from L, to L, and hence the is a compact subset in L, ( see e.g. [2]). Using interpolation's Sobolev class properties we can show that the Besov class Bi,r is also a compact subset in L, ( see [15]).The main results establishes sharp orders of entropy numbers. Theorem 1 Let p , q, s E R.Then en(wi, L,) >> n-s/d,1 < p , q < 00, s > 0 , and e n ( V ; ,L,) << n-sld, 1 < p , q < 00, s > d. Theorem 1 Let p , q, s, r E R. If 1 < p , q < 00, s > d and 1 5 r 5 00, then L,) =: n-31'. Remark 1 Sharp an power scale estimates of entropy numbers e n ( w i , L,), n -+ 00 have been obtained in [12, 131.
wg
1 Entropy Numbers of Sobolev Classes Let cu = (a1,...,a n ) ,P = (PI, ...,Pn) E IR" and (a,P) = Cr=l akPk. Let 1110111 = (a,cu)'/' be the Euclidean norm on IR", S"-l = { a E R" : IIIallI = 1 ) be the unit sphere in R", Bz = { a E R" : IIIaIII 5 1) be the unit ball in R" and Vo1,A be the standard n-dimensional volume of a subset A in R". Let us fix a norm II.II on R" and denote by E the Banach space E = (IR", II.II) with unit ball BE. The Levy mean M(R", I] . 11) is defined by
where d p denotes the normalized rotation invariant measure on S"-l. For a convex centrally symmetric body V c IR" we define the polar body V o of V bv
The dual space E" = (IR",I( - 11") is endowed with the norm JIaJl"= sup{l(a,P)I : P E B E ) . Theorem 1.1 (see e.g. [16, p. 6391) For the Banach space E = (R", (1 - 11) we have N L ( B E 111 , 111) 5 2cn(M(11.110))2, n E OV, where C i s a n absolute constant. Theorem 1.2 (see e.g. [3, p. 3201) There exists a n absolute constant C > 0 such that f o r all convex centrally symmetric bounded and absorbing set V in R",(VoI, V . Vol, V o . (Val, BF)-')~/"2 C, n E OV. Theorem 1.3 (see e.g. [4, p. 2941) Let {sn} denote any of the sequences {d"} or {dn}. For every w > 0 there exists C, > 0 such that for any compact subset A in X we have
92
A. Kushpel and S. Tozoni
{&}z=l
Let us consider an arbitrary system of functions in L,. We demand that the system {&};=1 be orthonormal in L2. Set En = span { & , ...,&} and let J : IR" + En be the coordinate isomorphism that assigns to a = (a1,..., a,) E IR" the function 6" = ELZla& E En.The definition llall(,) := llJall, induces a norm on R".We will denote the unit ball of (IR", 11 . by B;,. The following result gives estimates for the Levy means
M(IR"7 II . II(P))'
{&}E=l
Theorem 1.4 (see [14]) Let be a n arbitrary system of orthonormal harmonics an TN = @fzI=oHk,n = dimTN. Then there exists an absolute constant C > 0 such that
We remark that different estimates of Levy means have been obtained in [8] - [13] to calculate n-widths and entropy of sets of smooth functions.
Theorem 1.5 (see [14]) If2 5 q < 00 and s > d / 2 , then
d,(W;, L,)
=: n-'ld.
Theorem 1.6 (see e.g. [7]) Let f E L,, 1 5 p in L,. Then inf
UEL
where l / p
Ilf - 41,
I 00
and let L be a subspace
= sup{I((f,y))I : I l Y l l P ~ I 1, Y J-
J%
+ l/p' = 1 and y IL means ( ( u y)) , = 0 for all u E L .
Theorem 1.7 (see e.g. [18, p. 341) For 1 l/p' = 1 and l / q + l/q' = 1 we have
dn(W%, L,)
< p , p ' , q , q ' < 00 such that
= d,(Wi,,L,O,
nE
l/p+
N.
Theorem 1.8 (see e.g. [17, p. 154, 1691) Let 1 < p 5 2 I q < 00. Then e2n-l(q,
where s1 > d(l/p - 1/2), s2
--s
2
L,) i e n ( W ; L2) en(W2 L,), 7
> d(1/2 - l / q ) , s1+
7
s2
= s.
Proof of Theorem 1 (Lower bounds:) The problem of estimating the entropy numbers e n ( w i ,L,) from below usually splits into two parts: reduction
Entropy Numbers of Sobolev and Besov Classes on Homogeneous Spaces
93
to some finite-dimensional problem in a Euclidean space IR", and obtainment of a lower estimate for the volume of a special convex body in IR". As regards the first part of the problem, in many cases its solution is relatively simple; therefore the main difficulty is to obtain proper lower estimates of the volumes of special convex bodies in a Euclidean space which are connected with the structure of spherical harmonics on Add. We will show that for any n E N and E > 0 there are [~/CE]" functions f i , . . . , fp/c+ E 2I p < m, such that for all 1 5 k # I 5 [ ~ / C Ewe ]" have llfrc - fillp 2 Cn-S/dE2.
Ti,
For this purpose we take for the norm )I . 11 on IR" the particular choice 11 - 11 = (1 with 1 < p' 5 2, l / p l/p' = 1. Holder's inequality implies that
+
and using Theorem 1.4 we can conclude that
M(W,
11 l l ~ ~ ,5) )~ 1
p l / n ~ ,
00.
Now it follows from Theorem 1.1that for any fixed 2 5 p < 00 and all n E N
and hence
Comparing this estimate with Theorem 2.2 we obtain that Vol, ((Bi"p/))") L C"V0l" (B,") which implies that the cardinality N,((B&,))O)of a minimal E-net for (Bi"p,,)o in the Euclidean norm can be estimated as
N€((B&,))", Ill - Ill) 2 (C+"
> 0 and n E N. Consequently there exist [ ~ / C E points ]" a i , ...a[l/celn in (B$,,)O such that )~J(C - - LY YI~ ) ) / ~ 2 ~ / 2 ,1 5 k # 1 5 [ ~ / C E ]From " . the Theorem 1.6 it follows that for any E
94
A. Kushpel and S. Tozoni
This implies that for any a k E (BYpf))”, 1 5 k 5 [1/celn there exists U k E I; such that IIJak - 2Lkllp 5 1 and therefore f k := g s
* ( J a k - u k ) E mi.
Consider 1 5 k # I 5 [1/CeInand let 1 v k , l = - ( ( J a k - u k ) - (Jal - .I)) 2 Since (I$ok,lIlp llfk
.
I 1 , ( ( J c Y k , U l ) ) = o and n =: N d , we get
- fZllp’ 2
( ( f k - fl, p k , l ) )
1
= 5((gs/2
* (2Y’k,l),gs/2 * ( 2 p k , l ) ) )
1 - 1 - 2/1g~/2 ( J a k - Jal)lIz-k 211gS/2
*
1
2 2 ( N ( N+ a + p + l ) ) - ” / 2 1 1 J a k
* (uk - ‘U)llz
- Jallli
2 Cn-S/de2.
+
Then the balls f k (Cn-s/de2/2)Up~, 1 5 k 5 [1/CeIn are mutually disjoint. Putting E = 1/2C we get
en(TVi,L P / )2 Cn-s/d, Now given 1 < p , q embedding
n + oo.
< 00, let 2 I p < oo such that p 5 p and q 2 p’. Then by
en(Wi,L,) 2 en(Wi,Lpf)2 Cn-”jd, n --+ oo. ( U p p e r bounds:) Let 1 < p 5 2 5 q < 00, s Theorems 1.3, 1.5, 1.7 and 1.8 it follows that
> d , and w = s/2d. Then from
Entropy Numbers of Sobolev and Besov Classes on Homogeneous Spaces
95
and hence
en(TVi,L ~_<) Csn-sld. we take 1 < j j < 2 < < 00
Now given 1 < p,q < 00 q I 4 and hence by embedding
such that
p 5 p and
en(W;,L,) _< en(Wi,L ~5) Csn-sld. H 2 Entropy Numbers of Besov Classes Two complex Banach spaces A0 and A1 are called an interpolation pair A = (Ao,A l ) if there exists a Hausdorff topological vector space in which A0 and A1 are continuously embedded. We define:
Let 0 < 0 < 1, 1 5 T 5 00, and let GO,,. be the functional defined by @e,T(v(t)) :=
{
(Jr(t-"(t))'.$)l/T, 15 < T
esssup,,o t-'cp(t),
T
00,
= 00,
where cp is a non-negative measurable function. Given a E E(A) we define
I I ~ I I ( A ~ , A ~ ) ~:= , , . @e,AK(t,a ) ) . The set
(Ao,Ai)e,r := { a E x(4 : IIaII(Ao,Al)e,, < 0 0 ) is a Banach space with the norm 11 . ) I ( A ~ , A ~ ) ~ , ~ ,and is called the interpolation
space of the pair A by the K-method. More information about interpolation spaces can be found in [l,19, 151.
Theorem 2.1 (see [15]) If0 < SO < s1 then
B;,'T
If 1 I ro < T I 5 oc) then
c B;:,.,
1 2 p , r 5 00.
A. Kushpel and S. Tozoni
96
If A
= (Ao,A l ) is an interpolation pair and
(Ao,Al)e,l c A
c (Ao,Al)e+,,
we write A E N ( 8 , Ao, Al).
Theorem 2.2 (see [19]) Let A = (Ao,A l ) be a n interpolation pair of Banach spaces and let X be a Banach space such that Ao, A1 c X . Suppose that there exist ai 2 0 and Ci > 0, i = 0 , 1, such that
Hc(BAi,X)5 CiE-'lai, Suppose also that there exists
E
> 0 , i = 0,1.
e" E ( 0 , l ) and A E X ( 8 ,Ao, A l ) such that
H,(BA,X) 2 CE-lIa,
+
where 0 < c5 = (1 - 8)ao e " q , and C and a = (1 - 8)ao Oal, then
+
(5)
E
> 0,
> 0. If 8
H,(BA,X ) =:
E
(6)
E [0,1], A E N ( 8 ,Ao, A l ) ,
> 0.
(7)
Let A be a compact subset of a Banach space X and let n E IN, R,E > 0. Then the implications H,(A,X) 5 n + e,+l(A,X) 5 E , H,(A,X) > n - 1 + e,(A,X) 2 E , e,+l(A,X) < E + H,(A,X) < n, e,(A, X ) > E + H,(A, X ) > n - 1, follow from the definitions of €-entropy and entropy number. Therefore for a E R,a 2 0 we have e n ( A , X )=: T L - ~ , n E N if and only if H,(A, X ) x E > 0, and hence the corollary below follows from Theorem 2.2. E
E
Corollary 2.1 Let A = (Ao,A l ) be a n interpolation pair of Banach spaces and let X be a Banach space such that Ao, A1 C X . Suppose that there exist ai 2 0 and Ci > 0, i = O , l , such that
en(BAi,X)5 Cin-Qi, n E N, i = 0 , l . Suppose also that there exists
e" E ( 0 , l ) and A E N ( 8 , Ao, A l ) such that
e , ( B i , X ) 2 Cn-',
+
where 0 < c5 = (1 I!&, and C and a = (1 - 8)ao O c q , then
+
(8)
n E N,
> 0. If 8
(9)
E [0,1], A E N ( 8 , Ao, A l ) ,
Proof of Theorem 2 Consider 1 5 p , r 5 00, S , S O , S ~> 0, and 8 E (0, l), with s = (1 - 8)so 8sl. From ( 4 ) we have Bi,l = (WpSo,WpS1)e,l, Bi,oO= (WpSo, W ~ S ~ )and B ,hence, ~ , from (3),
+
Entropy Numbers of Sobolev and Besov Classes on Homogeneous Spaces
97
(wpSo,w.')e,ciwp" c (w,So, wpS1)e,oo. Therefore
w;E z(e,w;o, w;').
Now from (2) we have B;,l
c B;,T c B&,, and hence, from (4),
(wpSo,wpS1)e,ic B&. c (WpsO, wpS1)e,oo. Thus,
B ; , ~E q
e , w;o, wp~1).
Let s , p , q , be ~ as in Theorem 2 and choose SO,s1 E IR such that d < SO s < s1. Let 0 E ( 0 , l ) with s = (1 - B)so BSI . From Theorem 1 we have
+
en(TV:,
L ~5) Cn-'Jd,
n E N, i = 0, I .
<
(13)
From Theorem 1 we have,
en(TVi,L*) 2 Cn-'ld,
nE
IN.
(14)
Then the result of Theorem 2 follows from Corollary 2.1, ( l l ) ,(13) and (14).
Acknowledgments Partial funding of A. Kushpel's work was provided by FAPESP/Brazil, Grant 03/10393-8. S. Tozoni's research was supported in part by FAEPUNICAMP/Brazil, Grant 275/03, and by CAPES/Brazil, Grant AEX0710/032.
References 1. J. Bergh, J. Lofstrom. Interpolation Spaces, Springer-Verlag, 1976. 2. B. Bordin, A. K. Kushpel, J. Levesley, S. A. Tozoni. n-Widths of Multiplier Operators o n Two-Point Homogeneous Spaces, In Approximation Theory IX, v. 1: Theoretical Aspects ( C . K. Chui and L. L. Schumaker, eds.), 23-30, 1998, Vanderbilt Univ. Press, Nashville, TN. 3. J. Bourgain, V. D. Milman. New volume ratio properties f o r convex symmetric bodies in R",Invent. Math., 88:319-340, 1987. 4. B. Carl. Entropy numbers, s-numbers, and eigenvalue problems, J. Funct. Anal., 41 ~290-306,1981. 5 . S. Helgason. Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. 6. T. Koornwinder. T h e addition formula f o r Jacobi polynomials and spherical harmonics, SIAM J. Appl. Math., 25(2):236-246, 1973.
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7. N. P. Korneichuk. Exact Constants in Approximation Theory, Cambridge Univ. Press, 1991. 8. A. K. Kushpel. O n an estimate of Levy means and medians of some distributions on a sphere, In Fourier Series and their Applications (V. K. Dzyadyk, N. P. Korneichuk, eds.), 49-53, 1992, Inst. Math., Kiev. 9. A. K. Kushpel. Estimates of Bernstein's widths, Ukrain. Math. J., 45(1):54-59, 1993. 10. A. K. Kushpel, J. Levesley, K. Wilderotter. O n the asymptotically optimal rate of approximation of multiplier operators from L, into L,, Constr. Approx., 14(2) :169-185, 1998. 11. A. K. Kushpel. Levy means associated with two-point homogeneous spaces and applications, In Annals of the 49" Semintirio Brasileiro de Antilise, 807-823, 1999. 12. A. K. Kushpel. Estimates of n-widths and e-entropy of Sobolev's sets on compact globally symmetric spaces of rank 1, In Annals of the 50" Semindrio Brasileiro de Anblise, 53-66, 1999. 13. A. K. Kushpel. n- Widths of Sobolev's Classes on Compact Globally Symmetric Spaces of Rank 1, In Trends in Approximation Theory (K. Kopotun, T. Lyche, M. Neamtu eds.), 201-210, 2001, Vanderbilt Univ. Press, Nashville, TN. 14. A. K. Kushpel, S. A. Tozoni. Sharp Orders of n - Widths of Sobolev's Classes on Compact Globally Symmetric Spaces of Rank 1, In Annals of the 54" Semindrio Brasileiro de Anblise, 293-303, 2001. A preliminary version appeared as Relat6rio de Pesquisa 55/02 (2002), in www.ime.unicamp. br/rel-pesq/relatorio.html. 15. A. K. Kushpel, J. Levesley, S. A. Tozoni, Estimates of n-widths of Besov classes on two-point homogeneous manifolds, Technical Report 2002/34, Department of Mathematics and Computer Science, University of Leicester, Leicester, 2002. 16. A. Pajor, N. Tomczak-Jaegermann. Subspaces of small codimension of finitedimensional Banach spaces, Proc. AMS, 97:637-642, 1986. 17. A. Pietsch. Operator Ideals, Deutscher Verlag der Wissenschaften, Berlin and North-Holland Publ., Amsterdam, 1980. 18. A. Pinkus. n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985. 19. H. Triebel. Interpolation properties of e-entropy and diameters. Geometric characteristics of imbedding f o r function spaces of Sobolev-Besov type, Math. USSR Sbornik, 27:23-37, 1975. 20. H. C. Wang. Two-point homogeneous spaces, Ann. of Math., 55:177-191, 1952.
Operator Equations and Best Approximation Problems in Reproducing Kernel Hilbert Spaces with Tikhonov Regularization S. Saitohl, T. Matsuura2, and M. Asaduzzaman2 Department of Mathematics Faculty of Engineering Gunma University Kiryu 376-8515 Japan ssaitoh0math.sci.gunma-u.ac.jp
Department of Mechanical Engineering Faculty of Engineering Gunma University Kiryu 376-8515 Japan [email protected] asadQeng .gunma-u. ac .jp Summary. Let H K ( E )be a reproducing kernel Hilbert space comprising complexvalued functions {f} on E and Lj ( j = 1,2,.. .) be a bounded linear operator on H K ( E ) into a Hilbert space H j . Then, for d j E Hj we shall consider the simultaneous operator equations Lj f = dj ( j = 1,2,.. .) with the best approximation problem, for given d j E H j
Furthermore we shall give a general idea and method for approximations of L2 functions by Sobolev Hilbert spaces by using the Tikhonov regularization. We shall illustrate examples by figures for approximations of LZ functions by the first and second order Sobolev Hilbert spaces.
1 Introduction and Background Theorem We shall formulate our background theorem which has many concrete applications based on [2-61. Let H K be a Hilbert space comprising complex-valued functions {f} on a set E admitting a reproducing kernel K ( z ,y) and let L be a bounded linear operator on H K into a Hilbert space H . We introduce the inner product in the space H K , for any fixed X > 0
100
S. Saitoh, T. Matsuura, and M. Asaduzzaman
A(fl,f 2 ) H ~+ (Lfl, L f 2 ) H .
(1)
Then, it forms a Hilbert space and this Hilbert space H K ( L ; A )admits a reproducing kernel K L ( z ,y ; A) on E. Then, we have the relation of K ( z ,y ) and K L ( X , Y; A)
Theorem 1 T h e best approximation f:,g,fo in the sense, f o r any fo E HK and f o r a n y g E H
fx”,,,fo(X)
= X(fo(*), KL(.,2;A))HK
+ (g(.>,L K L ( *2;, A ) ) H .
(4)
As simple and typical reproducing kernel Hilbert spaces, we shall consider the Sobolev Hilbert spaces H K ~and H K ~admitting the reproducing kernels
and
The norms in H K ~and H K ~are given by
respectively. We shall examine the best approximation in (3) for some typical Hilbert spaces H and bounded linear operators L. In general, we are interested in the behaviours of the best approximation functions for X tending to zero from the viewpoint of the Tikhonov regularization. So, we wish to illustrate the behaviours of the best approximations for X tending to zero.
Reproducing Kernel Hilbert Spaces
101
2 Typical Examples See [3] for many concrete reproducing kernel forms for which Theorem 1 is applied. We can see a general example and a general approach for simultaneous linear partial differential equations in N. Aronszajn [l]who discussed deeply Green’s functions in connection with reproducing kernels. We shall give typical examples.
Let
2.1
1
G(X,Y)= -e-Iz-’l. 2
(7)
Then G(z,y) is the reproducing kernel for the Hilbert Sobolev space HG comprising all absolutely continuous functions f(x) on R with finite norms
Hence, we can examine the best approximation problem as follows: For any given FI,F2 E L2(R),
0.8 0.6 0.4 0.2 0 -0.2 -0.4
-2
-1
0
1
2
Fig. 1. Examples of approximated functions in (9). (a) Fl(z) = 0 and F~(S) = x[-1,11 (top thin curve). (b) Fl(z)= F2(z)= x[-1,11(middle bold curve). (c) A ( z ) = x[-1,11and Fz(z)= 0 ( bottom curve).
For the first order Sobolev Hilbert space H K we ~ shall consider the two bounded linear operators L1 : Hjy1 -+ Llf = f E L2(R) and L2 : H K ~ + L2 f = f’ E L2(R). Then, the associated reproducing kernels K1,1(x,y; A) and K1,2(2,y; A) for the RKHSs with the norms 2.2
Allfll”HK, + IIf112L2(R, and
102
S. Saitoh, T. Matsuura, and M. Asaduzzaman
&
A 11 f 11 % K ~+ II f 'II (R)
are given by K1,1(z,y;X) =
At2
+ (A + 1)4
and
respectively. Hence, the best approximate functions f;,l(z; A, g) and f:,2(z; A, g) in the senses, for any g E L2(R)
and
respectively. Note that fT,2(z;A, g) can be considered as an approximate and generalized solution of the differential equation y'= g(z) on R
(16)
in the first order Sobolev Hilbert space H K ~See . Figure 3. 2.3 For the second order Sobolev Hilbert space H K ~we shall consider the three bounded linear operators into L2(R) defined by
L1:f-f
-
Reproducing Kernel Hilbert Spaces L2:
f
and L3
:f
103
f’
f”.
Then, the reproducing kernels K2,1(2,y;A), the Hilbert spaces with the norms
K2,2(2,y; A)
and
K2,3(2,y; A)
for
A l l M f K , + IlfIl;,(R)? Allf l”H,, + Ilf’lI;,(R)r and
xllfll”H,, + llf”Il;,(R), are given by
and
Then, the corresponding best approximate functions f&(qA, g), f;,2(2; A, g), and f2*,3(2; X,g) are given by, for any g E L2(R)
and
respectively.
We shall give another type applications of Theorem 1. Note that 1 ~ ( 2 , y= ) -e-iz-yi (1 + Iz - YI) 4 is the reproducing kernel of the Sobolev space HK with finite norms
(23)
104
S. Saitoh, T. Matsuura, and M. Asaduzzaman
A
x = ,001
x = .0001 I
-4
-2
Fig. 2. Graphs of &(z;x,g) g(z) = X[-1,1].
I
2
4
in (14)(top) and f;,l(z;X,g) in (20) (bottom) for
e Reproducing Kernel Hilbert Spaces
1
0.5
105
x = .00001 x = .0001
x = .001
x = .01
5
10
106
S. Saitoh, T. Matsuura, and M. Asaduzzaman
Fig. 4. Graphs of &(z; X , g ) in (22) for g(z) = x[-1,11.
Therefore, we can examine the approximate problem as follows:
References 1. N. Aronszajn, Green’s functions and reproducing kernels, Proc. of the Symposium on Spectral Theory and Differential Problems, Oklahoma A. and M. College, Oklahoma (1951), 355-411. 2. D.-W. Byun and S. Saitoh, Best approximation in reproducing kernel Halbert spaces, Proc. of the 2nd International Colloquium on Numerical Analysis, VSP-Holland (1994), 55-61. 3. S. Saitoh, Integral Transforms, Reproducing Kernels and their Applications, Pitman Res. Notes in Math. Series 369,Addison Wesley Longman Ltd (1997),
UK.
Reproducing Kernel Hilbert Spaces
107
0.6
0.4 0.2
0
I
I
I
-0.2
-2
-1
0
1
2
3
4
5
Fig. 5 . Examples of approximated functions in (25). (a) Fl(z)= Fz(z)= F3(z)= x[-i,i](top bold curve). (b) Fi(z)= x[-1,11and Fz(z)= F3(z)= 0 (the second bold curve) (c) Fi(z)= F2(z)= 0 and F3(z) = x[-1,11 (the thin curve). (d) Fl(z)= = 0 and Fz(z) = X [ - ~ ,(the ~ I rest curve).
Saitoh, S., Approximate Real Inversion Formulas of the Gaussian Convolution, Applicable Analysis, (to appear). Saitoh, S., Matsuura, T., and Asaduzzaman, M., Operator Equations and Best Approximation Problems in Reproducing Kernel Hilbert Spaces, J. of Analysis and Applications, 1(2003), 131-142. S. Saitoh, Constructions by Reproducing Kernels of Approximate Solutions f o r Linear Differential Equations with L2 Integrable Coeficients, International J. of Math. Sci. (to appear).
Finite Fourier Transforms and the Zeros of the Riemann [-Function George Csordas’ and Chung-Chun Yang2 Department of Mathematics University of Hawaii Honolulu, HI 96822 USA [email protected] Department of Mathematics Hong Kong University of Science & Technology Clear Water Bay, Kowloon Hong Kong mayang(0ust.h.k
Summary. The distribution of zeros of entire functions represented by Fourier transforms are investigated. Applications include the Riemann &function and the Fourier transforms of kernels related to the Jacobi theta function.
1 Introduction Today, there are no known necessary and sufficient conditions that a “nice” kernel must satisfy in order that its Fourier transform have only real zeros. It is this fundamental issue that motivates us to consider questions and results dealing with the distribution of zeros of real entire functions represented by Fourier transforms. The specific entire functions we study, in the sequel, are intimately connected with the Riemann J-function. We recall that the Riemann &function admits the Fourier integral representation (see, for example, [19], [21, pp. 278-3081 or [7])
H ( z ) :=
1
(f) = 1 @(t)cos(zt) d t , 00
0
where
M
n=l
(called the Jacobi theta function) and where, for n = 1 , 2 , 3 . . . ,
110
George Csordas and Chung-Chun Yang 00
an(t) := x x n 2 ( 2 x n 2 e 4 t- 3) exp (5t - 7 ~ ~ ~ 6 (t 3 E ~ W ~). )
(1.3)
n=l
The Riemann Hypothesis is equivalent to the statement that all the zeros of H ( z ) are real (cf. [25, p. 2551). We also note that the Taylor series expansion of H ( z ) about the origin can be written in the form
c 00
H ( z ) := HJz)
:=
( - l l k b k 52k
k=O
(2k)!
7
where the bk’S are the moments (whose existence is assured by the properties of @(t)stated below) defined by bk =
lo
u ~ ~ @du, ( u ) k = 0 , 1 , 2 , .. . .
Now the ruison d’gtre for investigating the kernel @(t)(see Section 2) is that there is an intimate connection (the precise meaning of which is unknown) between the properties of @(t)and the distribution of the zeros of its Fourier transform. The present work is motivated by the following question that was raised in [3, p. 481. Does there exist a positive number R such that the entire function
1
R
H,(2)
:=
@(t)cos(2t)dt (R> 0 )
(1.6)
has some non-real zeros? While this problem remains open, our goal here is to relate the properties of certain admissible kernels (Definition 3.1) to the distribution of zeros of their Fourier transforms. Since our principal application involves the kernel Qi, in Section 2 we begin with a brief summary of results involving @ (Theorem 2.1) and we highlight some of the lesser known, recently established, convexity properties of Qi (Theorem 2.3 and Theorem 2.4). In Section 3, we first recall an elementary, albeit fundamental, lemma (Lemma 3.2) about the distribution of the zeros of Fourier transforms of admissible kernels. We use the properties of Qi(t),and those of related kernels, to analyze the distribution of zeros of their Fourier transforms (Proposition 3.3 and Theorem 3.4). The Laguerre inequalities, discussed in Section 3, provide the theoretical underpinning for some of our numerical work (Remark 3.5). With aid of some new Laguerre inequalities, we also formulate necessary and sufficient conditions for entire functions, represented by finite Fourier cosine transforms, to belong to the Laguerre-P6lya class (Definition 2.2 and Theorem 3.6).
2 A Survey of Properties of @ ( t ) For the reader’s convenience we begin with a brief review of the basic properties of @(t)defined by (1.2).
Finite Fourier Transforms and the Zeros of the Riemann &Function
111
Theorem 2.1 ([7, Theorem A]) The function @(t)of (1.2) satisfies the following properties: (i) f o r each n 1 1, a,(t) > 0 for all t 2 0, so that @(t) > 0 f o r all t 2 0; (ii)@(z)is analytic in the strip -n/8 < I m z < n/8; (iiip(t)is an even function, so that di(2m+1)(0)= 0 (rn = 0 ,1 ,. . . ); (iv)for any E > 0 , lim @")(t)exp [(n- &)e4'] = o t+M
for each n = 0 , 1 , . . . ; (v) @'(t)< 0 for all t > 0 .
The proofs of statements (i) - (iv) can be found in P6lya [19], whereas the proof of (v) is in Wintner [26] (see also Spira (241). 0 Let S ( T )denote the closed strip of width 27, 7 2 0, in the complex plane, C, symmetric about the real axis:
S ( r ) = { z E C : IIm(z)l 5 r } . Definition 2.2 Let 0 5 7 < 00. W e say that a real entire function f belongs to the class G ( r ) ,i f f is of the f o r m
where a {zk}&
cEl
z k E S(7) \ { O } , and 1 / l Z k ( ' < 00 and where the zeros o f f are counted according to multiplicity and are arranged so that
2 0,
0 < lzll 5 1221 5 . . . . W e allow functions in G ( T ) t o have only finitely m a n y zeros by letting, as usual, Zk = 00 and 0 = I/&, k 2 ko, so that the canonical product in (2.1) is a finite product. I f f E G ( r ) ,f o r some r 2 0, and i f f has only real zeros (i.e., i f r = 0), then f is said to belong t o the Laguerre-Pdlya class, and we write f E L-P. I n addition, we write f E L-P*, i f f = pg, where g E L-P and p is a real polynomial. Thus, f E L-P* i f and only i f f E 6 ( r ) , f o r some r 2 0 , and f has at most finitely many non-real zeros.
The relevance of the class L-P* in connection with the distribution of zeros of H R ( Z ) (cf. (1.6)) stems from the fact that by virtue of the P6lya-Wiman Theorem ([6] and [14]), for each R > 0, there exits a positive integer P O , where po depends on R, such that H t ) ( x )E L-P for all p 2 PO. In order to indicate the significance of the next theorem, we first recall that H ( z ) is an entire function of order one [25, p. 161 of maximal type (cf. [7, Appendix A]). Thus, with the above nomenclature the Riemann Hypothesis is true if and only if H E L-P. It is known ([25, p. 301 that all the zeros of H lie in the interior of the strip S(1),so that H ( z ) E 6 ( ~with ) , 7 = 1. The change of variable, z2 = -z in (1.4), yields the entire function
112
George Csordas and Chung-Chun Yang
Then it is easy to see that F ( x ) is an entire function of order and so the Riemann Hypothesis is equivalent to the statement that all the zeros of F ( a ) are real and negative. Now it is known (cf. Boas [l,p. 241 or P6lya and Schur [22]) that a necessary condition for F ( z ) to have only real zeros is that the moments b, (in (1.5)) satisfy the Turcin inequalities; that is, 2m-1
( m = 1 , 2 , 3 ,... ) .
b,-lZ1,+~>0 bkizl
These inequalities have been established (cf. [7] and [9]; see also [17]) as a consequence of one of the two properties ((a) or (b)) of @(t)stated in the following theorem.
Theorem 2.3 Let @(t)be defined by (1.2). Then @(t)satisfies the following concavity and convexity properties. (a) (“7, Proposition 2.11)
If
1
00
&(t) :=
@(@du
then log K$ ( t ) is strictly concave for t
(t 2 0 ) ,
> 0, that is,
d2 -logKG(t) < O for t > 0. dt2
(b) (19, Theorem 2.11) The function log@(&) is strictly concave f o r t (c) ([3, Theorem 2.121) @(&) is strictly convex for t > 0. 0 Now a calculation shows that log@(&) is strictly concave for t only if
+
g ( t ) := t [ ( ~ $ ‘ ( t) )@(t)@’’(t)] ~ @(t)@’(t) >0
for
t > 0.
> 0.
> 0 if and (2.4)
To express (2.4) in another way, we use the fact that @(t)> 0 (Theorem 2.1) and so we can write
@(t> = e-v(t) Since @’(t)< 0 for t
(t E IR) .
(2.5)
> 0, we see that
and hence, by (2.4) and (2.5), g ( t ) > 0 if and only if v’(t) is strictly increasing t for t > 0. Using some of the analysis in [9], Newman [18] proved the following stronger result.
Theorem 2.4 ([18, Theorem 11) Let v ( t ) defined by (2.5). Then ~ ” ’ ( t>) 0 for t > 0. 0
Finite Fourier Transforms and the Zeros of the Riemann &Function
113
In terms of @(t),Theorem 2.4 says that the function @’(t)/@(t) is strictly concave for t > 0. One consequence of part (c) of Theorem 2.3 is that the entire function represented by the Fourier cosine transform of @(&) cannot have any real zeros. More precisely, since @(&) is convex for t > 0, the following result holds.
Theorem 2.5 ([3, Corollary 2.131) With @(t)defined by (1.2), we have
irn
@(&) cos(zt)dt > 0
for all
zE
R.
3 The Distribution of Zeros of Fourier Transforms The convexity properties of @(t), summarized above, can be used to prove that some finite Fourier cosine transforms of @(t)have only real zeros. Indeed, @(t)> 0 and @’(t)< 0 for all t > 0 by Theorem 2.1 and a detailed (“hard”) analysis of @”(t)shows that @”(t)< 0 on the interval I = [0, 0.111 ([lo, Lemma 3.51). Thus, it follows from a classical theorem ([23, Part V, Problem #173]) that for all 0 < R 5 0.11
F,(z) :=
iR
@(t) cos(zt)dt E L-P,
(3.1)
see [lo,Lemma 3.51 for the details. We remark, parenthetically, that i f F,(z) E L-P for all sufficiently large R, then an easy argument would establish the validity of the Riemann Hypothesis. In order to expedite our presentation (and for the sake of simplicity), it will be convenient to introduce the following definition.
-
Definition 3.1 A function K : R R is termed a n admissible kernel, if K ( t ) satisfies (i) K ( t ) E Crn(R) and (ii) K(”)(t)= 0 (exp ( - lt12+a))as t -, 00 f o r some a > 0 and n = 0 , 1 , 2 , .. . .
The following elementary lemma highlights some of the differences that exist between the distribution of zeros of finite and infinite Fourier cosine transforms of admissible kernels.
Lemma 3.2 Let K ( t ) be an admissible kernel. For R > 0 , set F ( s ) :=
Irn
K ( t )cos(zt)dt
1
R
and
F,(z)
:=
K ( t )cos(zt)dt.
(3.2)
Then the following assertions are valid.
(1) If K’(0) # 0 , then the entire function F ( z ) has a n infinite number of non-real zeros and at most a finite number of real zeros.
114
George Csordas and Chung-Chun Yang
(2) If K ( R ) # 0, then the entire function of eqonentia2 type, FR(z),has an infinite number of real zeros and at most a finite number of non-real zeros; that is, with the notation introduced above, F, (x)E C-P*. Prooj It is easy to show that both F ( z ) and F,(x) are entire functions. Moreover, it is known that F ( z ) has order at most (2 a)/(l a ) < 2 ([20, p. 91) and that F,(z) is an entire function of order 1 and of exponential type. (1) First suppose that z # 0. Then using the properties of an admissible kernel and integrating by parts twice, we have
+
z 2 F ( z )= -K’(O)
+
Im
+
K”(t)cos(zt)dt.
(3-3)
Now, by the Riemann-Lebesgue Lemma (which is applicable since K ( t ) is an admissible kernel), the integral on the right-hand side of (3.2) tends to zero as 2 -+ f o o . This, together with the assumption that K‘(0) # 0, implies that F ( z ) has at most a finite number of real zeros. But F ( s ) is an even entire function of order less than 2, which tends to zero as z t f o o (by another application of the Riemann-Lebesgue Lemma) and thus we conclude (with an appeal to the Hadamard factorization theorem) that F ( z ) has an infinite number of zeros and all but a finite number of these zeros are non-real. (2) The proof of the assertion that F,(z) has an infinite number of real zeros is similar. Indeed, in this case we integrate by parts once to get
zFR(z)= K ( R )sin(zR) -
LR
K’(t)sin(zt) dt.
(3.4)
Since K ( R ) # 0, we once again infer from (3.4) and the Riemann-Lebesgue Lemma that F R ( z )has an infinite number of real zeros. For the proof that F,(z) has at most a finite number of non-real zeros we refer to [8, proof of 0 Corollary 2.71. A glance at the above proof shows that Lemma 3.2 remains valid for kernels which meet less stringent assumptions than those stipulated for admissible kernels. For related results, pertaining to the distribution of zeros of finite Fourier transforms, we refer to [23, Part I Problem #147, Part I11 Problems #149, and #205, Part V Problems #164, #170-1751 and [21, pp. 166-1971. We remark that an argument similar to the proof of part (1) of Lemma 3.2 shows that evenness of the kernel K ( t ) is a necessary condition for its infinite Fourier cosine transform to belong to L-P. In contrast, this requirement is not a necessary condition for finite Fourier transforms to be in C-P. In the sequel we will adopt the following notation. With a,(t) defined by (1.3), set N
00
a,(t)
s N ( t ) := n= 1
and
@j(t)
a,(t)
:=
for
j = 1 , 2 , 3 , .. . .
,=j+l
(3.5)
Finite Fourier Transforms and the Zeros of the Riemann [-F'unction
115
By Theorem 2.1, both S N ( t ) ( N 2 1) and @j(t)( j 2 1) are admissible kernels. Moreover, easy calculations show that ai(0) = 0.0394... > 0 and ai(0) < 0 for all n 2 2. But @'(O) = 0 (cf. Theorem 2.1) and so it follows that ~ " ( 0 )# 0. Now, fix positive integer N . Then by Lemma 3.2, the entire function
has an infinite number of non-real zeros and at most a finite number of real zeros. Since @(R)> 0 for all R > 0, Lemma 3.2 implies that, for each fixed R > 0, the finite Fourier transform
1
R
F, (z; S N ) :=
sN(t) cos(zt) dt
(3-7)
has an infinite number of real zeros and at most a finite number of non-real zeros (and in fact F R ( z ;S N ) E C-P". But does F R ( z ;S N ) have any non-real zeros? Information about the existence of non-real zeros of F, (z;S N )is given in the next proposition. Proposition 3.3 Fix a positive integer m. Then for each positive integer N , there exists a positive number Ro = Ro(m,N ) , depending on m and N , such that, for all R 2 Ro, the real entire function F R ( x ;S N ) (defined by (3.7)) has at least m.non-real zeros.
Proof. We have shown above that F ( z ;S N ) has an infinite number of non-real zeros. Fix a positive integer N and a compact subset, W , of C such that W contains in its interior m non-real zeros of F ( z ;S N ) . Choose r > 0 such that W C B := { z ( ( z ( 5 r } . Now it is easy to verify that F,(z;sN) +F ( z ; s ~ ) uniformly on as R + 00. Indeed, by Theorem 2.1 (iv), there exists a positive constant C1 such that 0 < @(t)5 C1 e ~ p { - $ e ~ ~for} all t 2 0. Also, an elementary argument shows that there exists a constant Cz > 0 such that 7r ertexp{--e4t} 5 C2e4te ~ p { - e ~ for ~ } all t 2 0. Hence, for all z E D, 2
n
~F'(Z;S N )- F ( x ;SN)J
5
Lm
@(t)ertdt 5 C2
=
SN(t) cos(zt) dt
km
e4t e ~ p { - e ~dt ~ }= CZ exp{-e
4R },
4
a
and whence F,(z; S N ) -+ F ( z ;S N )uniformly on as R + 00. Thus, it follows from the classical theorem of Hurwitz that for all R > Ro, Ro sufficiently large, the entire function F, (x;S N ) has at least m non-real zeros in for all R 2 Ro.
0 In sharp contrast to the result of Proposition 3.3, the (infinite) Fourier cosine transform of @j(t)(cf. (3.5)) has no real zeros.
116
George Csordas and Chung-Chun Yang
Theorem 3.4 ([lo, Theorem 3.41) With j = 1 , 2 , 3 . .. , and for all x E R,
@j(t)
defined by (3.5), we have for
Q j ( tcos(xt) ) cosh(yt) dt > 0 and for each fixed In particular, F,(z;
@j)
> 0 for all z E R and j
y E [0,26).
(3.8)
..
= 1,2,3..
The proof of Theorem 3.4 is based on a careful analysis of the convexity properties of @j(t).An interesting consequence of Theorem 3.4 is the following geometric interpretation of the distribution of zeros of the Riemann &function. With the notation of (1.1) and (3.6), we have, for each positive integer j = N , H ( x ) = F,(x; S N )+ F,(s; @ N ) . Since H ( x ) has an infinite number of real zeros ([25, p. 2561 or [21, p. 2421) and F,(x; S N ) has an infinite number of nonreal zeros and at most a finite number of real zeros, it follows that F,(x; S N ) is negative for 1x1 sufficiently large and the validity of the Riemann Hypothesis depends on the nature of the intersections of the curves y = F,(x; S N ) and y = -F,(x;
@N).
Remark 3.5 In light of Proposition 3.3 and Theorem 3.4, it is natural to inquire whether or not the entire function
1, R
A R ( ~u1) ; :=
u l ( t )cos(xt) dt,
(3.9)
has some non-real zeros for certain “small” values of R, where al(t) is defined by (1.3). For example, when R = 2/5, then A 2 / 5 ( ~ , u lhas ) at least two nonreal zeros, so that A 2 / 5 ( ~ , u l$) L-P. Our proof is based on the fact (see, for example, [4] or [8]) that a necessary condition for a real entire function, f(x),to belong to the Laguerre-PMya class is that f(x) satisfy the Laguerre inequalities; that is; for p = 1 , 2 , 3 , . . . ,
(3.10) While we are able to prove analytically that A 2 / 5 ( x , a l ) does not satisfy the Laguerre inequalities at x = 0, here, for the sake of brevity, we merely report ; the result of our numerical calculations. With R = 2/5, set A ( z ) := A R ( ~u1). Then, A’(0)2- A(O)A”(O)< -0.000275.... 0 We call attention to the following complex analog of the Laguerre inequal) ities which are both necessary and sufficient for a function f ( z ) E 6 ( ~(cf. Definition 2.2) to have only real zeros. If f ( z ) E S ( T ) ,then f E L-P if and only if lf’(.z)12 2 for all z E C. (3.11)
~(f(.)f”(.))
The proof of this result ([lo, Theorem 2.101) is based on the geometric iy)I2 is a convex function interpretation of inequality (3.11); namely,
.(fI
+
Finite Fourier Transforms and the Zeros of the Riemann &Function
117
of y. Another characterization of functions in L-P (often-rediscovered in connection with the Riemann Hypothesis) may be stated in the following form ([lo,Theorem 2.121).If f ( z ) E 6 ( ~then ) , f E C-P if and only if
1 -1m Y
{-f’(z)f(Z)}
2 o for all
z =z
+ iy E C,
y
+ 0.
(3.12)
In case of the functions H ( z ) (cf. (1.1)) and HR(x) (cf. (1.6)),the verification of the inequalities (3.11)and (3.12)appears to be extremely difficult. On the other hand, for entire functions represented by finite Fourier transforms, the following recently established result (cf. [8,Corollary 2.71)appears to be more tractable than the above cited Laguerre inequalities.
Theorem 3.6 Let K ( t ) be an even, positive admissible kernel. For R > 0, and fo r z, y E Iw, set F R ( z ;K ) :=
LR
K ( t )cos(zt) dt,
~ ( zy), := ~ ( zy;, R ) :=
and
LR
K(t) cos(zt) cosh(yt) dt.
(3.13)
(3.14)
Then FR(x;K ) E L-P i f and only if u(z, y; R ) satisfies the Laguerre inequality ( U Z h Y;
q2 - u(z, y; R)%Z(Z, Y; R ) > 0,
for all z E R and y E R - { O } .
(3.15)
0
The relationship between the functions (3.13) and (3.14) can be realized with the aid of the differential operator cos(yD), where D = d/dz denotes differentiation with respect to z. Indeed, for each fixed real number y, u(z,y) = 2cos(yD)FR(z;K)(see, for example, [5]for a treatment of this and other infinite-order differential operators). We also note that Theorem 3.6 remains valid when R = 00, provided that F,(z;K) E S ( T )and the zeros of F,(z; K ) satisfy a certain density condition (see [8, Theorem 3.31). Since the zeros of H ( z ) = F,(q @) meets these requirements ([8,p. 3961) by virtue of the Riemann-von Mangoldt formula (see [12, pp. 18, 19,3011, [13,Chapter lo], [15,Ch. V]or [25, Chapter IX]), it follows that the Riemann Hypothesis is true if and only if the harmonic function u(z,y; m) := cos(yD)F,(z; @) satisfies the Laguerre inequalities (3.15). Today, we still do not know whether or not H ( z ) (or H R ( z ) )satisfies the Laguerre inequalities (3.10)for all z E R. For z = 0, these inequalities are the Tur6n inequalities which have been proved for both H ( z ) and H,(z) as a consequence of Theorem 2.3 ([3],[7], [9]). The foregoing analysis leads to a number of unresolved problems and questions and here we mention only one conjecture pertaining to this circle of ideas.
118
George Csordas and Chung-Chun Yang
Conjecture 3.7 For each fixed y, IyI 2 1, and for all R suficiently large, the entire function
lo FR
FR(x; @) :=
@(t) cos(xt) cosh(yt) dt E L-P.
(3.16)
We remark that it follows from the Hermite-Biehler Theorem ([11, Proposition 3.11, [16, p. 3141 and [2, Theorem 9a] that for each fixed y, IyI 2 1,
1
00
F,(z; @) :=
@(t) cos(zt) cosh(yt) dt E C-P.
Moreover, the conjecture is valid if it can be shown that FR(x;dj) E E(T) for all R sufficiently large, where T 5 1. Conjecture 3.7 could have been also formulated for entire functions F , ( x ;K ) E 6 ( ~ whose ), zeros satisfy certain density conditions, where K ( t ) is an even, positive admissible kernel and Iyl 2 r. While the applications of our investigations have focused on the distribution of zeros of functions related to Riemann <-function, we must emphasize that the more general problem of characterizing the kernels whose (finite or infinite) Fourier transform is in L-P is of fundamental importance and of independent interest, whether or not the Riemann Hypothesis is true or false.
Acknowledgement The second author’s research was partially supported by a UGC grants of Hong Kong , Project No.: 604103.
References 1. Boas, R. P.: Entire Functions. Academic Press. New York 1954. 2. de Bruijn, N. G.: The roots of trigonometric integrals. Duke Math. J. 17 (1950), 197-226. 3. Csordas, G.: Convexity and the Riemann &function. Glas. Mat. Ser. I11 33(53) (1998), 37-50. 4. Craven, T., Csordas, G.: Jensen polynomials and the Tur6n and Laguerre inequalities. Pacific J. Math. 136 (1989), 241-260. 5. Craven, T., Csordas, G.: Differential operators of infinite order and the distribution of zeros of entire functions. J. Math. Anal. Appl. 186 (1994), 799-820. 6. Craven, T., Csordas, G., Smith, W.: The zeros of derivatives of entire functions and the P6lya-Wiman conjecture. Ann. of Math. 125 (1987) 405-431. 7. Csordas, G., Norfolk, T.S., Varga, R.S.: The Riemann hypothesis and the Tur6n inequalities. Trans. Amer. Math. SOC. 296 (1986), 521-541. 8. Csordas, G . , Smith, W., Varga, R.S.: Level sets of real entire functions and the Laguerre inequalities. Analysis 12 (1992), 377-402.
Finite Fourier Transforms and the Zeros of the Riemann &Function
119
9. Csordas, G., Varga, R.S.: Moment inequalities and the Riemann Hypothesis. Constr. Approx. 4 (1988), 175-198. 10. Csordas, G., Varga, R.S.: Necessary and sufficient conditions and the Riemann hypothesis. Adv. Appl. Math. 1 (1990), 328-357. 11. Csordas, G., Varga, R.S.: Fourier transforms and the Hermite-Biehler Theorem. Proc. Amer. Math. SOC.107 (1989), 645-652. 12. Edwards, H. M.: Riemann’s Zeta Function. Pure and Applied Mathematics, Vol. 58. Academic Press. New York 1974. 13. Ivid, A.: The Riemann Zeta-Function. The Theory of the Riemann ZetaFunction with Applications. John Wiley & Sons, Inc. New York 1985. 14. Ki, H., Kim, Y.-0.: On the number of nonreal zeros of real entire functions and the Fourier-Pblya conjecture. Duke Math. J. 104 (2000), 45-73. 15. Karatsuba, A. A., Voronin, S. M.: The Riemann Zeta-Function. Translated from the Russian by Neal Koblitz. Walter de Gruyter & Co. Berlin 1992. 16. Levin, B.Ja.: Distribution of Zeros of Entire Functions. Transl. Math. Mono. 5, Amer. Math. SOC.Providence, RI 1964; revised ed. 1980. 17. Matiyasevich, Yu. V.: Yet another machine experiment in support of Riemann’s conjecture. Cybernetics. 18 (1982), 705-707. 18. Newman, C. M.: The G H S inequality and the Riemann hypothesis. Constr. Approx. 7 (1991) 389-399. 19. Pdya, G.: Uber die algebraisch-funktionentheoretischen Untersuchungen von J.L.W.V. Jensen. Klg. Danske Vid. Sel. Math.-Fys. Medd. 17 (1927), 3-33. 20. Pblya, G.: Uber trigonometrische Integrale mit nur reellen Nullstellen. J . Reine Angew. Math. 158 (1927), 6-18. 21. Pdya, G.: Collected Papers, Vol. I1 Location of Zeros. (R.P. Boas, ed.) MIT Press. Cambridge, MA 1974. 22. Pdya, G., Schur, J.: Uber zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J . Reine Angew. Math. 144 (1914), 89-113. 23. Pblya, G., Szego, G.: Problems and Theorems in Analysis. vols. I and 11. Springer-Verlag. New York 1976. 24. Spira, R.: The integral representation for the Riemann E-function. J . Number Theory 3 (1971), 498-501. 25. Titchmarsh, E.C.: The Theory of the Riemann Zeta-function. 2nd. ed. (revised by D.R. Heath-Brown). Oxford University Press. Oxford 1986. 26. Wintner, A.: A note on the Riemann [-function J. London Math. Soc.10 (1935), 82-83.
BP,
Qp
Spaces and Harmonic Majorants
E. Ramirez de Arellano’, L. F. Reshdis 0.2,and L. M. Tovar S.3 Departamento de Matembticas, Cinvestav IPN Av. Instituto Politkcnico Nacional 2508, C.P. 07360, D.F. Mkxico eramirez(0math.cinvestav.mx Universidad Aut6noma Metropolitana, Unidad Azcapotzalco, C.B.I. Apartado Postal 16-306 C.P. 02200 Mkxico 16, D.F. Area de Anblisis Matembtico y sus Aplicaciones. [email protected]
Escuela Superior de Fisica y Matemdticas del IPN Edif. 9, Unidad ALM, Zacatenco del IPN., C.P. 07300, D.F., Mkxico. t0varQesfm.ipn.u
Summary. In this paper we show how the characterization of Q,-hnctions obtained in the previous work [3] gives us a way of translating the main results of Qp-theory in terms of harmonic majorants.
1 Introduction
(zI
Let A := { z E C : < 1) = A(0,l) be the open unit disk in the complex plane CC and T its boundary. Let A be the class of analytic functions on A and 0 5 p < 00. We define Besov-type spaces ([9], 141)
where
is a Mobius transformation of A, la[ < 1. Further, for 0 the Qp-spaces Qp
:= {f E A : (f}Q, := sup
aEA
5 p < 00, we define
If’(-z)J2g(z,a)pda:dg, < GO}
E. Ramirez de Arellano, L. F. Resendis O., and L. M. Tovar S.
122
and
QP,o := {f E A : lim lal+l
lf’(z)12g(z,a)Pdxdy= 0)
where g ( z , a )
is the Green function of the unit disk with logarithmic singularity at a E A . The Bloch space of analytic functions is
B
:= { f E
A : sup(1-
1-z12)1 f’(z)I
ZEA
< co} .
If 0 < p < q < 00, we know from [St] that BP
c BQ c B ,
B,P c B,4 c Bo
where Bo is the little Bloch space
Bo
:= {f E
A : lim (1 - 1-zI2)1f’(z)l = 0). l4-l
Aulaskari, Xiao and Zhao have shown [5] (Prop. 1)
Theorem 1.1 Let f be an analytic function in A and p 2 0 . i) f E Q p if and only if
ii) f E
Qp,O
i f and only if
Actually the proof of this result given by Aulaskari et a1 in this paper appears just for 0 5 p 5 1, but it is not hard to see that this result is true for every non-negative p. It follows from this result that if p < q then QPc QQ, and these authors have shown besides that the inclusion is strict. Consider the well-known Dirichlet space,
D={f~d:/~If’(z)~~d.-dy
(1)
and the BMOA space,
B M O A = { f E A : SUP aEA
SS,
lf’(z>I2 (1 - l4a(Z>12> dxdy
< +m)*
(2)
Thus, for 0 5 p 5 1, the Q p spaces form a family of spaces between D and BMOA; Aulaskari and Lappan [2] also have shown:
BP,Q,
Spaces and Harmonic Majorants
123
Theorem 1.2 Let f be an analytic function in A. Then the following conditions are equivalent. i) f E B; ia) f E Q,, f o r all p > 1; iii) f E Q p , for some p > 1. Aulaskari and Tovar have shown in Theorem 1 of [4] that
Theorem 1.3 The following inclusion holds
The aim of this paper is to show how the main results of &,-theory -quoted previously- have their counterparts in terms of the harmonic majorants introduced in [3]. In this work, the authors used as definition of Qp-function the integral expression of Theorem 1.1,where the term (1 - I& ( z )12)P appears, for obtaining its harmonic majorant. In section 2 of the present work we obtain the corresponding harmonic majorant associated to a function in Q, when we use the original definition of a Q, space, that is, the integral expression with the term g p ( z ,a) and we obtain a characterization of functions in Q p in terms of this new harmonic majorant. Thus we obtain that Theorem 1.1 can be translated in terms of the harmonic majorants associated to each one of the expressions involved in this theorem. In section 3 we see how Theorem 1.2 can be expressed in terms of harmonic majorants associated to a function in QP or B. Likewise in section 4, after considering the corresponding harmonic majorants that characterize functions in Q p or BP we see how Theorem 1.2 can be translated in terms of harmonic majorants. Finally in section 5 we show how this new line of studying functions in weighted spaces introduce new and interesting problems intrinsically associated to this approach.
2 Harmonic Majorants Associated to the Expression
J-lf’(429p(.,4 dx dY In this section we obtain the harmonic majorant associated to a function in QP when we consider the expression that involves the Green function g(z, a ) and we show how Theorem 1.1 i) can be formulated in terms of harmonic majorants. We recall the following results. Let h : T -+ R be an integrable function on T,then P[h]denote its Poisson extension on A , explicitly given by
/
P [ h ] ( z )= -!2n -=
Pl,,(Q - t ) h ( e i t )d t ,
(3)
124
E. Ramirez de Arellano, L. F. Resendis O . , and L. M. Tovar S.
is the well-known Poisson kernel.
Proposition 2.1 [6] Let h : T + R be a n integrable function o n T , then its Poisson extension P[h]has nontangential limit h(ei8) at almost every ei8 E T . Proposition 2.2 [3] Let h : A --+ R be a continuous function and 0 < R < 1. Let hR : T + R be the function defined b y
6 E [0,27r).
hR(ei8) = O
Then h R i s a continuous function. Moreover, if 0 hR1(ei8) 5 hRz(ei8)
(4)
< R1 < R2 < 1, then
for every 6 E [O, 27r) .
Corollary 2.1 Let h : A + R be a continuous function. Let hl : T defined by h l ( e i 8 ) = sup h(rei8), B E [0,27r). o
(5) -+
R be (6)
If h l ( e i 8 ) < 00 then R+llim
hR(ei8) = hl(ei8).
(7)
Let h : A + R be a continuous function and 0 < R < 1. Let P [ ~ R: AUT ] be a function, that is harmonic on A and such that
--+
R
P [ ~ R ] I=ThR, where hR is defined by (4), that is, P [ ~ Ris] the classical solution of the Dirichlet problem with boundary values h R on T. For 0 < R1 < R2 < 1, by ( 5 ) , (3) and since P1,T(6)> 0 for all 6 , it holds
P[h~,](5 z ) P [ h ~ , ] ( z5) sup h ( z )
for all z E A.
(8)
%€A
With the above notation as a consequence of Harnack's theorem we have the following result.
Theorem 2.1 Let h : A + R be a continuous function. Let hl : T defined by (6) and let H1 : A + R be defined by
H l ( z ) = sup P [ h ~ ] ( z ) for every z E A. O
2
co or
H1
is a harmonic function on A and
where z = rei8 and H1 has nontangential limit lim Hl(reie) = hl(eie) z+ea@
a.e. o n
[-7r,7r]
.
--+
R be
BP,QPSpaces and Harmonic Majorants Proof. Suppose that
125
is not identically 00, therefore by (8) and Harnack's Theorem, H1 is a harmonic function on A . We consider a sequence { R k } C 1) with lim Rk = 1. Observe that hR1 is an integrable function. Then by H1
[i,
z
( 5 ) and the positiveness of Poisson kernel { h R k (eit)Pl,,(O-t)} is an increasing sequence of measurable functions with
for all t E
lim hRk(eit)Pl,T(O - t ) = hl(eit)Pl,,(O - t )
k+w
[-7r,7r].
Then by monotone convergence theorem for all z = reie E A we have
H l ( z ) = lim P l h ~ , ] ( z ) k-cc
Besides it follows that n-
7r
H l ( 0 )=
[
P1,o(B- t )h l (e i t )d t =
[
hl(eit)d t ,
J-7T
J-7r
that is hl is an integrable function on T . Then by Proposition (2.1), H I has nontangential limit hl(eie) at almost every eie E T . We also obtain a reciprocal result.
Theorem 2.2 Let h : A --t R be a continuous function. Let hl : T defined b y (6) and let H1 : A --+ R be defined by
-+
R be
H l ( z ) = sup P [ h ~ ] ( z ) for every z E A. O
If there exists g : T -+ R an integrable function such that Ihl(eie)l I g(eie) for almost everywhere ,ae E T then HI is a harmonic function on A and
where z = reie and H I has nontangential limit lim Hl(reie) = hl(e")
a x . on
[-7~,7r]
.
%-+ele
Pro05 We consider a sequence { R k } hl5 is an integrable function and
c [i,1) with lim Rk
= 1. Observe that
E. Ramirez de Arellano, L. F. Resendis O., and L. M. Tovar S.
126
)hRk(eit)p1,,-(e- t ) )I max{Ihg(eit)l, g(eit)}P1,,-(e - t )
for all t E [0,2n].Applying Lebesgue Theorem we have
< 00. Then H l ( z ) = lim P [ h ~ , ] ( < z )00 k+ca
and by Harnack’s Theorem H l ( z ) is a harmonic function. The existence of nontangential limits is justified as before. As a corollary we obtain Corollary 2.2 Let h : A --t R be a continuous function. Let hl : T defined by (6) and let H I : A -+ R be defined by H l ( z ) = sup P [ h R ] ( z ) O
-+
R be
for every z E A .
Then H I is a harmonic function i f and only i f hl a n integrable function o n T . Moreover
H ~ ( z=) P [ h l ] ( z )= 2n
Pl,,-(B- t ) h l ( e i t ) d t -K
where z = reie and H1 has nontangential limit a.e. o n
[-T,T]
.
We will apply the previous construction of harmonic majorants to the integral expression that defines &,-spaces in terms of the Green function g ( z , a ) . Let 0
< r < 1 and a E A ( r ) . Define 0 < 7 = - l a ’ and 0 as the compact 3
set [0,2n]x [0,7]x Z ( a , q ) .
Proposition 2.3 Let f : A h : 0 + @ by
for p
n.
-+
C be an analytic function and a E A. Define
> 0 and h(O,O,b) = 0, where 0 < p . T h e n h
is uniformly continuous o n
BP,QP Spaces and Harmonic Majorants
JL
( c ,PO 1
If’(z)12g(z,c)Pdxdy
127
<E .
E
By Proposition 2.3, given - > 0 there exists 6 > 0 such that I ( 0 , p , b ) 2x17 ( e l , p’, c )I < b implies
Therefore
Let a E A(r) := { z E C : IzI < r } be fixed. We know that i f f E QP then
By the absolute continuity of the integral, given E > 0 there exists r] > 0 such that E lf’(z)I2g(z, u)“dx dy < - . (9) (a,l7) 6
sf.-
<- “I. Without loss of generality, by Theorem 2.3 3 r] there exists 0 < 6’ < such that Ib - a1 < 6’ implies We can suppose that 77 2
We consider now the function defined on
( z ,b)
+
z)] x a(a,z) by
D = P(T) - A(a,
lf’(z>I29(z, b)P
*
128
E. Ramirez de Arellano, L. F. Reskndis O., and L. M. Tovar S.
This function is uniformly continuous on D so, without loss of generality, there such that ( b- a(< S” implies exists 0 < S” < min{6’, 6
z}
Since A ( a ,
z ) C A ( a , q ) by
(9)
In a similar way, since A ( u ,
z ) c A(b,2)by
(9) and (10)
So we have we have proved the following theorem. Theorem 2.4 The function h : AT -+ R defined by
in continuous in A(?-). Following the same line used in [3], consider an analytic function f E &, 0 < R < 1, and a = reie, with 0 < r < R. We introduce the family of functions hR : T + R defined by
Since f E Q,, the function hR is well defined and moreover, by the previous theorem each function is continuous. Besides we have for 0 < R1 < R2 < 1,
Let H R be the corresponding harmonic function given by (3), that is, the solution of the Dirichlet Problem associated to hR. It follows from Theorem 2.4 and Corollary 2.1 that if H I is the function defined in Theorem 2.2 then
and we are in position to give the equivalent to the Theorem 3.1 in [3] when we consider the Green function in the definition of &?,-functions.
BP,Q P Spaces and Harmonic Majorants
129
Theorem 2.5 Let f : A + C be an analytic function and 0 < p < 00. Then f belongs to the class Q p if and only if hl : T 3 R defined by hl(eie) =
sup a = r e i @ ,O < r < 1
ss,
I f w I 2 g P ( Z , a)da: dY
is bounded. H I is then the Dirichlet solution with boundary values given by hl. Besides H I is the least harmonic majorant of the family of harmonic majorants {HR}, 0 < R < 1. Let f E Q p , and let H t be the harmonic majorant associated to f through the Theorem 3.1 in [ 3 ] ,that is, the harmonic majorant associated to the expression SUP
aEA
//
lf’(z>12(1- I4a12)Pda:dy .
Thus Hf is obtained through the classical solution of the Dirichlet problem with boundary values denoted by h?. In the same way let H: denote the harmonic majorant associated to f through the previous theorem, that is, the harmonic majorant associated to the expression
Thus Hf is obtained through the classical solution of the Dirichlet problem with boundary values denoted by hy. Then after Theorem 1.1 and Theorem 3.1 in [3] and the previous theorem, it follows immediately
Theorem 2.6 Let f be an analytic function in A and p 2 0 . Then the following conditions are equivalent: i ) f E Q,; ii) ht i s bounded. Besides H t is the classical Dirichlet solution with boundary values given by h,6 ; iii) h! is bounded. Besides H: is the classical Dirichlet solution with boundary values given by h:.
3 Bloch Spaces and Harmonic Majorants In this section we are going to see what Theorem 1.2 looks like when we consider the harmonic majorants that characterize functions in Q p and B , through the integral definition which involves the expression (1 - 1 4 a ( ~ ) 1 2 ) P . Observe that given the function f E B , the function defined in A by
h ( 4 := (1 - 1~I2>1f’(Z>1 is clearly continuous. Using the same construction proposed in [3], we can associate corresponding functions hf and HF such that:
130
E. Ramirez de Arellano, L. F. Resendis O., and L. M. Tovar S.
1) hf is defined and bounded on T by 11 f I I B ~ ~ ~ ~ : =f I Ila, ~ 2) HF results harmonic on A and can be obtained through the classical solution of the Dirichlet’s Problem with boundary values given by hp. We need several estimations between the expression that defines the semin~~m I1 fS Ila and VIQ; Proposition 3.1 Let f : A
+C
and p
> 0. Then
for all a E A .
ProoJ Let U ( a , z ) := { z E C : I&(z)l < R } be a pseudohyperbolic disk with center at a and radius R > 0. Then
Thus
for all a E and so
D.The function m 2 ( 1 - z2)P takes its maximum at
z =f
l dim’
BP,Q, Spaces and Harmonic Majorants Corollary 3.1 Let f : A
+ CC
be a n analytic function and p 2 2
BW2= SUP(1 - I4 ) aEA
1(p
+
I-
=
l)P+’
PP
131
> 0. Then
If I (.)I2
If 1
QP
For every p > 0, let H B be the harmonic majorant associated to the seminorm B ( f ) and pH1 the corresponding one associated to the seminorm [ f ] .~Then P we have
Corollary 3.2 Iff E
then
Qp
)I
HB(f)I KdP) IPHl(f
where K l ( p ) =
7
-.
Proposition 3.2 Let f : A + C be an analytic function and p > 1, then there exist constants C1 and C2 such that
P H l ( f )I C l H B ( f )L c 2 [ P H l ( f )-l Proof. We know that for every a and z in A, 1 Then by Proposition 2 in [2]and Corollary 3.2 we have
5 2 9 ( ~a,) (see [ 5 ] ) .
where
So the result follows from these inequalities. rn Theorem 3.1 Let f : A conditions are equivalent:
-+
C be an analytic function. Then the following
i) H d f ) < 00, ii) p H ~ ( f<) 00 f o r all p > 1, iii) p H l ( f ) < 00 for some p > 1. Proof i) implies ii). This follows from the previous proposition. ii) implies iii) is trivial. iii) implies i). This follows from Corollary 3.2.
132
E. Ramirez de Arellano, L. F. ResQndisO., and L. M. Tovar S.
4 The Relationship between the B P and the Q,
Harmonic Majorants In this section we will translate Theorem 1.3 to the “language” of harmonic majorants characterizing functions in the BP and Qp-spaces. First, we need to check that we can associate to any function in BP the same construction carried out in [3] to get a harmonic majorant characterizing f in a similar way to Theorem 3.1 in [3]. We have to prove: Proposition 4.1 Let f E BP, 0 < p < 2 and let pg : A
+R
be defined by
T h e n ,g is a continuous function o n A.
Proof Let a E A be fixed and let 6 > 0 be such that a ( a , 6 ) C A. The function I : x a ( a ,6) + IR defined by
a
a
is uniformly continuous on x a ( a , b ) . Then for given p > 0 such that if 12’ - zI < p and I(’ - ( 1 < p then
Note that
ss,lf’(4I”
2 P-1
(1 - I4
)
d x d y = ,g(O)
E
> 0, there exists
< 00
because f E B,. Therefore
By Proposition 3.1 in [3] if f E Qq, 0 defined bv
Iq <
1, the function
qh
:
A
+
is continuous on A too. For 0 < R 5 1, 0 5 q , p , consider the following functions defined in T associated to a function in &, or BP, respectively,
Both functions are well defined and satisfy for 0 < R1
< R2 5 1,
qhRl(ei8)5 qhR2(ei8)I qhl(ei8)5 sup q h l ( e i 8 )= 0<8<2.rr
[f]b 4
and
From Proposition 3.15 in [3],and the previous theorem it follows now:
Corollary 4.1 Let 0 for every 0 5 p , q.
< R < 1. The functions qhR and
Let now qhR and
be the corresponding harmonic functions given by
pgR
pgR
are continuous
where z = rei8, 0 5 r < 1, 0 5 0 5 27r. It follows from Harnack’s Theorem and Theorem 2.1 in [3] that qH1 and pG1 are harmonic functions on A and satisfy
and
Consider now Corollary 3.2 in [3]
Corollary 4.2 Let f E Q q , O . Then there exists R = R ( f ) E ( 0 , l ) such that qhR = qhRt, f o r all R‘ E [R,1 ) . In particular, i f qhl : T --f R is the function defined by (6), then ,hl is a continuous function and qH1 : A + R is given by 1 27r Pl,T(O - t ) M e i t ) d t , q m - 4 =
1
where z = rei8, 0
5 r < 1, 0 5 0 5 2 ~ .
134
E. Ramirez de Arellano, L. F. Reskndis O., and L. M. Tovar S.
Proof. See Corollary 3.2 in [3]. In a similar way we can obtain
Corollary 4.3 Let g E B,P. T h e n there exists R = R ( f ) E ( 0 , l ) such that pgR = pgR', f o r all R' E [R,1). In particular, if pgl : T --+ R i s the function defined by (13), then pgl i s a continuous function and pG1 : A t R i s given
where z = reie, 0 5 r
< 1, 0 5 0 5 2n
Proof. Similar to that of the previous Corollary.
Now we have all the elements to reproduce for a function f E BP an equivalent result to Theorem 3.1 a) in [3J: Theorem 4.1 Let f : A --+ C be a n analytic function, 0 < p < 00. T h e n the function f belongs t o the class BP i f and only i f pgl : T --+ R defined by p g l ( e i e ) :=
sup a=reie,O
1,
I ~ ' ( . > I P ( ~- 1 z 1 2 ) ~ - 2 ( 1 -
l$a(Z>I2)dxdy (14)
i s bounded. Besides, pgl i s a well defined bounded f u n c t i o n o n T and its corresponding Dirichlet solution pG1 : A --+ C i s the least harmonic majorant of the family of harmonic functions { P G ~ 0} < , R < 1.
Consider now Theorem 1.3. In particular it follows from the proof of this theorem (see Theorem 1 in [4]) that for every a E A and 0 < R < 1, there exists a constant C > 0 such that for any fixed q with 0 < q < 1,
Then pgR(eie) 5 C ( , h R ( e i e ) ) % .
So we have also that pgl(eie) 5 C(,hl(eie))%.
Consider now the following well-known inequality: If r > 1 and a , b > 0 , then ab 5 ar
+ b",
where r' = 5 .
(15)
If we take now the Dirichlet solutions pG1 and ,HI associated to p g l and qhl, respectively, and if z = reie then
B P , QP Spaces and Harmonic Majorants
135
I 2P*C(qH1(Z)+ 11, where we have applied (15) with So we arrive in the next result.
T
= 2 / p , a = (,hl(eie>)gand b = 1.
Theorem 4.2 Let 0 < p < 2, 0 < q < 1, and let f : A + C be an analytic function. Let besides qH1, pG1 be the associated functions through the corresponding seminorms in BP and Q p . Thus iff E Q p , then
+
L ~ " C [ , H ~ ( Z 1)1.
PGI(%)
(16)
From Theorem 3.1 in [3] it follows that if qH1 is bounded, inequality (16) implies Theorem 1 in [4]. Thus we can say that Theorem 4.2 and Theorem 1.3 i) are equivalent.
5 The Analytic Completion of a Harmonic Majorant As we already know for every p _> 0, i f f E Q p then f is in the Bloch space. In section 3 we have obtained the harmonic majorant H1 that characterizes a function f in the Bloch space. Consider the harmonic conjugate of H I , which we will denote by B1. Then G := HI + ifi1 results an analytic function in A . The question is to which of the weighted spaces of functions quoted in Section 1 does G belong. In this section we will prove that if f is in the Bloch space, G will be in the Bloch space too. Thus we show that a new line of problem arises from this new way of characterizing weighted spaces of functions in terms of harmonic majorants. Let u : A + R be a harmonic function and p > 0. Define
(&l,lu(reie)lPdO) ', n
M p ( r , u ):=
0
< r < 1.
The function u is said to be of class h P if MP(r,u ) is bounded for all 0 < T < 1. We observe that if u is bounded then u E hP for all p > 0. An analytic function belongs to the Hardy space H p , 0 < p , if its real and imaginary parts belong to h p .
Theorem 5.1 [12] Let u : A -+ R be a harmonic function. a) (M. Riesz) If u E h P for some p E (1, ca) then its harmonic conjugate ii is also of class
hP.
136
E. Ramirez de Arellano, L. F. Reskndis O., and L. M. Tovar S.
b) (Kolmogorov) If u E h l , then its conjugate ii E h p for all p < 1. c ) (Zygmund) If u E hP f o r some p E ( 1 , ~then ) its harmonic conjugate 21 is also of class h’.
We get immediately Theorem 5.2 Let h : A ---f I% be a bounded continuous function and let H I : A -+ R be the harmonic function defined f o r z = reie as
P1,T(O- t ) sup h(reit)dt . o
Then H I , H 1 E hP f o r all p > 0. In particular, f o r each p function H I iR1 E HP, where HP is the Hardy space.
+
> 0 , the holomorphic
The following theorem of Zygmund gives us more information about the analytic function H I iRl.
+
Theorem 5.3 [12] Let u : A 4 R be a harmonic function and 13 its conjugate. Iff( z ) = u ( z ) i i i ( z ) and z = reie then
+
r)
where u * ( z ) is the Poisson integral of IuI. Theorem 5.4 Let h : A -+ R be a bounded continuous function and let H1 : A + R be defined by (17). Then the holomorphic function H I ifil belongs t o the Bloch space.
+
Corollary 5.1 Let f : A + C be an analytic function belonging t o the Bloch space and let H1 and G = Hl+iH1 be the harmonic majorant and the analytic completion of H1 respectively. Then G belongs t o the Bloch space too. Open Problem: If f belongs to Q p (or BP) and H I is the corresponding harmonic majorant then G = H1 t i f i 1 belongs to Q p (or B P ) too?
Acknowledgements E. Ramirez de Arellano was partially supported by CONACYT 327263. L. F. Resendis 0. was partially supported by CONACYT and was Visiting Scholar at Michigan University. L. M. Tovar S. was partially supported by CONACYT, Becario de COFAA-IPN.
BP,Q P Spaces and Harmonic Majorants
137
References 1. R. Aulaskari, O n Q p functions, Complex Analysis and Related Topics, Operator Theory, Advances and Applications, Vol 114, Birkhauser, Basel 2000. 2. R. Aulaskari and P. Lappan, Criteria for a n analytic function to be Bloch and a harmonic or meromorphic function to be normal, Complex analysis and its applications, Pitman Research Notes in Mathematics, 305, Longman Sci.& Tech., Harlow (1994), 136-146. 3. R. Aulaskari, L. F. Reskndis 0. and L.M. Tovar S.,Q, spaces and Harmonic Majorants, Complex Variables Vol. 49, No. 4 (2004), 241-256. 4. R. Aulaskari, L. M. Tovar,On the function spaces BP and Q p , Bull. Hong Kong, Math. SOC.Vol. 1, (1997), 203-208. 5. R. Aulaskari, J. Xiao and R. Zhao O n subspaces and subsets of B M O A and UBC, Analysis Vol. 15 (1995), 101-121. 6. S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Graduate Texts in Mathematics 137, Springer Verlag, Berlin 2001, Second Edition. 7. S. Kobayashi, Range sets and B M O norms of analytic functions, Can. J. Math. VO~.XXXVI, No.4. (1984) 747-755. 8. P. Lappan, A survey of Q p spaces, Complex Analysis and Related Topics, Operator Theory, Adv. Appli., Vol 114, Birkhauser, Basel, 2000, 147-154. 9. K. Stroethoff, Besov-type characterizations for the Bloch space, Bull. Austral. Math. SOC.Vol. 39 (1989), 405-420. 10. J. Xiao, Carleson measuresJ atomic decomposition and free interpolation f r o m Bloch space, Ann. Acad. Sci. Fenn. Ser. A1 Math. 19 (1994), 35-46. 11. K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, Inc. 1990. New York . 12. A. Zygmund, Trigonometric Series. Cambridge Univ. Press, London and New York, 1959.
The ( t ,a)-Lattice and Decomposition Theory for Function Spaces Irfan U1-hag' and Zhijian Wu2 Department of Mathematics Prairie View A&M University Prairie View, TX 77446
USA i r f anxlhaq(0pvamu.edu Department of Mathematics The University of Alabama Tuscaloosa, AL 35487
USA zwu(0bama.ua. edu
Summary. For Q > 1 and 0 < t < 1, define B,(z,t) = {w : \ z - w ( < t(1- I z I ) " } . We use the lattice, called (t,a)-lattice, constructed by the above discs to obtain decomposition theorems for CJ,, spaces where -1 < p < 00, 1 5 q < 00, and 0 5 y < 00. Certain values of p , q and y give some well-known function spaces such as Bergman spaces, Hardy space, Dirichlet (Besov) spaces, BMOA space, Morrey space, Q p and Bloch spaces.
1 Introduction Decomposition of functions has long been studied. It is a useful tool in studying function spaces and their properties. The decomposition results for functions in holomorphic spaces are also used in studying operators such as Hankel and Toeplitz operators, and approximation by rational functions. For further details we refer the reader to the references at the end of this article. The main purpose of this article is to establish the decomposition results for C& spaces (defined later in this section) using the theory developed in [8]. In particular we use the lattice constructed in 181 for the unit disc D in the complex plane C. For this let a 1 1 and 0 < t < 1 and define the disc El,(%; t ) = {w E
D : 1% - W J< t(l -
%
E D.
By a @,a)-latticewe mean a family of points { z j } y in D such that the collection of discs { B a ( z j , t / 3 ) } Tforms an open cover for ID,and
140
Irfan U1-haq and Zhijian Wu
B a ( z j ,t/16) n B a ( z k t/16) , For a
= 0 for j
# k.
> 1, the existence of such a lattice is proved in [8]. When
a = 1, a
( t ,1)-lattice is the usual lattice related to the Bergman metric [2]. One can also construct a partition
{Dj}of D such that
B a ( z j ,t/l6) c Djc B a ( z j , t / 3 ) and that {Dj} is a disjoint cover for D. For a given arc I on aD, the boundary of based on I , i.e.,
D,let S ( I ) be the Carleson box
S ( I ) = {reie : 1 - 1 1 5 r < l,eie E I } , where 111 is the normalized arclength of I . Let y E [0,00). A non-negative measure p on Carleson measure provided that
D is
called a bounded y-
P ( S ( I ) )= O(lIl')
for all arcs I on aD.When y = 1, p is just the usual Carleson measure. The measure p is called a compact y-Carleson measure if P ( S ( I ) )= o(III'>
for all arcs I on aD. Denote by dA the normalized area measure on D. For 0 < q < 00, -1 < p < 00 and 0 I y < 00, let be the space of analytic functions F in D satisfying IF'(z)Iq(l - I ~ l ~ ) ~ d 5 A (CIIIy, z) L I ,
for any arc I c 8D.Denoted by l l F / l p p ,is the q-th root of the best constant in the above inequality. Moreover we denote F E VC:,p if F E C:,p and
The space C:,p was first studied by R. Zhao in [12] where he employed a different notation ( C i P= F ( q , p - y,y) -the notation in [12]). For certain values of p , q and y, C:,p corresponds to some well known function spaces. Here we list some of them (see [lo] and [12]). 0
For y > 1 , C9y,p = F - B l o c h space and VC:,p = little y - B l o c h space. For 0 < y 5 1, C2,,y= Qr space and VC2,y= Qr,o space (see, for example, [l],[lo] and [12]). In particular Cj,l = &I = B M O A space and VC;,, = QI,= ~ VMOA space.
The ( t ,a)-Lattice and Decomposition Theory for Function Spaces
141
0
For 0 < y 5 1, C& = analytic Morrey space C2iY and VC& = littleanalytic Morrey space .@? (see [lo]). C& = H 2 ( Hardy space ). Co,q= AQ ( Bergman space ). C8,, = DQ ( Dirichlet space ).
0
Ci,q-2= Besov Space, for 1
0
0 0
< q < 00.
Space Ci,,pis a Banach space for q
2 1 with the norm
I t is easy to see that for p + 2 < y, C:,p is the space of constant functions, therefore the interesting range of y for C&, is 0 5 y I p 2. The main results of this paper are
+
Theorem 1.1 Suppose p > -1, q >_ 1, a b>
p
2 1, 0 5 y 5 p + 2
and
+ max(1, Q - 1)
There exists a to > 0 such that for any ( t ,a)-lattice { z j } in D with 0 < t < to, we have
(a) If F E C;,p, then
F ( z )=
cxj
b-l+2n-
(1 - M2) b (1 - qz)
and
IXjlQ 5 ClIlr for any arc I , then the function F ,
(b) If { X j } satisfies ZjES(I)
defined above, is in Ci,p and
Theorem 1.2 Suppose p > -1, q 2 1, b>
(Y
2 1, 0 5 y 5 p
+ 2 and
p+max(l,q- 1) 4
There exists a to > 0 such that for any ( t ,&)-lattice { z j ) in D with 0 < t < to, we have
Irfan U1-haq and Zhijian Wu
142
(a) If F E VC,',,, then
F(2)=
cxj
(1 - 1%12)
b-l+2a-
(1 - q z )b
j
and zj
(b) If { X j } satisfies
C
IXj1q
ES(Z)
= o(III')
f o r any arc I , then the function F ,
ZjES(I)
defined above, is in VC,',p. In the case a = 1, the above theorems give us the decomposition results for BMOA, VMOA, Q p , Qp,o,Bloch and little Bloch spaces in the Bergman metric ~ 5 ,1[GI , ~91).
Theorem 1.3 Suppose p > -1, q 2 1, 0 5 y 5 p S >
max(q - 1 , 1 , q
+ 2 and
+ y - 2) - p
4
Then
c:,~
c:,,~+~;
(a) F E e F(") E (b) F E VC,',p e F(")E VC;,,+,.
Here F(")denotes the usual s-th order derivative if s is a positive integer. We will define later the fractional derivative F(") for non-integer s. Theorem 1.3 reveals that the parameters p and q in the space C:,p (or VC,',,) are related by derivative. Therefore there are only two independent parameters in the scale space C:,p (or VC,',,,). In section 2 we prove some preliminary lemmas and in section 3 we give the proofs of the main results. First we will prove Theorem 1.3 as we need this for the proofs of Theorem 1.1 and Theorem 1.2. The letter "C" denotes the positive constant which may vary at each occurrence but is independent of the essential variables and quantities. The notation =: means comparable, with the constants independent of the functions and quantities involved. For q 2 1, q' denotes the conjugate of q.
2 Some Preliminary Results The following lemma is standard (see, for example, [13]).
Lemma 2.1 For c > 1 and b > 0 =: (1 - I z ] 2 ) 1 - u .
The (I!,a)-Lattice and Decomposition Theory for hnction Spaces
143
For fixed b > 0, we define the linear operator Tp by
Theorem 2.2 Suppose q 1 1, p > -1, 0 I y 5 p
P>
max(q - 1,1,q 4
+ 2,
+ y - 2) - p , b > p + max(1, q - 1) 4
and 1c, is a measurable function o n D. (a) If
Ss(,) l$(.z)lq(l- I ~ l ~ ) ~ d A5 (CIII7, z ) f o r any I c am, then
holds f o r any I c XD. (b) If Js(r)11c,(.z)lq(l - ( ~ ( ~ ) ~ d=A o(lIly), ( z ) f o r any I
holds f o r any I
c BD, then
c BD.
Remark: 0
0
0
The parameter can be set to 1 if p 2 0, or 2 if p < 0. For p = 1 and q = 2, the result is proved in [7]. For /3 = 1 and q = 2, the result is proved in Ill] and for q = 2, the result is proved in 191. The following proof has its root in [9]. A similar theorem to Theorem 2.2 is stated in 191.
Proof of Theorem 2.2 For part (a),it is sufficient to show that for any arc I c aD I%W)lQ(1 - lwl )q(P-l)+PdA(w) 5 C(IIy.
s,,)
For any non-negative integer n 5 Zogzh, let 2"I be the arc on 8ilD with the same center as I and the length 2"111. We have the following estimate:
144
= El
Irfan U1-haq and Zhijian Wu
+ E2.
To estimate El, we first show that the linear operator B : Lq(D) defined by
-+
Lq(D)
is a bounded operator. Here
K (w ,2 ) =
('-lz1*)6-'-P/q
( l - l w / y l + p / q
I1-7w Ib+P In fact, if q = 1 then by Lemma 2.1, we have
If q > 1, consider g(z) = (1 - I z 1 2 ) s . It is easy to verify that the estimates
J, K ( w ,z)gq/(w)dA(w) 5 CgQ/(z) hold for all z , w E D. By Schur's theorem, we know that B is a bounded operator. Let h ( 4 = IWP - I ~ 1 2 ) % s ( 2 r ) ( ~ ) v z E D,
The ( t ,a)-Lattice and Decomposition Theory for Function Spaces
145
we clearly have h ( z ) E Lq(D) and
Therefore we can estimate El by
= 24-111w411;q
6 cllhll;'l <
To estimate the inequality
E2,
CJIJY.
we first note that (see, for example, [4,p. 2391) for n 2 0 11 - zwl 2 C(2nIII)
holds if w E S ( I ) , z E S(2"+'I)\S(2"1>. Direct computation yields also that for any fixed a > 1,we have .f,s(2nI)
(1 - I ~ l ~ ) " - ~ d A I( wC)(2n111)a.
Hence, rewriting the set D\S(21) as the disjoint union we can estimate E2 by
Since the measure \$(z)14 (1 - 1 we have the following: If q = 1, then
-
S(2n+11)\S(2nI),
~ 1 ~ ) dA(z) " is a bounded y-Carleson measure,
where the second last inequality holds because of b If q > 1, then
Therefore, for q
2
> 1+ p .
1, we can continue the estimate of
E2
as follows (note
p - l + - > o 9)
This proves part (a). For part (b) we only need to modify the proof in part (a). First note that, by assumption, for any E > 0 there exists 6 > 0 such that
IWl"1 - 142)PdA(4< 411Y L I )
holds when 111 < 6. We have therefore
The ( t ,0)-Lattice and Decomposition Theory for Function Spaces
Let N be the largest integer satisfying N I log, when n I N . Hence we have Js(2,,+11)
l+(z)lq(l
-1
147
&.We have that 2n+1 111 5 6
~1~)I " E (2n+1111)Y
for all
IN.
Since a compact y-Carleson measure is a bounded y-Carleson, we have
1+(41"1 - I4"" I c (2n+1111)y
JS(2"+'1)
for all n.
Thus, from the previous estimate for E,, if we let
s
. . . ~ A ( z=)
5(2"+'1)
5(2"+'1)
then
The last inequality follows from the fact that N
+ 1 2 log, &.In summary,
we have for /I1< $
I El +E2 q(P-l)+P+2-7
< ClIlY
-
(€+(?)
).
148
Irfan U1-haq and Zhijian Wu 2
dP--l)+P
This is enough to conclude that the measure ITp$(z)lq(l - IzI ) is a compact y-Carleson measure.
dA(4
For a fixed b > 0, let
Lemma 2.3 Suppose a 2 1. There exists C > 0 such that f o r any zo E and f o r any z E B a ( z o , t ) where 0 < t < 1, we have IKw(z) - Kw(.%)l i Ct ( 1 - Izol)-l
Ircu(z)I
Proof: For any w E D and for any z E B, (zo,t ) ,we have
Now
l1
1- E z - l-E&
l=lL 1 o w ( 2 - zo)
Also
From the above inequalities, we also have the following
and
w E D.
D
The ( t ,a)-Lattice and Decomposition Theory for Function Spaces
149
Now
< 2b t (1 - JZoJ)a-l -
.
Also
ds
< -
C t (1 - / z 0 y - l .
Therefore from above we have
Lemma 2.4 Let 0 < t < 115. There exists C > 0 such that for a n y analytic function F on D, the following estimate holds for all j .
Proof By the change of variable z = 4(w - 20) (1 - Jzol)" ' it is enough to show that
150
Irfan U1-haq and Zhijian Wu
By the reproducing formula
we have
Rewriting F ( z ) - F ( 0 ) as
JizF'(sz)ds, we obtain
This is enough.
Lemma 2.5 Let 0 < t < 1/4, q 2 1 and { z j } be a (t,a)-lattice. There exists a positive integer r = O ( t - 2 ) such that each point of D lies an at most r discs Ba(zj,1/4). Furthermore, if b > 0 and F is analytic o n D,then
Proof Let z E ID be fixed and let J , be the set of all j such that z lies in B,(zj, 1/4), i.e., JZ { j : IZ - z j l < (1 - Izjl)" /4} .
The (t,a)-Lattice and Decomposition Theory for Function Spaces
Also, let
JzkJ= minjEj, First we claim that
151
(Izjl).
In fact, for any w E B,(zj, t/16), we have
+
Since { B , ( z j ,t/16)}jE J , is a set of disjoint discs in B, ( z k ,t/16 1/2), we have that the sum of the areas of the discs B, ( z j , t/16), j E J, is less than the area of the disc B, ( z k ,t/16 1/2). This gives the estimate
+
Since z j = z k
+ z j - z + z - z k and therefore
Hence
Now we can continue the above estimate by
which implies
Let r = supzEDI J,I. Now it is clear that
The proof is completed.
152
Irfan U1-haq and Zhijian Wu
3 Proof of the Main Results Let b > 0 and s be any non-negative number. For any analytic function F on is defined by
D,the s-derivative of F
Here r(.)is the usual Gamma function and [s] denotes the greatest integer function of s. It is easy to check that
Therefore F(“) is the usual s-th order derivative if s is a positive integer. Moreover,
F(”+1) =
(F‘s’)’.
Proof of Theorem 1.3 Suppose F E Cz,p (or VCz,p).Applying Theorem 2.2 to the representation
F(’)(z) =
+
s
1 8 1 ,
-
r(b r(b) D (1 -;iliz)b+S
(1 - I
w ~ ~ ) ~ dA(w), -~
we obtain the desired result. Suppose F E C:qs+p(or VC;,,+,). If 0 < s < 3, consider the representation r(b+1) J T D ~ - [ ~ ] F ( ~ ) b+s-2 dA(z). F‘(z) = (1 - lWI2) r(b s - 1) D (1 - ,Z)b+l
+
The desired result again follows from Theorem 2.2. If s >_ 3, we only need to show that there is a positive integer k so that This can be done by repeating the 0 < s - k < 3 and F(s-k) E Cy Q,P ( s - k ) +P * following process and applying Theorem 2.2 to each step,
The proof is complete. For a (t,a)-lattice { z j } ? , recall that the family of discs {Ba(zj,t/3)}?
The ( t ,a)-Lattice and Decomposition Theory for Function Spaces
153
Continuing in this way, we get a partition of D. Clearly Dj has the following properties zj
E B,(zj,t/16)
c Dj c B,(zj,t/3)
and
lDjl x t2(1 - I ~ j l ) ~ ~ .
Proof of Theorem 1.1 For the ( t ,a)-lattice { z j } , without loss of generality, we assume 0 < t < 1/5 and lzjl > 0 for all j. Let F E C4y,,p. By the reproducing formula we have (for b > 0)
Recall that for the (t,a)-lattice { z j } , there is a partition { D j } of D which satisfies the conditions described in Section 1. Therefore F’(w) can be represented as
Therefore F ( w ) can be approximated by
where lDjl = JD3 dA is the normalized area of Dj. We now consider the error F - A ( F ) of this approximation. Note that
F’(w) - A(F)’(w)= b): j
/
Dj
2 b-1
F’(z) d A ( z ) (’ (l-zw)b+l
Therefore
By Lemma 2.3, we have for every z j and z E Dj c B, ( z j ,t )
Irfan U1-haq and Zhijian Wu
154
AlsobyLemma2.3andLemma2.4 (since Dj have
<
C t3
/
c B a ( z j , t )c B a ( z j ,1/4)),we
IF’(z)IIE(,(z)I d A ( z ) .
Ba(z,,1/4)
Therefore by Lemma 2.5, we obtain (note that F’(z)K,(z) is analytic on
D)
In summary we have
Applying Theorem 2.2(a) to the above estimate, we get
IIF - A(F>llc;,, I c t llFllC;,p v
F
q p .
Note that we apply Theorem 2.2(a) here with /3 = 1 which is fine if p 2 0. In case -1 < p < 0, we can work with F’ which is in C;q+p by Corollary 1.3. In this situation we can pick /3 = 2. Now we can choose a small to 5 1/5 so that Ct in the above estimate is less than 1/2 when 0 < t < to. Therefore the operator A is invertible and
A-’ =
C (Id
- A)”
n=O
is bounded (by 1/(1 - Ct,) 5 2 ) , where I d is the identity operator on C4y,p.
The ( t ,a)-Lattice and Decomposition Theory for Function Spaces
155
We have constructed an approximation operator A with bounded inverse. Therefore for any F E CSy,”, we can write
F(z)=AA-l(F)(z)
where
x , - ( d - ’ ( F ) ’ )( z j ) lDjl (1 -
3 -
zj
It remains to show that the measure Cj (XjlQ6z, is a bounded y-Carleson measure. In fact, by the mean value theorem, we have for a analytic function 9 on Ba ( Z j , t/16)
Therefore, we obtain
5 Ct~-211A-1(F)llC;,,I~I~
I c tq- IIF IIc;,,I I I-/. This completes the proof of (a). For part (b), it suffices to show that the measure IF’(z)lQ(l- I ~ 1 ~ ) ” d A ( z ) is a bounded y-Carleson measure. By the given formula we have F’(w) = b
1
AjTj
j
(1 - 1%12>
b-1+2cr-F
(1 - ZjW)b+l
We can find a positive constant Cj such that
156
Irfan U1-haq and Zhijian Wu
Therefore we have
By Theorem 2.2(a), we only need to show that the measure
is a bounded y-Carleson measure. Since {Cj} is bounded and B, ( z j , t/16) is a set of disjoint discs, we have
c
5C
I& IQ.
j : s (I)nB, (z j ,t/ 16)#0
Without loss of generality, we assume 111 5 1/4. We claim that if S ( I ) n &(zj,t/16) # 4 then z j E S(21). Therefore we can continue the above estimate as follows
Ic
c
(Xj(q
ZjES(2I)
5 CIIIY. The last inequality holds because the measure C j lXjlQSzj is a bounded y-Carleson measure. The proof of the theorem will be completed once we establish our claim. Note that it is enough to show that 1 - lzjl 5 2111
and
5 E 2111 .
Assume z E S(1) n B,(zj, t/16), then
Iz
- zjl
I k(l - Izjl)" , 1 - IzI 5 111 and
fi E I
.
The ( t ,a)-Lattice and Decomposition Theory for Function Spaces
157
Therefore
Hence
t 1 - IZj( - -(I 16
-
IZjI)"
5 I - (Z( 5 (I(
Also
This is enough to conclude that
a
E 21. H
Theorem 1.2 can be proved by modifying the proof of Theorem 1.1.
Acknowledgement Zhiian Wu's work was supported in part by NSF DMS 0200587.
References 1. R. Aulaskari, J. Xiao, R. Zhao, O n subspaces and subsets of B M O A and UBC, Analysis, (2) 15, 1995, 101-121. 2 . . R. R. Coifman and R. Rochberg, Representation theorems f o r holomorphic and harmonic functions in L p , Asthrisque, 1980. 3. P.L. Duren, Theory of H P Spaces, Academic Press, New York, 1970. 4. J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
158
Irfan U1-haq and Zhijian Wu
5. R. Rochberg, Decomposition Theorems f o r Bergman Spaces and their applications, Operator and Function Theory, S.C. Power Ed., Reidel, Dordrecht 1985, 225-278. 6. R. Rochberg and Stephen Semmes, A decomposition theorem f o r BMO and applications, J. Funct. Anal., 167, 1986, 228-263. 7. R. Rochberg, Z. Wu, A New Characterization of Dirichlet type spaces and A p plications, Illinois J. Math. (1)37, 1993, 101-122. 8. I. U1-haq, Z. Wu BMO; spaces and their properties, submitted 9. Z. Wu, C. Xie, Decomposition Theorem f o r Q p spaces, Ark. Mat. 40, 2003, 383401. 10. Z. Wu, C. Xie Q p Space and Morrey Spaces, J. Funct. Anal. 201, 2003, 282-297. 11. J. Xiao, Holomorphic Q classes, Lecture Notes in Math. 1767, Springer-Verlag, Berlin-Heidelberg, 2001. 12. R. Zhao O n a general fa m ily of fu n ctio n spaces, Ann. Acad. Sci. Fenn. Math. Dissertations 105(1996). 13. K. Zhu, Operator Theory in Function Spaces , Marcel Dekker, Inc., New York and Basel. 1990.
On a Generalization of the N. A. Davydov Theorem Oleg F. Gerus’ and Michael Shapiro2* Zhitomir State Pedagogical University Velyka Berdychivska str. 40 Zhitomir 10008 Ukraine ofg0com.zt.ua Departamento de Matemgticas ESFM del IPN Mexico City 07338 Mexico shapiro0esfm.ipn.mx Summary. We consider a Cauchy-type integral for a class of holomorphic-like functions acting from C to C2 whose components satisfy the Helmholtz, not the Laplace, equation. We determine sufficient conditions for its boundary values to exist uniformly and prove Sokhotsky-Plemelj-type formulae for them.
1 Introduction In classical complex analysis, the Cauchy kernel has the following fundamental properties allowing us to use the Cauchy-type integral for constructing holomorphic function theory: 0 0
0
it is holomorphic, reproducing property (any holomorphic function is determined by its restriction onto the boundary of a domain), universal property (the Cauchy integral representation holds for a wide class of domains of any shape, with the boundary smooth enough).
A Cauchy kernel for functions acting from R2 to C4 have been constructed in [7] (see also [6]). It has all the above-cited properties (with “ahyperholomorphy” instead of “holomorphy”). The parameter a defines the wave number, a2,of the Helmholtz operator AWz a2, thus the a-hyperholomorphic function theory plays the same role for the Helmholtz equation as the usual holomorphic function theory plays for the Laplace equation.
+
’The work was partially supported by CONACYT projects as well as by Insituto Politkcnico Nacional in the framework of COFAA and CGPI programs.
160
Oleg F. Gerus and Michael Shapiro
In the paper [3]the Sokhotsky-Plemelj formulae and the Plemelj-Privalovtype theorem for boundary values of the Cauchy-type integral with a-hyperholomorphic Cauchy kernel for Holder functions on the closed piecewise Liapunov curves have been proved. However, in classical analysis more general formulae have been obtained by N. A. Davydov in [5].They are valid for boundary values of the Cauchytype integral on any closed rectifiable Jordan curve. The formulae generalize the well-known formulae of Sokhotsky-Plemelj, which hold, in their classical form, for piece-wise smooth curves only, the latter being a proper sub-set of all rectifiable curves. In the paper [l](see also [2]) we have proved an analogous theorem in the theory of a-hyperholomorphic functions. We consider the functions from R2 to the algebra of complex quaternions . . W(@) := { a = C k3= O a k i k : ak E C, 2: = -1, 2122 = -2221 = is, i 2 i 3 = -2322
. .
= 21, i 3 i l = -2123
= 22).
W(@) is a complex, non-commutative,
associative algebra with zero divisors. In the case of arc E R we have the algebra of real quaternions W(R) which has no zero divisors. Let r C R2 denote a closed rectifiable Jordan curve and let O+ and Oare the interior and exterior domains respectively with the common boundary r. Let L? denote either O+ or O-. Reasoning analogously to [7] and [6],let us imbed R2 into the set of real quaternions so that z := z yi3 E R2 and define the Cauchy-type integral in the following way:
+
where K , is a quaternionic function playing the role of the Cauchy kernel in this theory. We shall write @: [f]and @; [f]for the respective restrictions of @, [f]onto O+ and W . In this paper, we establish sufficient conditions for boundary values of the above-mentioned generalization of the Cauchy-type integral to exist and obtain the formulae for them.
2 Quaternionic a-hyperholomorphic functions The choice of the Cauchy kernel K , as well as the definition of a-hyperholomorphic functions is based on the Helmholtz operator
A,z := A,z
+ “‘M,
where Q E @, ,‘Mf := a2f(see [7], [S]). It is factorizable:
A,z
= D, 0 (-Da),
On a Generalization of the N. A. Davydov Theorem
where
. a
D , := z2-
ax
a
- il-
dy
161
+ i 3 ,M.
The operator D, serves as an analog of the Cauchy-Riemann operator classical complex analysis.
Definition 1 ([7]) Let 0 be a domain in R2.A function f : 0 called a-hyperholomorphic if D , f = 0 in 0.
--+
3 from
W(@) is
Similarly to the classical case, we define the quaternionic Cauchy kernel K , as a fundamental solution of the operator D , (i.e. Da[K,](z) = b ( z ) , where S(z) is the Dirac delta-function). Due to the factorization (2), the Cauchy kernel K , can be calculated by the formula
K,(Z) := -D,[e,](Z), where Q,(z) is a fundamental solution of the operator A,2, which can be expressed in the following form:
1 1 -No(a(z()- - J o ( a ( Z I ) ( C - lad, 4 2n where C is the Euler constant and for an integer n, J, and N , are the Bessel &(Z)
=
and Neumann functions of order n respectively. Later on we will need the following their properties (see [4]): d --No@)
at
= --N1(t),
tJ&) = 2 J l ( t ) - t J o ( t ) . Using the equalities (3) and (4),we obtain K,(z) := - Da[8,](z) =
where
The next theorem is a quaternionic generalization of the N. A. Davydov theorem (see [5]).It is equivalent t o Theorem 3.2 of the paper [ l ](see also Theorem 3.2 of the paper [2], where a more general case a E H(@) have been considered) :
162
Oleg F. Gems and Michael Shapiro
Theorem 1 Let a E @, r be a closed rectifiable Jordan curve, f : I' -+ W(@) be a continuous function, and let the integral 6-0 lim
1
tE
I L ( C - t)l ldCl If([) - f ( t ) l , *
*
r,
r\rt,a where rt,6:= { C E Then the integral
r:
- tl
6 S } , exists uniformly with respect to t
E
r.
r
exists; moreover, the functions @,[f] extend continuously onto r, and the following analogues of the Sokhotsky-Plemelj formulas hold:
@L[f],
@ : [ f ] (= t ) F,[f](t) - (ICY,r(t)+ 1) il f ( t ) ,
@,[f](t) = F,[fl(t) - L,r(t)iif ( t ) , where @ $ [ f ] ( t:= ) lim @,[f](z), and
tE
tE
r,
r,
(8) (9)
f2f3z-4
ICY,r(t) := -a
/I
R-t
ITCY(<- t )i 3 dJdq.
3 C2-valued a-hyperholomorphic functions in C
zkzO
In the particular case f = 3 f k zk to be a real-quaternion-valued function (i.e. fk are real-valued functions) let us represent the function f in the form f = 91 + g 2 i 1 , where g1 := fo f 3 i 3 , g2 := fl f 2 i 3 , and let a be a real number. Thus the function f becomes a @'-valued function defined in C. We have
+
where
a - is-a a := -
+
and the bar denotes complex conjugation. Therefore ay the condition of a-hyperholomorphy in a domain 0 c C is equivalent t o the system of equalities:
ax
O n a Generalization of the N. A. Davydov Theorem
163
Remark The case of complex a in the system (10) requires a separate study using the methods from the paper ([2]). This will be done elsewhere. 0-hyperholomorphy of the function f in 0 is equivalent to anti-holomorphy of the complex functions 91, g2 in 0 , i.e.
D o [ f ] ( z= ) 0 (V2 E 0 ) e
{
%l(.) Q2(Z>
=0 ' = 0,
(VZ
E 0).
By formula (1) the Cauchy-type integral Qio is thus represented in the form:
where
is the classical Cauchy-type integral. So Theorem 1 in this case is reduced to the N. A. Davydov theorem for the functions 91, g2. Let C2 be a complex linear space with the basis el,e2. Interpreting a quaternion-valued function f = g1 g2il as a C2-valued function g = g l e l + g2e2 in the domain 0 c @, we realize the condition (10) as a definition of a-hyperholomorphic function g : 0 H C2:
+
where
164
Oleg F. Gerus and Michael Shapiro
r
I-+ CC2 we have the a-hyperholomorphic CauchyFor a function g : type integral @,[g] := @,,1[g]e1 @ , , 2 [ g ] e ~where , @,,k[g] := @,,k[f], k E {1,2}. On the basis of the equality (11) this formula can be rewritten in the traditional form:
+
@a[91(4:= / K d C
- 4 ( g ( 0d o
r with the matrix Cauchy kernel
where 2 denotes the operator of complex conjugation. Denote also by @: [g]and @: [g]the respective restrictions of I,[g]onto fl+ and 52-. And now we obtain from Theorem 1 the following analog of the N. A. Davydov theorem:
Theorem 2 Let a E R, CC 3 r be a closed rectifiable Jordan curve, g : I' CC2 be a continuous function, and let the integral lim
6+0
J
I l K Y ( C - t>II
*
-
119(S>- g(t>ll IdCI,
tE
H
r,
wt,&
exist uniformly with respect to t E
r. Then the integral (12)
lim F , [ g ] ( t ) := 6-0
r\rt.s exists; moreover, the functions @:[gJ extend continuously onto following formulae hold:
where, f o r k E { 1,a},
r
and the
On a Generalization of the N. A. Davydov Theorem
165
4 Proofs Proof of Theorem 1 follows from Theorem 3.2 of the paper [l]which contains the function K , + g , in the role of the Cauchy kernel, where
r Due to the continuity of the function k, in R2 and using the Green’s formula and the equalities (4)- (6) we get:
where
IQ,r(t):= -a//
ZQ(C- t ) i3 dJdq.
R+
i?,
Thus, the summand in the Cauchy kernel does not change the form of the formulas ( 8 ) , (9). Theorem 1 is proved. Proof of Theorem 2 goes over the cC2-form of the functions Fa, I,,r, @: in Theorem 1, taking into account the equalities (ll),(12) and the requirements a E W and f t o be real-quaternion-valued. And so we arrive at Theorem 2 as a reformulation of Theorem 1.
References Gerus, O.F, Shapiro, M.: On a Cauchy-type integral related to the Helmholtz operator in R2.Boletin de la Sociedad Matemiitica Mexicana 10 (2004), 63-82. Gerus, O.F, Shapiro, M.: On the boundary values of a quaternionic generalization of the Cauchy-type integral in R2 for rectifiable curves. Journal of Natural Geometry 24 (2003), 120-136. Gerus, O.F., Schneider, B., Shapiro, M.: On boundary properties of ahyperholomorphic functions in domains of R2 with the piece-wise Liapunov boundary. Progress in Analysis, v. 1, (Proceedings of 3rd International ISAAC Congress, Berlin, August 20-25, 2001, Eds: H. G. W. Begehr, R. P. Gilbert and M. W. Wong), World Scientific, 2003, 375-382.
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4. Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. 6th ed. San Diego, CA: Academic Press. xlvii. 5. Davydov, N.A.: The continuity of the Cauchy-type integral in a closed domain. Dokl. Akad. Nauk SSSR 64 (1949), 759-762 (Russian). 6. Kravchenko, V.V., Shapiro, M.V.: Integral representations for spatial models of mathematical physics. Addison Wesley Longman, Pitman Research Notes in Mathematics Series 351 (1996). 7. Shapiro, M., Tovar, L.M.: Two-dimensional Helmholtz operator and its hyperholomorlhic solutions. Journal of Natural Geometry 11 (1997), 77-100.
Dual Integral Equations Method for Some Mixed Boundary Value Problems Juri M. RapPoport* Russian Academy of Sciences Building 27, Apt.8 Vlasov Street Moscow 117335 Russian Federation jmrap0landau.ac.n
Summary. New applications of the modified KONTOROVITCH-LEBEDEV integral transforms for the solution of some problems of mathematical physics are given. An algorithm for numerical solution of some mixed boundary value problems for the HELMHOLTZ equation in wedge domains by means of dual integral equations method is developed.
1 Introduction The method of dual integral equations [1,4,10] is one of the effective approaches for the solution of boundary value problems of mathematical physics. The dual integral equations may be reduced to FREDHOLM integral equations or infinite systems of linear algebraic equations. The special emphasis in this paper is on dual integral equations with modified Bessel function of imaginary index KiT(z)in the kernel. The problems of computation of this function are considered in [2,3,7].But the theory and applications of this type of dual integral equations have elaborated quite insufficiently until now. The paper presents important results for the numerical solution of these types of problems.
2 Application of the Dual Integral Equations Method for Some Mixed Boundary Value Problems The modified KONTOROVITCH-LEBEDEV integral transforms [5,8]with kernels *Partial funding was provded by CRDF grant RM1-361.
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and where K,(z) is MACDONALD’S function, are of great importance in solving some problems of mathematical physics, in particular mixed boundary value problems for the HELMHOLTZ equation in wedge and cone domains. It is necessary to compute ReK1,2+ip(z) and ImK1/z+ip(z) [6,7,9] to use this transforms in practice. These functions also occur in solving some classes of dual integral equations with kernels which contain MACDONALD’S function of imaginary index Kip(z). Therefore the computation of the modified Bessel function K 1 1 ~ + ~ pof ( zthe ) second kind is considered in more detail [6,7,9]. We cite the definition of two pairs of direct and inverse modified KONTOROVITCH-LEBEDEV integral transforms [5]:
Sufficient conditions for the existence of these transforms and the validity of the inversion formulas are given. It is shown that the inversion formulas for the modified KONTOROVITCH-LEBEDEV integral transforms can be deduced from the inversion fortransforms and the corremulas of the “usual” KONTOROVITCH-LEBEDEV sponding theorem is proven. For the case of nonnegative finite functions with restricted variation the conditions of present theorem are reduced to one condition, which is then necessary and sufficient. The verification of the solution of singular integral equations of the form
where f(z) is a given function and X is a parameter, satisfying X < 1/n, is transforms. The proof given by means of modified KONTOROVITCH-LEBEDEV of the PARSEVAL equalities for these transforms is given. The problem of the evaluation of the modified KONTOROVITCHLEBEDEVtransforms is greatly simplified by means of their decomposition into the form of compositions of simpler integral transforms, in particular FOURIER
Dual Integral Equations
169
and LAPLACE transforms. The expression of the modified KONTOROVITCHLEBEDEVintegral transforms over the general MEYERintegral transforms of special index and argument is given. The dual integral equations with MACDONALD'S function of imaginary order Ki7(z)in the kernel of the following form were introduced by LEBEDEV and SKALSKAYA [4]
M ( r ) rtanh(ar)Ki,(kr)dr
= rg(r), 0
< T < a,
10
lm
M(r)&(kr)dT = f ( r ) , r
> a,
where g ( r ) and f ( r ) are given functions. They showed [4]that solutions of these equations may be determined in the form of single quadratures from solutions of second kind FREDHOLM integral equations with symmetric kernel containing MACDONALD'S function of complex order Kl,2+i7 (z)
$ ( t ) = h(t)-
IW
K(s,t)$(s)ds, a
I t < 00,
0
function of complex where Re K 1 / 2 + i 7 ( ~ is) the real part of MACDONALD'S order 1/2 ir. In the case g(r) = 0
+
h(t)=
fiekt d IT
dt
1
O0
e-"f(r) dr7
The numerical solution is presented. Economical methods for the evaluation of kernels of the integral equations based on GAUSSquadrature formulas on LAGUERRE polynomial knots are proposed. Preliminary transformation of integrals and extraction of the singularity in the integrand are used to increase the accuracy and speed of algorithms. The cases of dual integral equations admitting complete analytical solution are considered. Examples demonstrate the efficiency of this approach to the numerical solution of mixed boundary value problems of elasticity and combustion in the wedge domains [l]. The application of the KONTOROVITCH-LEBEDEV integral transforms and dual integral equations to the solution of the mixed boundary value problems are considered. The diffusion and elastic problems reduce to the solution of the proper mixed boundary value problems for the HELMHOLTZ equation. Mixed boundary value problems for the HELMHOLTZ equation [4] Au - k2u = 0
(1)
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Juri M. Rappoport
arise in some fields of mathematical physics. The solution of this type of problem in wedge domains is determined by the following way in the form of the KONTOROVITCH-LEBEDEV integral
where M ( T ) is the solution of a dual integral equation. It is shown that the solution of the above-mentioned problem for the HELMHOLTZ equation are present in the form of a single quadrature from the integral equation type. The dimension of the problem solution of FREDHOLM is lowered one unit by this approach, which is the essential advantage of this method. Examples permitting the complete analytic solution of the problem are given. The numerical solution of the mixed boundary value problems and the dual integral equations obtained is carried out. It consists of numerical solution integral equation of the second kind with symmetric kernels of FREDHOLM and the subsequent taking of quadratures from their solution. Estimation of error is given. Test calculations results give the precision for the solution to 6-7 digits after the decimal point. The examples considered demonstrate the efficiency of the dual integral equations method in the solution of the mixed boundary value problems for the HELMHOLTZ equation in wedge domains. We use the following notations here and in what follows: r,cp - polar coordinates of the point; a - angle of the sectorial domain; u - desired function; - normal to the boundary. The numerical solution of some boundary value problems for the equation of the form (1) in arbitrary sectorial domains is considered in our work under the assumption that the function ulr is known on part of the boundary and the normal derivative is known on the remainder of the boundary. The KONTOROVITCH-LEBEDEV integral transforms [4] and dual integral equations method [4,10] are used for finding the solution. Let's consider the symmetric case to simplify the calculations ( AU - k2u = 0,
IU I , - , ~ u\,O
- restricted, - restricted.
The solution of (2) is determined in terms of KONTOROVITCH-LEBEDEV integral transforms [4]
where M ( T ) is the solution of the dual integral equation
Dual Integral Equations
M(-r)-rtanh(a-r)Ki,(kr)d-r
=rg(T), 0
171
< T < a,
M(7)KiT(kT)dT = f(r), T > a,
(4)
where g(r) and f ( r ) are given functions and K,(z) is the modified BESSEL function (MACDONALD function) of imaginary order. The dimension of the problem is lowered one unit by this approach as can be seen easily. Dual integral equations of this type were considered in [4].It was shown in [4]that the solutions of these equations may be determined in the form of single quadratures from auxiliary functions satisfying the second kind FREDHOLM integral equations with symmetric kernel containing MACDONALD’S function Kl/2+i.r(z)of complex order. The general case is reduced to the case g ( r ) = 0, as follows from [4]. For simplicity, let us consider this case from this point on. Let us denote
fiekt d h(t) = --7r dt
1
O0
e-kTf(T)
dr7
is the real part of MACDONALD’S function of complex where order 1/2 i-r. Then we obtain the following procedure for the determination of M ( T ) , on the basis of [4]:
+
where $(t) is a solution of the integral FREDHOLM equation of the second kind k w $(t) = h(t) - ; K ( s ,t)$(s)ds, a 5 t < m. (7)
1 a
It is useful under the decision of boundary value problems to find the solution u on the boundary of the sectorial domain
Substituting expression (6) for M ( T ) into (8) and interchanging the orders of integration we obtain
where
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Juri M. Rappoport
So the numerical solution of the boundary value problem (2) consists of equation of the second kind the numerical solution of an integral FREDHOLM with symmetric kernel and the subsequent quadrature of the solution. Let us truncate the integral equation (7) in the following way:
/
k b $(t)= h(t)- ; K(s, t)$(s)ds ,u5 t 5 b. a
The estimations obtained show that we don't suffer any loss of accuracy in the bounds under the truncation of the integral equation (7) for b 2 10 in view of the fast decrease of the kernel K ( s , t ) for s , t + 00. The method of mechanical quadratures with the use of the combined SIMPSON formula with instant integration step is one of the most convenient integral equation of the methods of numerical solution of the FREDHOLM second kind. formula, possible in the case The application of the combined SIMPSON of fixed step At and odd number of steps, leads to the linear inhomogeneous system of algebraic equations $i
+
N
At k --xAjKij$j 3 T
= hi,i = 1, ..., N ,
j=1
where
t j =U+
(j- l ) A t , j = 1,..., N ,
Kij = K ( s ~t j ,) , hi = h(ti),i, j = 1, ...,N , $i
- approximate values $(ti),i = 1, ...,N , Aj=
{
1 for j = 1 , j= N , 4 f o r j = 2 , 4 , 6,...,N - 1 , 2 for j = 3,5,7, ..., N - 2.
It is convenient to use the Gaussian elimination for the solution ( 1 2 ) . The speed and operating memory of the mainframe computer BESM-6 made it possible t o use up to N NN 150 knots in the calculations. The solution of the system (12) gives values $1, ...,$n. The approximate solution of the integral equation (7) on the whole interval [u,b]is found by means of interpolation over these values &, i = 1,...,N . For the analytic expression of the approximate solution we take
Dual Integral Equations
173
having the values $1, ...,I)" at the points of interpolation. We obtain the greatest accuracy in this case compared to linear or quadratic interpolation. Furthermore, the solution of the dual equation was computed by the formulas (6) with the use of codes and routines for the computation of ReK1/2+iT(x) [6,7,91. The error estimation of the numerical solution scheme (12) - (13) may be found. The integrals ( 5 ) , (10) may be expressed through known functions for special values of the angle a , in particular for a = lr/n, n = 1,2, ... We compute the truncated integrals in fact by the computations of integrals ( 5 ) , (10) on the computer: the integration is carried on over some interval [O,B].In view of this fact it's important to choose the truncation interval [0, B] correctly, ensuring the computation of the integrals mentioned with the necessary precision without the expenditure of unnecessary computer time. The estimations of the error
arising from the truncation are useful for this purpose. On the basis of inequalities from [4,8]for the functions K i T ( z )and Re K 1 / 2 + i T ( xwe ) obtain e-2aB B 1 KB(s,t) 5 ~ ~ k - ~ / ~ ( s t )2-a~ /(B2 ~-+a + GI,
where A and c are some positive constants having the multiplicity of a unit. As it can be seen from (14) and (15) it is necessary t o extend the interval [0,B] for smaller values of the angle a. Let us consider the example admitting the complete analytic solution of the problem (2)
lr
g ( r ) = 0,a = -. 4 Then we obtain, on the basis of the relevant calculations [4], that
h(t) = eWk'
1 + -e-ka IT
+ +
+ 4,
+
K ( s ,t ) = Ko(k(s t ) ) Kl(k(s t ) ) ,
Juri M. Rappoport
174
and +(t)= e-kt. Substituting (16) in (9) and performing some calculations we obtain for T < a
and for
T
>a
(verification of the conditions of the problem). We obtained the precision in 7-8 significant digits for the solution of the dual integral equation (computation of the values (cosh ? ) - ~ M ( T )) so for a = 1.0, k = 1, (cosh % ) - l M ( 3 ) = .928825310 - 01. We obtained the precision of 6-7 digits after the decimal point in the calculation of values u I r ( r ) so for a = 1.0, k = 1, u l r ( 2 ) = .174544410 00. The different preliminary procedures of the separation of singularity or transformation of the integral t o an integral without singularity are useful for the computation of integral (9). Let us consider the calculations for the case cx = in detail [4]. We introduce the functions
+
r
+t)
and
Then GT(t)= Gl,(t)
+ Gz,-(t) and
The formula (17) is more convenient than formula ( 9 ) for the application of then formula ( 9 ) numerical integration procedures. Here the function GI,@)has no singularities for T , t > 0 and the function Gz,(t) has no singularities for t 2 a , <~ a. For T 2 a,the integrand of the second integral in (17) has a singularity for t = T and the integral itself is equal to
Let us make the change of variables tl =
,/s
and introduce the function
Dual Integral Equations
175
Then the second integral in (17) is equal to 2 S , 1 g ( t l ) d t l , where the latter integral has no singularities in the integrand. It is very efficient to use of numerical integration procedures for the transformed integral. The accuracy of computations is increased and the computer time is shortened by this approach.
3 Summary The dual integral equations with MACDONALD’S function of imaginary order K i T ( x )in the kernel are considered. The solutions of these equations and proper mixed boundary value problems are determined in the form of single quadratures from solutions of FREDHOLM integral equations of the second kind. The numerical solution is carried and problems of the computational methodology are discussed. Examples demonstrate the efficiency of the dual integral method in the numerical solution of the mixed boundary value problems of elasticity, combustion and electrostatics in wedge domains.
Acknowledgments The author wishes to thank the organizers of the 4th ISAAC Congress for their encouragement and support.
References 1. Alexandrov, V.M., Pozharskii D.A. Nonclassical spatial problems of the mechanics of contact interactions of elastic bodies. Moscow, Factorial, 1998.-288pp. (in Russian). 2. Ehrenmark, U.T. The numerical inversion of two classes of KONTOROVICHLEBEDEVtransform of direct quadrature. J. Comp. Appl. Math., 61 (1995), 43-72. 3. Fabijonas, B.R., Lozier, D.W., and Rappoport J.M. Algorithms and codes for the Macdonald function: recent progress and comparisons. J. Comp. Appl. Math., 161(2003), 179-192. 4. Lebedev, N.N., and Skalskaya, I.P. Dual integral equations connected with KONTOROVICH-LEBEDEV integral transform. Prikl. Matem. Mechan., 38(1974), N 6, 1090-1097 (in Russian). 5. Lebedev, N.N., and Skalskaya, I.P. Some integral transforms related KONTOROVICH-LEBEDEV transform. The Problems of Mathematical Physics. Leningrad: Nauka, Leningr. otd., 1976, 68-79 (in Russian). 6. Rappoport, J.M. Tables of Modified BESSELFunctions K 1 p + i T ( x ). MOSCOW: Nauka, 1979.-338pp. (in Russian).
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7. Rappoport, J.M. The programs and some methods of the MACDONALD function computation. Akad. Nauk SSSR, OVM, Moscow, 1991.-16pp. (in Russian). 8. Rappoport, J.M. Some results for modified KONTOROVICH-LEBEDEV transforms. Proceedings of the 7th International Colloquium on Finite or Infinite Dimensional Complex Analysis. Marcel Dekker Inc., 2000, pp.473-477. 9. Rappoport, 3.M. The canonical vector-polynomials at computation of the BESSELfunctions of the complex order. Computers and Mathematics with Applications, 2001, ~01.41,No.3/4, pp.399-406. 10. Ufliand, I.S. Method of dual equations in the mathematical physics problems. Leningrad: Nauka, Leningr. otd., 1977. - 220pp. (in Russian).
One Parameter-Dependent Nonlinear Elliptic Boundary Value Problems Arising in Population Dynamics Kenichiro Umezu Faculty of Engineering Maebashi Institute of Technology Maebashi 371-0816 Japan kentlmaebashi-it.ac.jp Summary. The author considers a local bifurcation problem for a type of one parameter-dependent nonlinear elliptic boundary value problems arising in population dynamics, with a nonlinear boundary condition. Some types of bifurcation phenomena, tangential as well as transcritical and pitchfork bifurcations can occur as the growth rate of the nonlinearity on the boundary of the domain is varied. The proofs are based on the local bifurcation theory from simple eigenvalues due to [4].
1 Introduction In this paper we consider the existence of positive solutions of the following nonlinear elliptic boundary value problem:
on d D . Here D is a bounded domain in RN, N 2 2, with smooth boundary d o , X is a nonnegative parameter, m E Ce(B)satisfies m(z0) > 0 for some z o E D , b E C1+'(dD) satisfies b 2 0 on d D , p > 1 is a constant, and n is the unit exterior normal t o dD. The equation -nu = X(m(rc)u-u2)in D arises in population dynamics. If X > 0, then the equation denotes the steady state of the population density of some species with diffusion 1 / X and growth rate m ( z ) ,where it is understood that the region where m ( z )< 0 is unfavorabIe for the species. The nonlinearity -u2 represents the crowding effect, a negative effect for the species. If we recognize our boundary condition as given
(X-' V u ). n = b(z)uP on d D ,
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Kenichiro Umezu
then this means that the population inflow into the region D at the border dD is governed nonlinearly by b(x)up. Note that the boundary condition is approximated by the Neumann zero condition at u small enough. A solution u E C2(D) of (1) is called positive if u > 0 in D. However, by the strong maximum principle and boundary point lemma (cf. [ 6 ] ) ,we have u > 0 in B if u is a positive solution of (1). In the sequel we call (X,u) a solution of (1) if u is a solution of (1) for some X 1: 0. Clearly, (A, u)= (A, 0) for X 2 0 is a solution of (1). (X,u) = (0,c) for c E R is also a solution of (l),which means the solution line (X,u) = ( 0 , c ) bifurcates at the origin (X,u) = (0,O) from the solution line (X,u) = (X,O). This paper is devoted to a local analysis for (1) of bifurcation to the region X > 0 of positive solutions from the two solution lines above. The bifurcation to the region X > 0 is from the viewpoint of population dynamics that X-' is the reciprocal of the diffusion rate of the species. The linearized eigenvalue problem at (X,u) = (X,O) is given
on BD. By pl(X) we mean the principal eigenvalue of (2). Here an eigenvalue of (2) is called principal if it has positive eigenfunctions. It is well-known that p1 (A) is simple, unique and the smallest among the eigenvalues. Clearly, we have p l ( 0 ) = 0. It has been proved (cf. [l])that pl(X) < 0 for all X > 0 if JD m d x 2 0, and that there exists a unique A1 > 0 such that pl(X1) = 0 if JD m d x < 0. If a bifurcation occurs at (X,u) = (X,O), then pl(X) = 0 (cf. [2]). From this fact, the following three types of bifurcation curves of positive solutions for (1) may occur: 0 0
0
bifurcation curves to the right at the origin (see Figure l ) , secondary bifurcation curves at some (X,u) = (O,c),where c > 0 is a constant (see Figure 2), bifurcation curves at (X,u) = (X1,O) when JD m d x < 0 (see Figure 3).
If there exists a sequence { ( A j , u j ) } of positive solutions of (1) satisfying X j 10 and uj+ 0 in C ( B )as j + 00, then the origin (A, u)= (0,O) is called a bifurcation point to the right for (1).Now we can state: Theorem 1.1 ([7]) The origin (X,u) = (0,O) is a bifurcation point t o the right f o r (1) if we assume JD m d x = 0, and assume either p > 2, or p = 2 and ID1 > JaD b d a . Otherwise, this i s not the case. Remark 1.1 (1) Let p be an integer greater than 1. If (X,u) = (0,O) is a bifurcation point to the right for ( l ) ,then we can prove that there exist a constant E > 0 and analytic functions A(-) : ( - E , E ) + R, v1(.) : ( - E , E ) t X , where X = {w E C2+'(B) : JD v d x = 0 } , satisfying X(0) = 0, X(a) > 0 for 0 < cy < E , q ( 0 ) = 0, and (X(cx),a(l v l ( a ) ) )is a positive solution of (I) for
+
One Parameter-Dependent Nonlinear Elliptic Problems
179
0
Fig. 1. Bifurcation to the right.
PI
x t
0
Fig. 2. Secondary bifurcation.
each 0 < (I! < E . Moreover, if (X,u) is a positive solution of (1) for X > 0 and I1uIIcz+s(D) both small enough, then (X,u) = (X(a),a ( l v ~ ( ( I ! for ) ) ) some
+
O
(2) Let p > 1 be not an integer. If (X,u) = (0,O) is a bifurcation point to the right for (l),then we can prove that there exists a positive solution (X,u(X)) of (1) for any X > 0 small and u(X)-+ 0 in C2+e(D)as X J. 0.
Xj
If there exists a sequence { ( X j , u j ) } of positive solutions of (1) such that converges to some positive constant c as j 00, then (A, u)=
1 0 and uj
--f
(0,c) is called a secondary bifurcation point for (1).Green's formula shows the following condition for (A, u ) = (0,c) to be a secondary bifurcation point for (1) is necessary:
m dz Now we can state:
- clDl+ cP-'
Kenichiro Umezu
180
I 0
A1
Fig. 3. Bifurcation at (X1,O).
Theorem 1.2 ([7]) (I) Assume J D m d x > 0 . If either 1 < p < 2 , or i f p = 2 and (Dl> JaD b d a , then problem (1) has a unique secondary bifurcation point (0, c1). If either p = 2 and ID1 5 JaD b d a , or i f p > 2 and ID1 < mpJaD b d a , then there is n o secondary bifurcation point for (1). Here mp is a positive constant given by mp= ( p - 1)p-l
(p7ii2)p-2 -
where rn = JD m d x / l D J .Finally i f p > 2 and ID1 > mpJaD b d a , then problem (1) has exactly two secondary bifurcation points (0, c1) and (0, c 2 ) . (11) Assume JD m d x < 0. If either p > 2, or i f p = 2 and JDJ < JaD b d a , then problem (1) has a unique secondary bifurcation point ( 0 , c l ) . If either p = 2 and (Dl2 JaD b d u , or i f 1 < p < 2 and (Dl> mpJaD b d a , then there is n o secondary bifurcation point for (1). Finally i f 1 < p < 2 and ID( < mpJaD b d a , then problem (1) has exactly two secondary bifurcation points (0,q)and ( 0 , ~ ) . (111) Assume JD m d x = 0. If p # 2, then problem (1) has a unique secondary bifurcation point ( 0 ,c l ), whereas there is n o secondary bifurcation point for (1) i f p = 2 and ID1 # JaD bda.
Remark 1.2 ( 1 ) Indeed, it can be verified that there exist a constant x j > 0 and a continuous function : [0, S;j) C2+'(D) satisfying u(0)= c j , such .(a)
--f
that (A,u(X)) is a positive solution of ( 1 ) for each 0 < A < x j . Moreover, if (A, u),X > 0, is a positive solution of (1) near (0, cj), then (A, u)= (A, .(A)) for some 0 < A < ( 2 ) The classification on the possibility for such a secondary bifurcation point is not complete. Theorem 1.2 does not deal with the following cases:
5.
> 0 , p > 2 and ID1 = mpJaDb d a ,
JD
mdx
JD
m d x = 0 , p = 2 and
IDm d x < 0 , 1 < p < 2 and IDI = mpJaD b da, IDI = JaD b da.
One Parameter-Dependent Nonlinear Elliptic Problems
181
The main purpose of this paper is to discuss the bifurcation at ( A , u ) = Our main results are given in the next section.
(A1,O).
2 Statement of Main Results In the sequel we consider bifurcation of nontrivial solutions at (A, u)= ( X I , 0 ) from the solution line (A, u)= (A, 0) for the problem
(-nu = A(m(x)u- u 2 )
in D, on d D ,
where A, b and p have the same properties as in Section 1, but JD m d x < 0. Note that (A,u) is a positive solution of (1) of Section 1 if and only if it is a positive solution of (1).We mean by 41 E C2+'(D)a positive eigenfunction of (2) of Section 1 corresponding to the principal eigenvalue p l ( A 1 ) = 0. It is verified by use of Green's formula that JD m& d x > 0 . First we prove the following existence result:
Theorem 2.1 Assume JD m d x < 0. If 2 is any complement of (41)in C(D), then there exist a constant E > 0 and continuous functions y(.) : ( - E , E ) + R, z ( . ) : ( - E , E ) + 2 such that y(0) = 0, z(0) = 0, and ( A ( s ) , u ( s ) ) = (XI y(s), s(d1 z ( s ) ) ) is a positive solution of ( I ) for each s E ( - - E , E ) . Here (41) = (t41 : t E R}. Moreover, i f ( A , u ) is a nontrivial solution of ( I ) in a neighborhood of (A1,O) in R x C ( D ) , then (A,u) = ( A ( s ) , u ( s ) ) f o r some
+
+
s E
(-E,E).
Secondly we study the direction of the bifurcation curve (A(S),ZL(S)) at = 0 the bifurcation is transcritical, more precisely y is continuously differentiable in ( - E , E ) and y'(0) > 0 (cf. [5]).When b $ 0, we prove the following: s = 0. In the case b
Theorem 2.2 Let (A(s), u ( s ) ) be the bifurcation curue emanating f r o m (A1,O) given by Theorem 2.1. Then the following assertions hold true: (1) If 1 < p
< 2, then it is a tangential bifurcation. More precisely,
(see Figure 4). (2) If p = 2 and JD 4: d x = JaD b&da, then it is a subcritical pitchfork bifurcation. More precisely, y is analytic at s = 0 and we have
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Kenichiro Umezu
j=O
with 70 = y1 = 0 and 72 < 0 (see Figure 6). (3) If either p > 2, or if p = 2 and f D & d x # f a D b&dg, then it is a transcritical bifurcation. More precisely, y i s continuously differentiable in ( - E , E ) and we have Y’(0)
{ <>
0 if p = 2 and JDrp;dx 0 otherwise
< JaD b&da1
(see Figures 5 and 7).
Fig. 4. tangential bifurcation.
A1
Fig. 5 . transcritical bifurcation with y’(0) < 0.
The rest of this paper is organized as follows. Our bifurcation analysis is based on the local bifurcation theory from simple eigenvalues in [4].Note that the local bifurcation theory does not work in the framework of Holder space C2+e(n)when 1 < p < 2 for the lack of regularity. To overcome the difficulty, we reduce (1) to a compact operator equation in C(D)in Section 3. Section 4 is devoted to the proof of Theorem 2.1 and we prove assertion (1) of Theorem 2.2 in Section 5 .
3 Reduction In this section we reduce (1) of Section 2 to an operator equation in G(D). Consider
One Parameter-Dependent Nonlinear Elliptic Problems
183
hi
Fig. 6. subcritical pitchfork bifurcation.
A1
Fig. 7. transcritical bifurcation with y'(0) > 0.
(-A
+ 1).
= f(x) in
on
D,
aD.
It is well-known that the resolvent K O : LP(D) --t W , 3 p ( Dof) (1)is isomorphic for any 1 < p < 00, where
By the Sobolev imbedding theorem, it follows that K O : C(n)--+ compact for any 0 < a < 1. Consider
cl+cy(n) is
It is well-known that the resolvent Kr : C1+*(aD) + C2+a(n) of (2) is isomorphic for any 0' < a c 1. By using the a priori estimate (see (31)
the linear operator Kr : C ( a D ) 4 W1>P(D)with domain C1+"(dD) is uniquely extended to a bounded linear operator of C ( a D ) into W17P(D) for any 1 < p < 00, since C1+"(aD) is dense in C ( a D ) .Again using the Sobolev imbedding theorem, Kr : C ( a D )+ Ca(D) is compact for any 0 < a < 1. Now we can reduce (1) of Section 2 to the compact operator equation
184
Kenichiro Umezu
+
u = K D [ u X(mu - u’)]
+ K,[Xbi(g(u))]
in
C(D)
(3)
with g ( u ) = I u I P . Here i : C(b) -+ C(6’D) is the usual trace operator.
4 Proof of Theorem 2.1 This section is devoted to the proof of Theorem 2.1. The proof is based on the following local bifurcation theory (see [4,Theorem 1.71):
Theorem 4.1 Let X and Y be Banach spaces, and let V a neighborhood of 0 in X . Assume the function
F : ( - 1 , l ) x V + Y ; ( t , z )H F ( t , z ) has partial Frkchet derivatives Ft, F, and F,, and these are all continuous. In addition, assume that 0 0 0
F(t,O) = 0 for t E (-1, I), the null space N[Fx(0,O)]has dimension 1, the range R[F,(O,O)] has co-dimension 1, Ftx(0,O ) ~ O@ R[Fz(O,O)],where N[Fx(O,O)]= ( ~ 0 ) .
If Z i s any complement ofN[F,(O, O)] in X , then there exist a constant E > 0 and continuous functions t(.) : ( - E , E ) + EX, z ( . ) : ( - E , E ) + 2 such that
t ( 0 ) = 0 , z ( 0 ) = 0 , and F ( t ( s ) ,s(zg + z ( s ) ) ) = 0 for each s E ( - E , E ) . Moreover, If ( t , x ) attains F ( t , z ) = 0 in some neighborhood of (0,O) in R x X , then (t,x) = (t(s),s(z0 ~ ( s ) ) )for some s E ( - E , E ) .
+
-
Now we associate with (3) of Section 3 a nonlinear mapping F as given
F :R x C(B) ( X , U ) +-+
C(D)
+
u - K D [ U X ( ~ U- u’)] - l C r [ X b i ( g ( ~ ) ) ] .
Note that the FrBchet derivatives F,, FA and FA, exist and are all continuous. Indeed, F, is given
+
F,(X,u)v = v - KO[V X(m - ~ u ) v-]Kr [Xbi(g’(u)v)], where g’(u) is continuous in
b for any u E C@), explicitly given
Substitute (A,u) = ( X 1 , O ) for (1). Then we have
F,(X~,O)V = 21 - lC~[(1+ X l m ) ~= ] 21 - Tv, where T = K,[(l+ Aim).]. Hence the null space N[F,(Xi,O)] is given
(1)
One Parameter-Dependent Nonlinear Elliptic Problems = (41).
"U(X1,O)I
185
(2)
Moreover, since T is compact in C ( D ) ,the range RIFu(X1,O)]is closed and the Fredholm alternative shows
Now it remains to verify
Note that FX~(X1,0)41= -xD[m4l]* Assume to the contrary that (4) breaks down, that is, there exists w E C ( D ) such that FU(X1,O)V
= -b[m41].
By elliptic regularity we have v E C2(D),satisfying
(-A
-
X1m)w = -m+1
in D, on aD,
from which JD m& dx = 0 by using Green's formula, a contradiction. Assertion (4) has been verified. Assertions (2), (3) and (4) prove the theorem, based on Theorem 4.1. 0
5 Proof of Assertion (l), Theorem 2.2 In this section we prove assertion (1) of Theorem 2.2. When 1 show how to prove the bifurcation curve
( A 7 4 = ( A 1 + y(s), 441 + z ( s ) ) ) ,
sE
< p < 2, we
(-€,E)
is tangential at s = 0. Note that z(s) E C 2 ( D )and y, z satisfy
-A2 = h ( m z - s(41
{g=
+ z y ) + y(m(41 +
( X i +y)b(signs)ls(P-'l41
2)
- s(41
+z)2)
in D, on aD.
+ZIP
Using Green's formula leads to - sX1
J, 41(41+
.)2
dJ:+ y
J,( m h ( 4 1 + z )
- S41(4l
+ z ) 2 )da:
186
Kenichiro Umezu
Note that s = (signs)lsl, a n d then
is continuous, z ( 0 ) = 0 and $0) = 0, this equality Since z : ( - - E , E ) + C(D) proves assertion (1) of Theorem 2.2. 17 A more complete description of the proof of Theorem 2.2 will appear in a forthcoming paper (see [S]).
References 1. G. A. Afrouzi and K. J. Brown (1999), On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions, Proc. Amer. Math. SOC.127 (1999), 125-130. 2. H. Amann (1976a), Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. 3. H. Amann (1976b), Nonlinear elliptic equations with nonlinear boundary conditions, New Developments in differential equations (W. Eckhaus ed.), pp. 43-63, Math. Studies Vol. 21, North-Holland, Amsterdam, 1976. 4. M. G. Crandall and P. H. Rabinowitz (1971), Bifurcation from simple eigenvalues, J. F‘unct. Anal. 8 (1971), 321-340. 5. P. Hess (1991), Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Math. Series Vol. 247, Longman Scientific & Technical, Harlow, Essex, 1991. 6. M. H. Protter and H. F. Weinberger (1967), Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs New Jersey, 1967. 7. K. Umezu (to appear), Multiplicity of positive solutions under nonlinear boundary conditions for diffusive logistic equations, Proc. Edinburgh Math. SOC.,to appear. 8. K. Umezu (in preparation).
Combined Integral Representations Heinrich Begehr I. Mathematisches Institut F'reie Universitat Berlin Arnimallee 3 D-14195 Berlin Germany begehr(0math.fu-berlin.de
Summary. Modified Cauchy-Pompeiu representations for complex functions in the unit disc of the complex plane are given in relation to higher order Dirichlet and Neumann boundary value problems.
1 Introduction One of the simplest ways to deduce the Taylor formula under slightly stronger assumptions for functions of one real variable is to iterate the representation formula given by the fundamental theorem of calculus and using integration byparts. For f E C'((a,b);IW)nC([a,b],R) andx,xo E ( a , b ) the fundamental theorem states X r
Applying this formula under proper assumptions to f ' also and inserting this into the preceding equation gives after integration by parts 5
Continuing inductively the Taylor formula
is attained for any f E Cn+l((a,b ) ;R) n Cn([a,b];R). This Taylor formula (2) is proper for solving the inhomogeneous initial value problem
188
Heinrich Begehr
f ( " f l ) ( x= ) n!g(x)in [a,b] f'"'(X0)
= v!a,,
0 5 v 5 72,
for a, E R and g E C ( [ a b]; , R). The unique solution obviously is n
f(z)=
C u=o
U,(X
- xO),
+
j
(X - t)"g(t)dt.
(3)
20
If the differential equation has lower order terms included then the form ( 3 ) will transform the initial value problem into a Volterra type integral equation of second kind for g . This procedure can be imitated for integral representation formulas in many situations in real and complex analysis, in quaternionic, octonionic and Clifford analysis, see [1]-[17]. The basic formula is again the fundamental theorem of calculus which however in the case of several variables in form of the Gaui3 divergence theorem has to be modified by introducing some fundamental solution to the differential operator involved in order to lead to a proper integral representation formula. These are representations of Cauchy-Pompeiu type. Replacing the fundamental solutions by kernel functions adjusted to boundary value problems leads to solutions to certain model equations satisfying these boundary conditions [7, 11, 15, 161. The Green and the Neumann functions are such kernel functions.
2 Modified Cauchy-Pompeiu representat ions The higher order Cauchy-Pompeiu representation formulas for w E C(n)(D; C)n dn-')(D;C ) easily derived from the G a d theorem [3]
and
just being some special cases of a general one related to the differential operator a:-'@,O 5 k 5 n, can be modified using the higher order Green and
Combined Integral Representations
189
Neumann functions. In order to be explicit special domains like the unit disc or the upper half plane have to be considered. However, the considerations hold true for any domain regular for the Dirichlet and Neumann problem. The higher order Green’s function for the unit disc D is, see [7], for 1 5 n
The Neumann functions are given iteratively, see [16], via
Nl(Z,C) =log[lC-~1211-~T121, (7)
They satisfy the conditions
C E B, z # c, a,”G,(z, C) = 0 on X D for C E D, 0 5 v I n - 1,
(aza-i->nGn(z, C) = 0 in D for
and
c)
(az&)~n(z, =~
C)
~ - 1 ( z ,in
D for
< E D,
n
s
dz
Nn(t,c)z
= 0 for
CED
aD
+
respectively. Here dVldenotes the outward normal derivative i.e. = zi3, Both kernel functions are, moreover, symmetric in z and C,i.e. they are satisfying
Z&.
Gn(Z,
C) = Gn(C,z ) , Nn(z,<) = %(C,
). for 375 E B, z #
c.
An explicit formula for Nn(z,C ) besides the case n = 1 is not yet known. Observing
190
Heinrich Begehr
the representations (4),(5) can be modified to
and
just by applying the G a d theorem, see [7]. Another special Cauchy-Pompeiu representation besides (4) and (5) is e.g., see [14],
n- 1
u= 1
Introducing G,(z,
"-l1 aD
C) in the case D = D leads to
"
Combined Integral Representations
Using the notation O$$u+l(z, see [7], Lemma 2, for 1 I p 5 v
C) = (I C l2 -
C) for any v E N and,
-l)’-lgl(z,
1
and -
1 z ( v - l)!v! a’+lyGu+l(.Z, c C) = (1 - zC)u+l > 1 a~+la;+lGu+l(z, C) = 0, ( v - l)!v! c so that on
I<\=
1 for 1s p 5 v - 1
191
192
Heinrich Begehr
n-1
[;I-1 p=l
n-1
v=2p+l
x gX(z, <)(I-
v-2p
.
1
(
p-1
X=l 12
I’),”ara;+’w(C)dC}
’J
- 7r
Gn(z,C)dgnyw(<)dtdq.
D
This representation is related to the higher order Dirichlet problem
Combined Integral Representations
193
Its Green function has the properties, see [7],
<
for any E D. Applying the Neumann function rather than the Green function (11) for n = 1 can be written as W(.>
1 =27ri / { w ( C ) - 1%
dC I C - z l2 av,wo} 7
ann
+;
'S D
(14) log
I (C -
-
.T> l2 a,+(odJd%
i.e.
+'7r J ~ i ( zC, ) a c v ( < ) d t d v . ED
This formula (15) provides a solution to the Neumann problem d,w(z) = y(z) on BD, = f in D,
a,&
if and only if the solvability condition
is satisfied. The solution can be uniquely determined by fixing
(15)
194
Heinrich Begehr
In a similar way (11) can be transformed into
On this basis in [16] it is shown that the higher order Neumann problem
for proper right-hand sides is uniquely solvable if and only if the solvability conditions
with
a2 = 2
and for 3 5 k k-2
&k =
-
c
p!2
(k - l)!(k - 1 - p)!2(2p - k + l)! %+'
p=[$1
are satisfied. The solution then is given as
iD
with
1 pH(.> = 2 &,N,(z,c) on
For a proof see [16].
aD for 1 5 p 5 n and z E D.
Combined Integral Representations
195
References 1. Begehr, H.: Iterated integral operators in Clifford analysis. ZAA 18 (1999), 361377. 2. Begehr, H.: Integral representations for differentiable functions. Problemi attuali dell’analisi e della fisica matematica, ed. P.E. Ricci, Aracne, Roma, 2000, 111130. 3. Begehr, H.: Integral decomposition of differentiable functions. Proc. 2nd ISAAC Congress, Fukuoka, 1999, Vol. 2, eds. H. Begehr, R.P. Gilbert, J. Kajiwara, Kluwer, Dordrecht, 2000, 1301-1312. 4. Begehr, H.: Cauchy-Pompeiu representations. Problems in Differential Equations, Analysis and Algebra, Actobe, 1999, eds. K.K. Kenchebaev et. al., Actobe Univ., Actobe, 2000, 218-221. 5. Begehr, H.: Pompeiu operators in complex, hypercomplex and Clifford analysis. Revue Roumaine Math. Pure Appl. 46 (2001), 1-11. 6. Begehr, H., Dai, D.Q., Li, X.: Integral representation formulas in polydomains. Complex Var., Theory Appl. 47 (2002), 463-484. 7. Begehr, H.: Orthogonal decompositions of the function space L I ( ~C). ; J . Reine Angew. Math. 549 (2002), 191-219. 8. Begehr, H.: Biharmonic decompositions of the Hilbert space Lz. Revue Roumaine Math. Pure Appl. 47 (2002), 559-570. 9. Begehr, H.: Representation formulas in Clifford Analysis. Acoustics, Mechanics and the Related Topics of Math. Analysis, ed. A. Wirgin, World Sci., New Jersey, 2002, 8-13. 10. Begehr, H., Du, J.-Y., Zhang, Z.-X.: On Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra. Acta Math. Scientia (China) 23B (2003), 95-103. 11. Begehr, H . , Du, J.-Y., Zhang, Z.-X.: On higher order Cauchy-Pompeiu formula in Clifford analysis and its applications. General Math.11, no. 3-4 (2003), 5-26. 12. Begehr, H., Dubinskii, Ju.: Orthogonal decompositions of Sobolev spaces in Clifford analysis. Ann. Mat. Pura Appl. 181 (2002), 55-71. 13. Begehr, H., Gackstatter, F., Krausz, A.: Integral representations in octonionic analysis. Proc. 10th Intern. Conf. Complex Analysis, eds. J. Kajiwara, K.W. Kim, K.H. Shon, Busan, Korea, 2002, 1-7. 14. Begehr, H., Hile, G.H.: A hierarchy of integral operators. Rocky Mountain J. Math. 27 (1997), 669-706. 15. Begehr, H., Kumar,A.: Boundary value problems for bi-polyanalytic functions. Preprint, FU Berlin, Berlin, 2003, Appl. Anal. (to appear). 16. Begehr, H., Vanegas, C. J.: Iterated Neumann problem for the higher order Poisson equation. Preprint, FU Berlin, Berlin, 2003, Math. Nach. (to appear). 17. Vu, Thi Ngoc Ha: Integral representations in quaternionic analysis related to Helmholtz operator. Complex Var., Theory Appl. 48(2003), 1005-1023.
Remarks on Quantum Differential Operators Robert Carroll Department of Mathematics University of Illinois Urbana, IL 61801 USA rcarrollQmath.uiuc.edu
Summary. Some aspects of q-differential operators and their origins are sketched.
1 Introduction We do not deal here with the qKP or qKdV hierarchy pictures but will indicate some other origins of quantum differential operators such as quantum groups, differential calculi, zero curvature relations, Maurer-Cartan equations, q-Virasoro ideas, Moyal quantizations, etc. One remarks that some quantum differential operators do not involve q and some q-differential equations are not clearly of quantum nature (e.g. various q-special function equations). The role of q in quantum theory seems in fact to be somewhat elusive; sometimes it seems to really involve physics in a fundamental manner but other times it seems to be only a way to diagonalize a Hamiltonian for example and thus appears purely calculational in nature.
2 Origins of q-Equations 2.1 General Remarks In [4, 61 we sketched a number of sources for linear q-differential equations in various contexts (see the references there for details and cf. also [4, 13, 401). This includes many relations involving q-special functions, Casimir operators, quantum groups, and representation theory with linear q-differential equations arising naturally. A good source of general information with Lie theoretic background is [20, 21, 22, 23, 241 and for references to special functions (hypergeometric, Bessel, Jacobi, etc.) we cite [41, 42, 631 (this is already an industry with many practitioners so no list will suffice). There is also a development involving special functions and q-special functions where tau functions
198
Robert Carroll
arise as generating functions for matrix elements in general group representations (cf. [15, 27, 28, 35, 36, 39, 49, 50, 51, 52, 531). A priori given any classical differential equation one can write down many q-versions but most of them are probably meaningless and one feels that any interesting differential equation necessarily arises from some symmetry connection. Consequently any interesting q-differential equation should arise from some corresponding q-symmetry connection. This means that our q-equations should arise from quantum groups or have meaning on quantum spaces connected to quantum groups (QG). However we do not even know if this is true in the hierarchy picture treated above. We recall that QG are quasitriangular Hopf algebras and one point of origin was the study of quantum integrable systems (cf. [3, 41). We assume the basic Hopf algebra material is known and recall that there is no exponential map in QG theory but analogous constructions can be achieved via the duality between Fun(G) and U(g) extended to U,(g) and Fun,(G). Remark 2.1 A synopsis relating classical and quantum integrable systems goes as follows (following [48]. 0
0
Classical situations involve &L = [MIL]qn = Tr(L") with { L I ,L 2 ) = [r12,LiLaj, and [ T 1 2 , T 1 3 ] [ T 1 2 , T 2 3 ] -k [ T 3 2 , T 1 3 ] = 0 , anf = {qn, f}, Mr = -nTT2(?-12L;), anL = [Mn,L] Put at lattice sites n = l , . . ., N we get equations Ti = L i ( N ) . . . L 1(2)Li (l ),{ T I ,T2) = [ q 2 , TlT2] in addition to
+
M f ( n ) = -pTrz [ L z ( N ) -
a
.
L2(n)rlzLz(n- 1)
+
0
.
Lz(1) . TZp-',
and aPL(n)= M p ( n 1 ) L(n) - L ( n ) MP(n) For quantization one goes to (A) R 1 p 5 - L = ~ L2mLlR12 with the quantum Yang Baxter equation (QYBE) R12R13&!23 = R23R13R12. Write R12 = Fg'F12 with F.1 = P12F12P12 so R satisfies QYBE and set L1 * F L2 = F12L1 L2Fz1 from which L1 * F L2 = L2 * F L1. Quantum analogue of qn is Qn = T ~ i . . . ~ ( & -* l * ,A12L1. ~ * * L,) where A 1 2 = P12R12. One gets Qn . Qm = Q m Qn. Now write (A) as L 1 - L2F,i1 = RTiL2. L1 and for 0;= Trz(L2RTl) one has L 1 . 0 ; 0;. Ijl Think of Qn -ha, flow and set fiM; = U; - 1 1 . Q1 to get Lax equation dlL1 = [A@,L1] where [ , ] involves the numerical matrix product and with anL1 = [h;rf,L1], the quantum group product. Can extend to etc. rn
-
9
-
= A
0
N
&lr
We note that the QYBE equation corresponds to the Artin braid group equation and monodromy representations of the braid group correspond to representations of the braid group along solutions of the KZB equation (which are equivalent to representations from quantum groups - Drinfeld-Kohno theorem); in fact the KZB equation is the differential equation satisfied by the partition function of the WZW model (cf. [4, 541 for more detail). There are q-differential equations arising in an algebraic context over general fields (cf.
Remarks on Quantum Differential Operators
199
[32, 471). Further one has a Galois theory of Fuchsian q-differential equations (cf. [56]) and a study of p-adic q-differential equations in [19]. Finally we mention a very interesting series of papers by G. Vitiello and collaborators (see e.g. [12, 31, 641) in which damped quantum harmonic oscillators are studied by doubling the phase space variables (creating a heat bath) and relating states to thermal field theory situations. The Fock-Bargman framework is used and the algebraic structure of thermal field dynamics involves the q-deformations of the algebra of creation and annihilation operators. The canonical quantization is studied in terms of q-differential operators and the time evolution involves tunnelling between unitary inequivalent representations (UIR) of the canonical commutation relations (CCR) (Weyl- Heisenberg algebra). Via examination of certain Bogoliubov transformations one can relate the q parameter to I't via 4 ( t ) = exp(Ft) and this labels the UIR. Matters of entropy, coherent states, Bose statistics, etc. are developed. Remark 2.2 The development in [20] for example emphasizes the ideas of intertwining and group invariant differential operators in a general Lie theoretic context with representations involving Verma modules (cf. also [21, 22, 23, 241). One arrives e.g. at a Maxwell hierarchy and its quantum group version in the context of a particular q-Minkowski space-time. The spirit is that of QG symmetries described via invariant q-differential operators and here the equations are also q-conformally invariant. Other work involving Dirac equations is described in [4]for example and the differential operators involved seem to be primarly linear. However the treatment in [25] provides interesting nonlinear differential operators in the context of multilinear intertwining differential operators. Bilinear and trilinear formulas for example are written out in [25] for d ( 2 ) and there is an interesting connection to KdV. Thus let G be a Lie group with representations T , T' in spaces C, C'. An intertwining operator 3 is a continuous linear map (A4) 3 : C t C' : f --+ j such that JoT(g)= T'(g) 03. The equation (A5)3f = j is then a G-invariant equation. Omitting discussion of integral intertwining operators one defines multilinear intertwining differential operators and as an example take k = 2 with G = SL(2, R) and consider C" functions with
where a6 - & = 1. The operator (A6) 23(f) = f"'f' - (3/2)(f'')2 has the intertwining properties 23
:
f €3 f
+
j; f E
co,j E c8;2 3 0 (TO(g)€3 TO(g)) = T8(g) 02 3
(2.2)
where C" denotes the space of functions with (2.1) as transformation rule. In particular this leads to an infinite hierarchy of even order intertwining differential operators producing (n/2)-differentials (n E 4N)with the ordinary Schwarzian derivative being the case n = 4. One specifies 230,(4)= qMzq5 = q5q5' where a! d ( 2 ) root and
-
Robert Carroll
200
Further note 0
23,,(40)
= 0;
ax-y
40
=; nE2 S-PX
+ 2N; S%,(+)
1
= -2J:a(4)
(2.4)
(@I2
(generalized Schwarzian) and recall that the KdV equation can be written =4(f)f’ = 0 (note S%4(f) = in the Krichever-Novikov form (A7) &f S(f) is the Schwarz derivative. Then to pass to the standard KdV form (A8) &U+U”’6uu’ = 0 one uses the substitutiion u = -(l/2)Sch4(f). It is conjectured in [25]that the equations (A9) &f+S%,(f)f’ = 0 (TI E 2N+2) are integrable, and if so they should coincide with the KdV hierarchy. One expects that a similar analysis in the q-context would be of great interest in understanding qKdV.
+
-
2.2 Integrable Systems When we come to integrable equations such as K P or KdV there are a number of classical derivations of intrinsic geometrical or algebraic interest which should morally have a q-version. Moreover one would expect the q-versions to bear some natural or canonical relation to the hierarchy version described earlier. In [4, 5, 6, 71 we examined some such derivations but without being able to establish a clear connection to the hierarchy picture (see however [8]where a possible connection occurs - sketched below). The constructions however seem interesting enough and well modeled on meaningful classical situations so some further study is indicated.
Differential Calculi First in a somewhat experimental manner consider a differential calculus approach following [4,5 , 7, 17, 181. Thus
Example 1. Consider a calculus based on (A) dt2 = dx2 = dxdt + dtdz = 0 (B) [dt,t] = [dx,t] = [ d t , ~=] 0 and [dx,x] = qdt. Assumingt the Leibnitz rule d ( f g ) = ( d f ) g f(dg) for functions and d2 = 0 one obtains (A10) df = f,dx (ft (1/2)qfZz)dt.For a connection A = wdt +udx the zero curvature condition F = d A A2 = 0 leads to ( A l l ) (ut - w, (q/2)uz, r/uu, = 0
+ +
+ +
+
+
which for w, = 0 is a form of Burger’s equation.
Example 2. Next consider (A) [dt,t] = [dx,t]= [dt,x] = [dy,t] = [dt,y] = [dy,y] = 0 with (B) [dx,4 = 2bdy and [dx,y] = [dy,x] = 3adt. Further (C) dt2 = dy2 = dtdx + dxdt = dydt dtdy = dydx + dxdy = 0. Then (A12) df =
+
Remarks on Quantum Differential Operators
+ +
+ +
+
f d x (f, bf,,)dy (ft 3af,, abf,,,dt). finds that d A A2 = F = 0 implies U,
+
+
+ w d t + u d y one
+2bvv,; wX = 3av,, +abv,,, + + 3 ~ v (+~bv,,); , (2.5) + bwzz = + 3auXy+ u~u,,, + - v[2bwX- 3a(uY + buzz)]
= vUy buzz W,
For A = v d x
201
~UUU,
~t
~UUU,
Taking e.g. w, = (3a/2b)uy+ (3u/2)uZzin the last equation to decouple one Sauu,) = (3a/2b)uY,; for suitable a, b arrives at (A13) &(ut - (ab/2)u,,, this is KP. If the equation is independent of y we obtain a version of KdV. H
+
It is surprisingly difficult to convert these examples into meaningful q-calculus equations and in that spirit for guidance we were motivated to develop many formulas concerning qKP, qKdV, etc. We recall first the first order differential calculus (FODC) r+from [40] on a quantum plane or Manin plane (cf. also [4]);this is based on x p = qpx with standard formulas as in [40]. In this FODC the partial derivatives ai of r+act on U = formal power series with x , p ordering, via (apxn= qnxnap and axpn = q"p"8,)
%(f( X ) h ( P ) )= P ; 2 f ( x ) ) h ( P ) ; a p ( f ( x ) h ( P ) )= (Tqf(x))(D;2h ( P ) )
(2.6)
q2" - 1 a,(~")= Dq2xn = [ [ T z ] ] ~ ~ x " - ' ; = ; appn = [[n]],ppn-' q2 - 1
(2.7)
[[~t]]~2
Example 3. The q-plane itself is not immediately fruitful however so consider the generalized q-plane with an algebra generated by x , y , x - l , y-' where x y = q y x , x d x = q d x x , y d x = q-'dxy, x d y = q d y x , and y d y = q-'dyy. Also from qdya: = d x y we have qdydx = d x d y and a little calculation yields (A14) da:" = [ ( l -q - " ) / ( l - q-l)]xn-'dx with dy" = [ ( l - q " ) / ( l - q)]y"-'dy. Working from f = C a n m x " y m one obtains then (note dxy" = qmymdx) df = DYDZ-1 f d x
+
+ D: f d y
(2.8)
+
Set then A = w d y u d x with d A = D,D;-'wdxdy D,Yudydx and, noting that dyx" = q-"xndy, dyy" = qmymdy, and dxy" = qmy"dx with d x x n = q-nx"dx one calculates d A A2 = 0 which, setting e.g. qw = D;'D,2_,u, leads to
+
D;u
+ ( D ; - - I ) ~ U+ q-'(O;'D,u)(D,'D~-,u) + uD;'D;-lu = 0 (2.9) 1 we have (A15) u,+ u,, + 2uu, = 0 so this appears to be an exact
For q + q-form of Burger's equation.
Example 4. We will try now a somewhat different approach. First we take qderivatives only in x , as in the case of qKP for example and we know from Example 2.2 that (A12) leads to interesting consequences so begin with an assumption (fy = a f / a y , etc.) df = D , " f d x
+ + b ( D , " ) 2 f ) d y+ (ft + 3d,D," f + ~ b ( D , " ) ~ f ) d t(2.10) (fy
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Robert Carroll
Then we can determine what elementary commutation relations between the variables are consistent with (2.10). This is rather ad hoc but we stipulate z, y, t ordering and then there are relations
+ b[2],dy; y ydx + 3adt dyz = q2zdy + ~[3],dt;dtz = q3xdt; d ~ =
(2.11)
[dt,y] = [dy,y]= [dy,t]= [dx,t] = [dt,t]= [dz,y] = 0
(2.12)
dzz = qzdx
along with
which are determined by (2.10). The underlying structure for z , y , t is not visible from (2.10) but (2.11) - (2.12) do lead to (2.10) and whatever zero curvature equations subsequently arise (see [6,7]for details). Now assume first dz2 = dy2 = dt2 = 0 and take A = vdz wdt udy; after some computation modeled on Example 4.2 one arrives at a q-version of K P in the form
+
+
~t+”b(D;)~u - -D;Dx(D;)2~ [31 ab
PI 9 +[3]qauD%D;u - -(Dq) 3a PI9b
x -1
D,U~,= A(u,W)
+ B(u)
(2.13)
where (2.14)
A(u,W )
=W
D ~-UuD:w;
[31 a
3a
B(u) = L D , ~ , D ~+U- D ~ D , ~ , u
- 3aayDtu 1219 [21& Then A t 0 and B -+ 0 as q + 1 and the equation (2.14) goes to the standard KP form. We see no immediate connection to qKP however. For y independent u we obtain
+ ~ b ( D , “ )-~-D:D,(D:)’u [31 u 9ab + [3],auD~D;u PI 9 = [31 a ( D , D ; ~ D ~ u~ ~ p ~ ; u )
ut
(2.15)
[21&
which is a quantum KdV equation.
rn
Geometry We indicate two “geometrical” contexts in a classical vein and subsequently give “quantum” versions of these.
Remarks on Quantum Differential Operators
203
Example 5. One can devise a procedure directly from [14]. Thus look at
SL(2,R) with matrices X =
(z i)
, ad
-
bc = 1. The right invariant
+
w$ = 0. Maurer-Cartan (MC) form is w = dXX-’ = ( w j ) where w i The structure equation of SL(2,R) or MC equation is (A16) du = w A w or explicitly (A17) dwi = wf A w i ; dwf = 2wi A w f ; dwi = 2~2 A w1. Now let U be a neighborhood in the ( x , t ) plane and consider a smooth map f : U + SL(2,R). The pullback of the MC form can be written as (AM) w i qdx Adt; w f Qdx Bdt; wf N rdx Cdt with coefficient functions of x,t. The equations (A19) become N
+
N
+
+
1. -Q + A x - Q C + r B = 0 2. -Qt Bx- 2qB + 2 Q A = 0 3. -rt Cx - 2rA 2qC = 0
+ +
+
Take T = 1 with q independent of ( x ,t ) and set Q = u(z,t). Then from (1) and (3) one gets (A19) A = qC+ 3Cx; B = uC-qC, - %Cxx. Putting this in the ( 2 ) above yields ut = K ( u )where (A20) K ( u ) = uxC+2uCx+2q2Cx-~Cxxx. In the special case C = q2 - ( 1 / 2 ) u one gets the KdV equation (A21) ut = 1 3 p x x x - pux.
Example 6. Following [l],let Vec(S1) denote the Lie algebra of smooth vector fields on S1 and then the Virasoro algebra is Vir = Vec(S1) @ R = H7 @ R with (note the minus sign convention involving f ’ g - fg’) (2.16)
(B? Witt algebra). Here Jsl f’gl’dx is called the Gelfand-Fuks cocycle, where a cocycle on a Lie algebra g is a bilinear skew symmetric form c(.,-) satisfying (A22) C c([f,g ] ,h) = 0 over cyclic permutations of f , g, h. This means that 6 = g @ R (central extension) with commutator [(f,a ) , (9,b)] = ([f,g ] ,c(f,9 ) ) satisfies the Jacobi identity of a Lie algebra. Now the Euler equation corresponding to geodesic flow is a 1-parameter family of KdV equations. To see how this arises consider (A23) Vir* = {(u(x)dz2,c);u smooth on S1 and c E R}. Then (A24) < (w(x)ax,a ) , ( u ( x ) d x 2c) , >= Jsl w(x)u(x)dx ac. The coadjoint action of (fax,a)E Vir on (udx2,c)E Vir* is (ad; : g* + g*, ad;w(u) = w(ad,u)) (A25) udTfaz,al(udz2,C) = ( 2 f l ~ + f ~ ’ + c f ’ ’ ’ )0) d~~, N
+
which arises from the identity (A26) < [(fa,, a ) , (ga,, b ) ] , (udx2,c) >=< (sax,b ) , adTfa, ,a)(udx2,c) >. Following [11 one arrives at an Euler equation m = -ad*A - ’ m m ( m E jj) which here takes the form (A27) &(udx2,c) = - u d ~ - l ( u ~ x 2 , c ) ( u d 2 2leading , c ) . to an equation (A28) atu = -2u’u - uu’ cut” = -3uu’ - cu”’where c is independent of time.
204
Robert Carroll
3 q-Versions In [8]I developed a q-version of this using a type of q-Virasoro algebra based on [45, 461 (cf. [8] for details and [lo] for a sketch). One works with dm = zrn+'aq where (A29) a, f = { f ( q z )- f ( q - 1 z } / ( q - q 4 - 1 ) z (so 8,z" = [m]z"-l). Then setting (A30) r(d,) =< m > d, = (qm +q-")d, a q-cocycle is constructed based on
and the cocycle is defined via
qqwa,, Wa,)
=a
J
W(a,"(T
+T - y ( a , V ) )
(3-3)
where a = 1/[2][3]and e.g. w = C wn+lzn+l. Then one arrives at the equation Ut
+.a:(,
+
T-')-'a,U
+ a,(UTU) + T-lUaqT-lU
(3-4)
+
Due to the expression (A31) (T ~ - l ) - l = T X ( - ~ ) " T ~ equation " (3.4) involves an infinite number of terms (which is similar to the qKdV hierarchy picture (cf. [5, 71). Remark 3.1 The Maurer-Cartan (MC) formulas were indicated in Example 2.5 and a version of this in a q-plane context appears in [40]. A version of this can also be developed directly from the discussion of duality (cf. [7] for details). A more interesting version comes from a quantum line Ri coupled with a time variable (e.g. A, = C(R) g~Ri) with e.g. z A = q A z ; z d z = qdzz; dzA = q A d q zdA = qdAz; ela: = q A z and e1A = 0; @ ( e l ) = e l f ; e2A = q A z ; e22 = 0; d f ( e 2 ) = e2f ( A is introduced for technical reasons (cf. [7]). The MC equations are (A32) dwo = Q ( w 0 A w l ) ; dwl = -wo A w2; d w 2 = Q(w1 A wa) (0 = q2 q4) and we must find expressions df and dw for functions and 1-forms. After considerable calculation based on zero curvature ideas (and some assumptions on A , dA) one can say that if u = u(z,t) does not depend on A a qKdV type equation based on the quantum line arises in the form (A33) ut = &a:-l(a;)2u
+
+
4 Moyal Approach One knows that Moyal quantization plays an important role in mathematical physics (for discussion cf. [3, 41). In fact KP can be identified with a Moyal
Remarks on Quantum Differential Operators
205
quantization of dKP (cf. [3,4, 111 for details). We mention also in this direction a recent preprint of Blaszak and Szablikowski [2] where one constructs soliton and lattice field equations from Poisson algebras of dispersionless systems. Generally speaking one writes
+ + -1
f * g = f e x p [~ ( 8 ~ 8 , spa, -
+ t g = f x + ~ a , , p - - ~ a ~ ) s ( x , (4.1) p)
(
and { f , g } ~= (f * g - g * f ) / 2 ~is the Moyal bracket. This leads to +-+ ++ (A34) {f,g}M = sin[^( a ,a - a d , ) ] g } . One can also write (using x , t variables and K 8/2) (A35) f * g = m o exp(8P/2)(f 8 9); m(f 8 g) = fg; P = dt @I a, - 8, @I at. Now go to [17, 181 as developed in [4] where one deals with bicomplexes (BC) M = eT$W with linear maps d, 6 : M r -+ AdT satisfying (A36)) d2 = b2 = d6 bd = 0 (note b is not the standard metric adjoint). In the spirit of Examples 2.1 and 2.2 one develops zero curvature equations to produce integrable equations such as KdV and KP.
,
k{f N
+
Example 7. Let M = Cm(R3)@ (12 ((12 @ : A j ) with (A37) df = (ftfzZz)7 (1/2)(fY - f,,)E and 6f = (3/2)(fy f,,)~f,E where 7, E E A’. Deform now (or “dress”) d to Df = df + S(vf) - vS(f) so N
+
of= [ft-fxzz+
+
+
(4.2)
(3/2)(vy+vx,)f+3v,f,]~+(1/2)(fy-f,,+2v,f)E
+
Then the required BC condition D2 = 0 becomes (A38) vZt - (1/4)vzzzz 3v,v,, - (3/4)vyy = 0 which is equivalent to K P for u -v,. There are also other forms of dressing of one BC to another while preserving the BC conditions. The underlying ideas here are zero curvature, cohomology (to get hierarchies), and gauge transformations (involving Seiberg-Witten (SW) maps - cf. [3, 4, 581). In particular SW maps preserve zero curvature and e.g. solutions of the KdV equation determine solutions of a noncommutative KdV (NCKdV) equation in a manner similar to what happens with the SW map between commutative and noncommutative gauge theories (cf. [3, 4, 581 for details). Thus take M = Cm(R2)@ (12 with T , E E (1’ satisfying T~ = E2 = TE ET = 0. One can then define d, b on M o = Cm(R2)and extend by linearity via d(f7 h t ) = ( d f ) ~ (dh)< for example. Start with (A39) df = -f,& ( f t 4f,,,)~ and 6f = f,E - 3fZ,7 (similar to Example 2.1). Apply a dressing to d in the form (u = 4,) N
+
+ +
+
+
of= df + 6(4 * f ) - 4 * 6f =
-(f,,
+ u * f)E + + 4fZII - 6u * f, - 321, * f (ft
) ~
(4.3)
(note d, and at are derivations for * which is a product as in (4.1) based on x,t). The only nontrivial BC equation is now D2 = 0 and this is equivalent to the NCKdV equation (A40) ut u,,, - 3(u * u, u, * u) = 0.
+
+
Remark 4.1 We refer to [43] for other aspects of noncommutativity and discretization in integrable systems. For further information on Moyal KdV
Robert Carroll
206
see also [4, 16, 611 (we do not attempt a complete survey here). However it is worth describing briefly the technique of [16, 61, 621 (cf. also [4] for a more detailed sketch of this). In these papers one goes back t o the fact that Moyal dKdV corresponds to KdV (seee [4] for an extensive discussion of this and see [3]for dKP and dKdV). Since [16]and [61] were written almost simultaneously we refer here to the Das-Popowicz-Tu algebra and follow [16]. Thus via (4.1) one has (A41) pn*pm =pm+, andp”*f(x) = C ~~)(-2n)’ff(’)(x)*pn-’. ,
One defines Lax type operators
I
+ u1(x) * pn-l + u2(x) * pn-2 + . . + un(x); A , = p” + u1(x) * p”-1 + + un(x) + u,+1* p-1 + - L,
= pn
(4.4)
*
* * *
*
and arrives at (A42) dA,/at’ = {A,, ( A k ’ n-) > m } ~(Ic # en) which provides a consistent Lax equation for m = 0,1,2 (( )+ = ( ) O is the standard projection). For KdV one has (A43) L = p2+u(z) and (L3/’)+ = p3+(3/2)u*p-rcu’ leading to (A44) ut = -(K.u”’ (3/2)uu’). We note that K. is seen to be proportional to the central charge (of a related CFT model) or equivalently to the Virasoro parameter and KdV itself appears from a “quantized” theory *dKdV so it is already a quantum equation in a specific sense. Further there is a Hamiltonian formulation via an action (A45) S = J dt(p * 2 - (I,-)+) with L = L(p, x) independent of t. Thus ( L w ) +plays the role of Hamiltonian on a (p,x) phase space and one has
+
(4.5) is given in (A34) (cf. [ll]for more on this). Remark 4.2 We add a few comments relative to [29, 30, 551 where some NC integrable systems are obtained. Omitting calculations one arrives at a NCKP equation where {
Ut
, }M
1 3 3 3 + -u,,, + -(u, * u + u * u,) + - a l l ~ y+v -[u,d F 1 ~ y ] = . 0 4 4 4 4
(4.6)
d;luy]* vanishes in the commutative The nontrivial deformation term [u, limit. The corresponding NCKdV equation has the form (A46) ut+(3/4)(u,* u u * u,) ( 1/4)u,,, = 0 which is equivalent to a dressing formula obtained already in [17, 181. There are also NC Burger’s equations (NCB) linearizable via NC Cole Hopf transformations. Thus
+
+
au,, ut - au,, Ut -
+ 2au, *
=0
+,
(u= - 2au * u, = 0 (u= +-I 21
+ +
+
* +-I); * $,)
(4.7)
More generally (A47) ut - au,, (1 a - b)u, * u (1- a - b)u * u, = 0 is an NCB equation but it may not be integrable for arbitrary constants a , b. In [30] one considers such matters in connection with NC solitons and D branes.
Remarks on Quantum Differential Operators
207
Remark 4.3 Still another quantum KdV equation arises from [44, 571. Thus recall the mKdV equation (A48)vt = (l/4)vzz, - (3/S)v2v, and the Miura transformation (A49)u = -(1/2)(v, - (1/2)v2) yielding (A50)ut = (1/4)uzzz - (3/2)uu, (see [57, 651 for philosophy). A critical ingredient here from the point of view of symmetry is the UrKdV equation (A51)qt = (1/4)(qzzz - (3q2,/2q,)) which gives a solution of mKdV via v = qZ,/q5. One writes S(f,z) = (f”‘/f’) - (3/2)(f”/f’)2 for the Schwartzian derivative and notes that u = -(1/2){q;x) is a solution of the KdV equation (A50).Now write (A52)tit = u,,, 32121, and remove the factor (1/4) in (A51).Define a geometric Virasoro action (Polyakov) via (A53)SO = - ( c / 4 8 ~ ) J dxdtqztg,,qz2 and set (A54)S = SO A, J dxdtp,[u] where the p , are standard conserved densities for KdV. Then S is a classical action yielding KdV hierarchy equations via
+
+ zy
Ut
=
2 4 ~ --(a3 + ua + au>-bS SU C
The c here again refers to a central charge in CFT language. This can then A4!,?], where the P, are normal ordered be quantized to (A55)ut = [u, densities and for A2 = 1 with all other A, = 0 one obtains the quantum KdV equation of [37] (cf. also [26]).
xT
References 1. V. Arnold and B. Khesin, Topological methods in hydrodynamics, Springer, 1998 2. M. Blaszak and B. Szablikowski, From dispersionless to soliton systems via Weyl-Moyal like deformations, to appear 3. R. Carroll, Quantum theory, deformation, and integrability, North-Holland, 2000 4. R. Carroll, Calculus revisited, Kluwer, 2002 5. R. Carroll, Complex Variables, 82 (2003), 759-786 6. R. Carroll, math.QA 0208170, 0211296, 0301361 7. R. Carroll, Inter. Jour. Pure Appl. Math., 5 (2003), 177-211 8. R. Carroll, math.QA 0303362 9. R. Carroll, On the emergence of q, in preparation 10. R. Carroll, Proc. Conf. Symmetry Nonlinear Math. Phys., Kiev, 2003, Part 1, 356-367. 11. R. Carroll, quant-ph 0309159 12. E. Celeghini, M. Rasetti, and G. VitieIIo, Ann. Phys., 215 (1992), 156-170 13. M. Chaichian and A. Demichev, Introduction to quantum groups, World Scientific, 1996 14. S. Chern and C. Peng, Manuscripta Math.,28 (1979), 207-217 15. A. Chervov, math.RT 9905005 16. A. Das and Z. Popowicz, hep-th 0103063 and 0104191 17. A. Dimakis and F. Muller-Hoissen, JPA, 29 (1996), 5007-5018; math-ph 9809023, 9908016, nlin.SI 0006029,0008016, 0008022,0104071; hep-th 0006005,0007015, 0007074, 0007160
208
Robert Carroll
18. A. Dimakis and F. Muller-Hoissen, hep-th 9401151, 9408114; q-alg 9707016; physics 9712002, 9712004 19. L. diVizio, math.NT 0211217 20. V. Dobrev, Habilitationsschrift, Clausthal, 1994 21. V. Dobrev, H. Doebner, and C. Mrugalla, Jour. Phys. A, 29 (1996), 5909-5918 22. V. Dobrev, Rept. Math. Phys., 25 (1988), 159-181; hep-th 9405150; Phys. Lett. B, 341 (1994), 133-138 23. V. Dobrev, P. Truini, and L. Biedenharn, J. Math. Phys., 35 (1994), 6058-6075 24. V. Dobrev, A. Mitov, and P. Truini, Jour. Math. Phys., 41 (2000), 7752-7768 25. V. Dobrev, hep-th 0303179; Jour. Geom. Phys., 25 (1998), 1-28 26. P. diF'rancesco, P. Mathieu, and D. SGnGchal, Conformal field theory, Springer, 1997 27. A. Gerasimov, S. Khoroshkin, D. Lebedev, A. Mironov and A. Morozov, hep-th 9405011 28. A. Gerasimov, S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and M. Olshanetsky, hep-th 9311142 29. M. Hamanaka and K. Toda, hep-th 0211148 and 0301213 30. M. Hamanaka, hep-th 0303256 31. A. Iorio and G. Vitiello, hpe-th 9503156; math-ph 0009035 and 0009036 32. U. Iyer and T. McCune, math.QA 0010041 and 0010042 33. B. JurEo and P. Schupp, solv-int 9701011, 9912002 34. B. JurEo and M. Schlieker, q-alg 9508001 35. S. Kharchev, A. Mironov, and A. Morozov, q-alg 9501013 36. S. Kharchev, S. Khoroshkin, and D. Lebedev, q-alg 9506022 37. B. Khesin and I. Zakharevich, hep-th 9311125; 9312088 38. B. Khesin, V. Lyubashenko, and C. Roger, Jour. Fnl. Anal., 143 (1997), 55-97 39. S. Khoroshkin, D. Lebedev, and S. Pakuliak, q-alg 9605039 40. A. Klimyk and K. Schudgen, Quantum groups and their representations, Springer, 1997 41. E. Koelink, q-alg 9608018 42. T. Koornwinder, math.CA 9403216; q-alg 9608008 43. B. Kupershmidt, Discrete Lax equations and differential-difference calculus, AstBrisque, 123, 1985; K P or mKP, Amer. Math. SOC.,2000 44. B. Kupershmidt and P. Mathieu, Phys. Lett. B, 227 (1989), 245-250 45. K. Liu, Jour. Algebra, 171 (1995), 606-630 46. K. Liu, CR Math. Rept. Acad. Sci. Canada, 13 (1991), 135-140; 14 (1992), 7-12 47. V. Lunts and A. Rosenberg, Selecta Math., 3 (1997), 335-359; 5 (1999), 123-159 48. J. Maillet, Phys. Lett. B, 245 (1990), 480-486 49. A. Mironov, hep-th 9409190 and 9607123 50. A. Mironov, q-alg 9711006 51. A. Mironov, A. Morozov, and L. Vinet, hep-th 9312213 52. A. Morozov, hep-th 9311142 53. A. Morozov and L. Vinet, hep-th 9309026 54. T. Otsuki, Quantum invariants, World Scientific, 2002 55. L. Paniak, hep-th 0105185 56. J. Sauloy, math.QA 0210221 57. J. Schiff, hep-th 9205105, 9210070; nlin.SI 0209040 58. N. Seiberg and E. Witten, hep-th 9908142 59. M. Tu, solv-int 9811010
Remarks on Quantum Differential Operators
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60. M. Tu, J. Shaw, and C. Lee, solv-int 9811004 61. M. Tu,hep-th 0103083 62. M. Tu, N. Lee, and Y. Chen, hep-th 0112262 63. N. Vileilkin and A. Klimyk, Special functions and the theory of group representations, Vols. 1-3, Moscow, 1991 64. G. Vitiello, hep-th 9503135 and 0110182 65. G. Wilson, Phys. Lett. A, 132 (19881, 445-450
The Inverse Monodromy Problem in a Class of Knizhnik-Zamolodchikov Equations Valentina Golubeva* Department of Mathematics All-Russian Institute for Scientific and Technical Information 20 Usievich Street Moscow 125315 Russian Federation golub0viniti.ru Summary. For a given semisimple complex Lie algebra g with a root system R(g) and with a set of reflection hyperplanes L = U L i in C" the inverse monodromy problem is stated in the following way: to characterize the representations P rl(C"
\ L , 20) -+ GL(p,C )
which can be realized as monodromy representation of some Pfaffian system of Fuchsian type, more precisely, of the Knizhnik-Zamolodchikovtype. For the case R = A,-1 the solution of this problem was given by Drinfeld and Kohno. After a short description of their result a sketch of development of investigations connected to this problem in a class of generalized Knizhnik-Zamolodchikov equations for different root systems is given. Problems are stated.
1 Introduction The paper is devoted to one of the possible multi-dimensional generalizations of the Hilbert's 21st problem, the so called Riemann-Hilbert problem consisting of a recovery or in a proof of existence of the Pfaffian system of Fuchsian type for a given divisor in C" and monodromy representation of the system. More exactly, let L = ULi be a reducible algebraic variety in @" of codimension one and X = @" \ L. For many problems in physics (in quantum field theory) and complex analysis (analytic theory of the ordinary and partial differential equations) the study of the different representations of the fundamental group 7r1(X,XO) is actual, that is the study of the the homomorphisms P7r
: T1 (X, T o ) -+ GL(P,@).
*The author is thankful to Steklov Mathematical Institute of Russian Academy of Sciences and RFBR for financial support in connection with the participation in the ISAAC Congress-2003 in Toronto.
212
Valentina Golubeva
The simplest and most investigated case of the structure of the fundamental group and its representations is the case where L is some hyperplane arrangement in C". In this paper, we consider some particular cases of such arrangements and more exactly the arangements with hyperplanes being the reflection hyperplanes of the corresponding Weyl group W ( g )of the root system R(g) in C" associated to a semisimple finite-dimensional complex Lie algebra 0. It is known that for R = A,-1 the fundamental group T I(X,xo) is the well known pure braid group. On the other hand, for the same Lie algebra g, the root system R(g) and L = U L i in C". Some Pfaffian system of Fuchsian type can be defined as follows: df = Of, where fl is the l-form having coefficients in different algebras connected in some way to the Lie algebra g and polar singularities of the first order on components of L. The form O of such a system has to satisfy the Frobenius integrability condition dO = f2 A f2 = 0, and f is a multi-valued (vector or matrix, etc.) function defined on the covering X of X. The ramification of f on L defines the monodromy representation pa of the fundamental group Tl
(X, 50) p n : .rrl(X,.o)
+
GL(p,C).
So, with the same Lie algebra g, R(g) and L = U L i in C" two representations p K and p a are connected. It is necessary to note that these representations can be constructed independently one from another. We can state two problems. Direct problem. Find the monodromy representation p a of a given Fuchsian system. Inverse problem. We are given some divisor L and a representation p : .rr1 (X, xo) + GL(p,C). Characterize those representations pT that can be realized as monodromy representation pn for some Fuchsian system.
2 Physical Background and Mathematical Approaches The inverse problem can be considered as a variant of a multi-dimensional problem of Riemann-Hilbert type. At the end of the 20th century, a class of differential equations in mathematical physics was discovered (for example, in conformal field theory and statistical mechanics), which appears closely connected to the classical Riemann-Hilbert problem and its multi-dimensional analogues. In 1965 T. Regge stated the problem of construction and investigation of the multi-dimensional analogues of the hypergeometric equations with singularities on reducible algebraic varieties being the ramification varieties of the Feynman integrals in quantum electrodynamics (Landau varieties). Later a similar problem was considered for correlation functions of physical models in conformal field theory. In the models of the Wess-Zumino-Novikov type as
The Inverse Monodromy Problem
213
singular varieties of correlation functions of fields it was natural to consider the hyperplanes arrangements associated to different root systems and even more general analytic images. For a root system of the A,-1 type, the corresponding differential equations were discovered as Ward conditions for the field functions (Knizhnik and Zamolodchikov, 1984). Now it is the well known KZ equation. But the physical models can have symmetries corresponding to different root systems. Therefore, after detailed study of the KZ equations of type A, the solution of direct and inverse problems for them, the theory of generalized KZ equations with symmetries corresponding to the root systems different from A,-1 starts its development. As for the case An-l, the following problems concerning the other root systems have to be stated and solved; - description of the pure braid groups in terms generators and relations; - geometrical realization of the braid groups; - investigation of the representations of the generalized braid groups; - construction of the generalized equations of the KZ type; - investigation of the monodromy representations for these KZ equations in classical terms; - construction of the theory of braided quasi-bialgebras for describing the monodromy representations of the KZ equations and for describing the representations of the braid groups; - proof of the theorems of the Drinfeld-Kohno type; - investigation of the connection of the monodromy representations of the generalized KZ equations to the representations of the braid groups considered by Burau, Kramers, Bigelow, R. Lawrens and others; - treatment of the monodromy propereties in terms of knot and link invariants; - different physical applications, such as investigation of the Potts, Gaudin, Ising, Hubbard and other models.
3 The Generalized Knizhnik-Zamolodchikov Equations 3.1 The KZ Equation of the
A,-1
Type
Let g be a semisimple Lie algebra, U ( g ) universal enveloping algebra, Uh(g) a p, A ) quantum universal enveloping algebra, where h is a parameter,(ll(g) [ [h]], a trivial deformation of U ( g )with multiplication p and A comultiplication A. Let t be the Belavin-Drinfeld tensor 1 t = - (A(c) - 1@ c - c @ 1) 2 where t E g 8 g, c is the Casimir element in g. Let A,-1 be a given root system in C" and WA,-~the corresponding Weyl group. Let H be the set of the reflection hyperplanes (the so-called hyperplane arrangement) of the Weyl group W A ~ -H~=, Ui,iHij where
214
Valentina Golubeva H i j = { ( ~ 1 , . . . , ~ :nz)i - z j = O } ,
for 1 5 i < j C", that is,
< n for the root system A,, X,
= ((21,.
. . ,z,)
E C."; zi
and let X , be its complement in
# z j for i # j ) .
The fundamental group n1 (X,, zo), zo E X,, is a pure braid group P,. Since the symmetric group Sn does not change the arrangement of hyperplanes, we can consider the Artin braid group.
B(A,-1)
= 7r1(Xn/Sn,zo)
for
zo E
X,.
The KZ equation of the A,-1 type has the form
where
x,
f is a holomorphic functions on the universal covering with values in (U(g)[[h]]@"", t i j are the elements of gB3" defined by the Belavin-Drinfeld tensor t E g@' (see Drinfeld, Kohno, Kassel). The integrability conditions of this equation are consequences of the relations dR = 0 and R A R = 0. Written in terms of the coefficients t i j these conditions are called the infinitesimal relations of the pure braid group of the An-1 type. The Drinfeld-Kohno theorem is an assertion about the equivalence of two quasi-bialgebra structures for the representations pK and pn (see Drinfeld, Kohno, Kassel). This theorem give a solution of the inverse monodromy problem. 3.2 The KZ Equations of the B, Type
Before continuing, it is necessary t o note that among the root systems connected to the semisimple Lie algebras there are systems all roots of which have one and the same length (such as, for example, the root systems of the types A,-l, E6, E7, Es) and the systems with roots of two different lengths (such as, for example, the root systems B,, Gz). In the last case it is natural to use , h z ) of the universal enveloping algebra. the two-parametric quantization U(hl Let H B be the union of the reflection hyperplanes of the Weyl group of the root system B, in 6"
H.= { Z i - z3' - 0, l < i < j < n } , 23 H$ = { z i + z j = 0 , 1 5i < j 5 n}, Hi
= {zk = 0,l
5 k 5 n}.
The Inverse Monodromy Problem
215
and let Y, be its complement in C", Y, = CC" \ H . The fundamental group Tl(Yn, yo), YO E Y,, is the generalized pure braid group P(B,). We consider the generalized K Z ( B )equation associated to the root system Bn dlk(z) = fl(B,)lk(z) where
The coefficients t;, t;, t: are defined by the elements t- E U ( g ) @ 2to , E U(g) and by the Weyl-Chevalley authomorphism aw : V ( g ) + U ( g ) . Let R be the set of root vectors of the Lie algebra g, and R+ the set of positive roots. We denote by e,,a E R, the root vectors, and by h j the basis of the Cartan subalgebra of g. Then the Weyl-Chevalley authomorphism ow acts by the rule: aW(e,) = e-,, aw(hi)= -hi, a(1) = 1. Let t be the Belavin-Drinfeld tensor written in basis (e,, h i ) , o E R , i = 1,.. . ,m, n
t=
C e, 8 e--cr+ C gijhi 8 h j ,
and to =
C (e,e-, + e-,e,) &R+
the Leibman element. Let further t- = t , t+ = ( a w 8 l ) t .Besides, by tij, t;, ty will be denoted the images of the elements t-, t+,to under the natural inclusions
U(gP2
+
U(LPn
on the ith and j t h factors for t* and on ith factor for t o respectively. The coefficients t;, t $ ,t: of the KZ equation of the Bn type satisfy the relations that are equivalent to the integrability condition dL! = flA L! = 0 for the l-form L!(B,). The l-form Ong,, is invariant with respect to the Weyl group WB,, = S, K (Zz)", where Sn is the permutation group acting on U ( g ) @ ,by transpositions of the tensor factors and the standard generators ~i of ( 2 2 ) " act as the Weyl-Chevalley authomorphism aw,i on the i-th tensor factor of U(gPn. For this equation the principal problems listed at the end of Section 2 are solved (see Golubeva, Leksin); for example the geometrical interpretation of the braid group of B, type is given, and the theorems of the DrinfeldKohno type are proved. But there are many interesting problems for future investigation (see also Golubeva, Leksin). The interesting paper of R. Haring-Oldenburg has to be mentioned. See references in [ 6 ] .
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Valentina Golubeva
3.3 The Connections of the KZ Type for the Root System Gz Now we consider the case of the root system G2. The root system consists of 12 vectors of R3:ei - e j and ei ej - 2ek where ei are the vectors of the canonical basis of R3 and 1 5 i < j 5 3, 1 5 k 5 3, k # i , k # j . The vectors a1 = el - e2 and a 2 = e2 + e3 2ek are basic vcctors of the root system Gz, and a1, a2, a1 + a 2 , 2 a l a 2 , 3 a l a 2 , 3 a l + 2 0 2 represent positive roots of the root system. The Weyl group W of the root system is generated by reflections in R3
+
+
~
+
The generators of this group are the reflections satisfying the relations 2 2 6 s1 = s2 = (s1sz) = 1.
s1 = s,,
and
s2 = saZ
The Weyl group is isomorphic to dihedral group 1 2 ( 6 ) . Now we represent the reflection hyperplanes of W on real plane R2. They have the equations X I = 0,xa = 0 , q f 8 x 2 = 0 , f i x 1 f 2 2 = 0 , among them the lines x2 = O,x2 f 4 x 1 = 0 are orthogonal to the short roots, and the other three lines to the long roots. Now we could write some equation of the KZ type with non specified coefficients and naturally with non-specified spaces of solutions. It contains two parameters k l , kz and has the following form dF = OF where
k2
+-(A421ra
dXl
+As
4x1 - 8 x 2 )
51 -4 x 2
XI
+
4x1
+ 8% ) >.
x1+ a x 2
The fiobenius integrability condition in this case is equivalent to the commutation relations of every coefficient with sum of all coefficients of the equation (Bolibrukh and Chernavski). If we make the change z = 2 1 / 2 2 and use the 3 6 AI,= 0 , we obtain an ordinary differential equation. relation Ci=lA i Ck=4 The vector form of the KZ equation in seven-dimensional representation was given in a paper by Zamakhovski and Leksin (see references). Now we obtain a tensor form of the KZ equation. For this it is more appropriate to use standard notation from Lie algebras theory. Let g be a semisimple finite-dimensional complex Lie algebra, U its universal enveloping algebra, (, ) a Killing form, H a Cartan subalgebra of g, R E H* be a system of the roots of g with respect to a Cartan subalgebra H , e,, Q E R a Weyl system of root vectors, such that (e,,e-,) = 1, [e,,e-,] = h, E H , [e,,ep] = N,,pe,+p for Q p # 0 with Na,p = N-,,-p E C. Let Hi = h,,,i = 1 , . . . , d , be a basis of H over C, Hi,j = (hi,h j ) , G = H - l . Introduce the following elements of U 8 U and of U :
+
+
The Inverse Monodromy Problem
217
d
r
=
C e, @ e-, + C
ffzo
Gijhi
@hj E
U @ U,
i,j=l d
-(x+ C 1
v = 5?
e2,
e-,e,)
E U.
Denote by u an authomorphism of U ,then a(e,) = -e,, o ( h , ) = -ha; x i : u - , u ~ 2 : x 1 ( u ) = 1 @ u , ~ 2 ( u ) = u ~: lu; @~ u~ ~, ju @ u ; x 1 , 2 ( u @ u ) =
=xi(.)
UgU.
( u @ u ) , x 2 , 1 ( u @ u V ) = ~ ~ u ; r i =, x ji , j ( r ) , p i , j = x i , j ( p ) , ~ i E These elements ri,j,pi,j and v i satisfy the commutation relations that
are consequences of the relations between elements e,, ha and h i (see Varadaradjan, Humpreys). Now we can write the equation of the KZ type associated with the root system G 2 in tensor form. Let V be a finite-dimensional space, p : g -+ EndV be a representation of g. The 1-form R has the form
where p 2 : U @ U
+ End(V
18V) is the representation constructed from p as
P(S1) @ P(S2).
From the commutation relations mentioned above for the elements ri,j,pi,j and vi, it follows that the connection dF = OF is integrable, that is the 1form R satisfies the condition R A R = 0. As follows from results of Hain, Golub, in the case of G 2 these conditions are equivalent to commutation of every coefficient of R with the sum of all other coefficients. It is necessary to note that the geometrical interpretation of braid group of the G 2 type is known (E. Suchilkina). It is generated by 01 and 02 satisfying the relations 01U2U1U2U1U2
= U2U102U102U1.
Here QI looks like a triple of the same ordinary generator ~1 of the Artin braid group of the type A,-1 disposed in the horizontal direction (and connecting the pairs of points one on the top horizontal line and the other on the bottom horizontal line and the line ltop-2bottom is over the line 2top-lbottom, etc). The generator ~2 looks as follows: the first top point is connected to the first bottom point, the line 2top-3bottom is under the line 3top-2bottom and sw, the sixth top point is connected to the sixth bottom point. The multiplication rule for such generators is defined as usual by means of superimposing the horizontal bands (see Fig. 1).
218
Valentina Golubeva
xss
Fig. 1. Braids ~
1 CTZ ,
and proof of the relation for them
It has to be added that the monodromy representation of the solutions of the G2 KZ equation has been investigated only for the particular case when this equation is reduced to the hypergeometric equation. The monodromy representation for general case has not been studied. It should be noted that there are many known representations of different orders for the Lie algebra G2. It is an interesting problem to connect such representations with some equations of the KZ type. But the braided bialgebra structures for the representations of the braid group of the Ga type and for the corresponding monodromy representations have not been investigated. Consequently, the theorems of the Drinfeld-Kohno type have not been proved.
4 Different Types of KZ Equations Above we characterized results connected with the solution of the inverse monodromy problem and to the generalization of the Drinfeld-Kohno theorem for several rational KZ equations. In the last short section we consider the state of investigations of the other possible forms of the differential equations of the KZ type. Among such we can mention two papers of V.Toledano Laredo where in sketch form results of Drinfeld-Kohno type are presented for a set of equations similar to KZ equations. The author calls these equations Casimir, Coxeter-KZ connection, etc. But in these papers, the detailed study, following a plan given V. Drinfeld and mentioned at the end of Section 1, is not given. Maybe such an analysis is not necessary for the considered equations. Above
The Inverse Monodromy Problem
219
we touched on the rational KZ equation associated with the root systems A, B , G2. Naturally, it would be interesting to study the cases of the rational KZ equations in tensor form for the other root systems F4, Es, E7, &. But in this direction there are no general results. At the present time, possibly with the single exception of the case F4, even the forms of the KZ equations associated with the root systems different from A , B , C, D,G2, are not known. Only a few of the structures of pure braid groups of the types F4, Es, E7, E8, have been studied. And, evidently, the braided quasi-bialgebra structures for these cases could not be given. It is worth mentioning that, besides the rational KZ equations, many of the papers are devoted to the trigonometric and elliptic KZ equations, and also the q-difference KZ equations have been studied, but we do not consider these results here. In conclusion I want to pay attention t o the original papers of J. F'uchs in which are written ordinary Fuchsian equations for some correlation functions with symmetries corresponding to the root systems B,, G2 and F4. Although this work does not fall in the purview of our review, but it appears me very perspective for using his approach in the study of the problems considered above.
Acknowledgements This paper was written under support by grants RFBR N 02-01-00721, grant of supports of leading Scientific schools N 457.2003.01 and grant CNRS N 03-01-22000.
References 1. Knizhnik, V.G. and Zamolodchikov, A.B. Current algebra and Wess-Zumino models in two dimensions. Nuclear Physics, B247:83-103, 1984. 2. Drinfeld, V. G. Quasi-Hopf algebras. Leningrad Mathematical Journal, 1(6):114148, 1989. 3. Drinfeld, V. G. On quasi-triangular quasi-Hopf algebras and a group closely connected with Gd(Q/Q). Leningrad Mathematical Journal, 2(4):829-860,1990. 4. Kohno, T. Linear representations of braid groups and classical Yang-Baxter equations. In Braids, Contemporary Mathematics, 78:339-363, 1988. 5. Kohno, T. Quantized universal envelopping algebras and monodromy of braid groups. Preprint Nagoya University, 1991. 6. Golubeva, V. A. and Lexin, V. P. Twc-parametric quantization of Lie bialgebras of the BN type. In H.G.W. Begehr, R.P. Gilbert and M.W. Wong, editors, Progress in Analysis, vol. 11, 827-835. 3rd International I S A A C Congress, Berlin, August, 2002. World Scientific, Singapore. 7. Golubeva, V. A. and Lexin, V. P. Algebraic characterization of the monodromy of the generalized Knizhnik-Zamolodchikov equations. In A.A. Bolibrukh and
220
Valentina Golubeva
C. Sabbah, editors,Monodromy in problems of algebraic geometry and differential equations: Collected papers. Proceedings V.A. Steklou Institute of Mathematics, 2003, uol. 238, Nauka PubLMAIK Nauka/Interperiodika, Moscow. 8. Leibman, A. Some monodromy of generalized braid groups. Communications in Mathematical Physics, 164:293-304, 1994. 9. Leibman, A. Fiber bundles with degenerations and their applications to computing fundamental groups. Geometriae Dedicata, 48:93-126, 1993. 10. Leibman, A. and Markushevich, D. G. The monodromy of the Brieskorn bundle. Contemporary Mathematics, 164:91-117,1994. 11. Bolibrukh, A.A. and A.V. Chernavski F'uchsian system on BbbC2 with constant coefficients on a divisor ( z - a l u ) . . . ( z - a l u ) = 0. In: Some questions in mathematics, in problems of physics, mechanics and economics. Moscow. Institut of Physics and Technology, 1990, 11-14 (In Russian). 12. Zamakhovskii, M. P. V. P. and Lexin Vector version of the Matsuo-Cherednik equations for the root system of the G2 type. Matematicheskie Zametki, 56:456460, 1995. 13. Varadarajan, V.S.. Lie algebras and their representations. Springer-Verlag, Berlin Heidelberg New York Tokyo, 1995. 14. Humpreys, J.E. Introduction to Lie algebras abd representation theory. Graduated texts in Mathematics, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1978. 15. Toledano Laredo, V. Flat connections and quantum groups. Acta Applicandae Mathematicae, 73:155-173, 2002. 16. Toledano Laredo, V. A Kohno-Drinfeld theorem for quantum Weyl groups. Duke Mathematical Journal, 112:421-451, 2002. 17. F'uchs J. Operator algebra from fusion rules: the infinite number of Ising theories. Nuclear Physics, B328:585-610, 1989. 18. F'uchs J. Superconformal Ward identities and the WZW model . Nuclear Physics, B286:455-484, 1987. 19. Fuchs J. A f i n e Lie algebras and quantum groups. Cambridge monographs on mathematical physics, Cambridge University Press, 1995.
Reduction of Two Dimensional Neumann and Mixed Boundary Value Problems to Dirichlet Boundary Value Problems M. Jahanshahi Department of Mathematics Azad Islamic University of Karadj P.O.BOX 31485-313 Ftajaee-Shahr, Karadj Iran jahan~nl5c8yahoo.corn
Summary. In the theory of boundary value problems the following three kinds of BVPs play an important role in this theory: 1. The Dirichlet problem, 2. the Neumann problem and 3. the mixed problem. In the first kind, the unknown function is given on the boundary and in the second kind the derivative of the unknown function with respect to the outward normal vector is given on the boundary. In the third kind a combination of the first and the second kind is given on the boundary. Because of the simplicity of the Dirichlet problem the question arises whether the second and the third of these problems can be reduced to the first kind (Dirichlet problem). In this paper it is shown that the second and the third kind of these problems with respect to the Laplace equation can be reduced to the Dirichlet problem.
1 Reduction of the Neumann to the Dirichlet Problem At first we consider the Neumann problem for the Laplace equation in the
We assume that the boundary of the region D is a Ljapunov curve, that is the normal vector n, varies continuously when the point z varies on all. We also assume cp(x) to be a continuous function. The harmonic conjugate of u(x)is denoted by ~ ( x )Then . v(x) is a harmonic function, and the Cauchy-Remann equations
222
M. Jahanshahi
hold. If the first and the second equation are multiplied by c o s ( n ~ , x i ) , c o s ( n ~x2) , respectively, and are added together, we have the relation
The left-hand side of this equation is the directional derivative of U(Z) with respect to the outward normal vector nc at the point (. Let aD be positively oriented. Then cos(nc, 21) = cos(rc, x2), cos(ng, x2) = - cos(rc, X I ) , where rc is the tangent vector to a D at the point t. By considering these relations we obtain the following relation
If x tends to ( from the inside of D, then from the boundary condition (2)
follows. If we integrate this equation on a D , with an initial point
where C is an arbitrary constant being equal to a Dirichlet problem for v ( x ) as follows:
~((0). Now,
t o , then
we have obtained
Au(x)= 0 , x = (XI,2 2 ) E D, v(x) = v(x0) +
s,:
cp(r)dr, x E aD.
If we calculate v ( x ) from this problem, then we can calculate u(x) from Cauchy-Riemann equations (3).
2 Reduction of the Mixed Problem to the Dirichlet
Problem For the reduction of the third kind of B.V.P (mixed type) to the Dirichlet problem, we first consider the following analytic function
Reduction of Boundary Value Problems to Dirichlet Problems
223
+
f ( x ) = b(x) i a ( x ) , x = ( 2 1 , x2) E R2,
where b(x) and a(.) are the real and the imaginary part of f(z) respectively. From the Cauchy-Riemann equations for an analytic function we have
We here assume that u ( x ) and v ( x ) satisfy the system
where a and b satisfy (7). Remark 1 If in this system, a(.) and b(z)are identically equal to zero, we will have the Cauchy-Riemann equations. If we differentiate the first equation of (8) with respect to x1 and the second equation of (8) with respect to 2 2 , we obtain @U(X) --
-
ax:
@v(x) axlax2
+ aa(x) ax1 ---U(X)
+ a(.)-
a44 ax1
By adding these relations we have
By observing (7), the first and the second bracket in the above relation are zero. If we calculate d v / d z l , dv/dx2 from ( 8 ) and substitute these values in the above relations, we have a 2 L
Au(x)= u(z)3x1 -a(.)
au + b ( ~ 8x2 ) - + b ( z ) [ ~ ( x ) u (x )u ( x ) ~ ( x )-
au [8x1
= b2(X)U(X)
.(X)U(.)
+ a2(x)u(.)
- b(z)v(z)]
224
M. Jahanshahi
Therefore, we have
AIL(.) = [a2(<.) + b2(x)]~ ( 2 ) .
(9)
By derivation of the first and second equation of (8) with respect t o x2 and x1 respectively we will have the relations
By subtracting these relations from one another and using (8) we obtain
=o By regarding (7) the first bracket is zero and the second bracket also is zero. After algebraic operations we have
A v ( x )= [u'(x) + b2(x)]v(x).
(10)
Now, we assume D c R2 be a bounded smooth domain and all be positively oriented. Then we have
If we consider the directional derivative of u(x) with respect t o n, and by using relations (8), then we obtain
Reduction of Boundary Value Problems to Dirichlet Problems
+u(x)u(x)cos(n,,
XI)
+ b(z)v(z)cos(n,,
225
21)
Therefore we have the relations
Remark 2 Note that if the functions u ( x ) and b(z)are identically zero, then this case will be the same as the previous case ( 5 ) , i.e. du(z)
-
dn,
dv(z) dTX
which was used in the reduction of the Neumann t o the Dirichlet problem. Now, we consider the third kind boundary value problem (mixed problem) in the form
nu(.) du(4 -dnx
= [u2(2)+ b 2 ( 2 ) ] u ( x ) , 2 E [a(.) cos(n,, X I )
D c R2,
+ b(z)cos(n,,
(12)
22)] u(x) = p(x)
, 2 E aD.
(13)
By considering the adjoint equation (10) for equation (12) and (11)this mixed problem can be rewritten as
+
Av(2) = [a2(<.) b 2 ( 2 ) ]v(x), dV(4 dTX
+
- [a(.)
COS(~,,Q)
(14)
+ b(z)
COS(T,,
XI)] V(Z)
= p ( ~ ,) 2 E
aD,
(15)
M. Jahanshahi
226
where rz is the tangent vector at the point x E aD. Assume that z o is an arbitrary point of the boundary aD,moreover, assume s is the arc length on aD between z o and z. Then we can write
v(z) = ?I(+ =) w) (s), dv(z) - dw(s) -dl-z ds * Hence, the boundary condition (15) reduces to the following ordinary differential equation
+
d w ( s ) a ( s ) w ( s )= cp(s). ds By multiplying with exp (- J : a ( ( ) d ( ) we have
where Q(Z)
= u(x)cos(r,, x2)
+ b(z)cos(r,,
z2).
Finally we obtain
w ( s ) = ceJ,”a(c)dc +
I’
(p(7)e-Js7 a(c)dcd7.
(16)
Now, if we replace (15) by (16), we will have a Dirichlet boundary condition for equation (14). Therefore by calculating the v(x) from this Dirichlet problem we can find u(z)from the equation (8) as the solution of the mixed problem. At the end we consider an example. Example The two boundary value problems
r < R,
Au = 0, I={ -au= - -au an
ar
-
f ( q ,7- = R,
A v = 0,
r
< R,
C
are equivalents, where C is the circle with radius are conjugate harmonic functions t o each other.
R.Note that
u(x) and v(z)
References 1. Sokolnikoff, I.S., Redheffer, R.M.: Mathematics of physics and modern engineering, McGraw-Hill, New York, 1958. 2. Myint-U, T., Debnath, L.: Partial differential equations, North Holland, Elsevier Publisher, New York, 1987. 3. Gonzalez-Velasco, E.A.: Fourier analysis and boundary value problems, Academic Press, San Diego, 1995. 4. Myint-U, T.: Ordinary differential equations, Elsevier Publisher, North Holland, New York, 1978.
Lipschitz Type Inequalities for a Domain Dependent Neumann Eigenvalue Problem for the Laplace Operator Pier Domenico Lamberti and Massimo Lanza de Cristoforis Dipartimento di Matematica Pura ed Applicata Universitb degli Studi di Padova Via Belzoni 7 35131 Padova Italy [email protected], mldc0math.unipd.it
Summary. In this paper we announce some results concerning the dependence of Neumann eigenvalues and eigenvectors of the Laplace operator upon domain perturbation. Let R be an open connected subset of R" of finite measure for which the Sobolev space W','( R) is compactly imbedded into the space L2(a).We consider the Laplace operator with Neumann boundary conditions in a class of domains 4(fl), where 4 is a locally Lipschitz continuous homeomorphism of fl onto the subset 4(R) of R". Then we present Lipschitz type inequalities for the reciprocals of the eigenvalues and for the projections onto the eigenspaces upon variation of 4.
Introduction This paper concerns the dependence of Neumann eigenvalues and eigenvectors of the Laplace operator upon domain perturbation, and it is based on the material of [5].We consider an open subset R of R" as in ( w ) in the following section, we deform R by a locally Lipschitz continuous homeomorphism q5 of a class @(O)introduced in (3) below, and we consider the weak formulation of the Neumann eigenvalue problem for the operator - A in the deformed domain $(Q). That is, we consider the problem
J,,,,
DvDw dy = X
vwdy
Vw E W1>2(q5(Q)),
(1)
in the unknowns v E W1*2(q5(R))and X E R. Under our assumptions on R and #I, such a problem is well known t o have a sequence of eigenvalues
228
Pier Domenico Lamberti and Massimo Lanza de Cristoforis
which we write as many times as their multiplicity. We are interested in the dependence of the eigenvalues X j [4], and of the corresponding projections onto the eigenspaces upon the deformation 4. For each pair 4, of elements of @(Q),we define a characteristic of closeness 6(4,4), and we show that the distance between the reciprocals of the eigenvalues in +(Q) and of those in &a), as well as the distance between the projections onto the eigenspaces can be estimated in terms of 6($,6). Actually 6 is a pseudometric on the class @(Q), and it can be easily estimated in terms of distance between the gradients of 4 and of However, as we can easily see from the definition of S, S($, depends only on the symmetric parts of the polar decomposition of the gradient matrices. In particular, 6 ( + , 4 ) vanishes if 4 is the composite of and of a rigid motion. We note that our results are 'global' in the sense they hold with no restriction on the size of 6(4,4). Spectral stability properties are known to be of help in the explicit computation of the eigenvalues, and are far less well understood when dealing with Neumann boundary conditions than when dealing with Dirichlet boundary conditions. In particular, our inequalities say that if 4 tends to then the distance between the reciprocals of the eigenvalues in $(Q) and of those in &a), as well as the distance between the projections onto the eigenspaces tends to zero with order at most S(4, We mention that stability properties of the eigenvalues of the Neumann Laplacian have been recently investigated by Burenkov and Davies [2]. We comment on their work in a special section below.
6
4)
4.
4
4,
6).
An Inequality for the Eigenvalues We denote by W ' ~ 2 ( ( n the ) Sobolev space of distributions in 0 which have weak derivatives up to the first order in the space L2(Q)of square-summable real-valued functions. We consider an open subset Q of R" satisfying the set of conditions
(4
Q is connected and of finite measure, and W 1 > 2 ( Qis)compactly imbedded in L2(Q).
We note that under assumptions ( w ) , the Poincard-Wirtinger constant c[O]of Q, ie., the best constant for which the inequality
holds, is finite. Here
Du denotes the gradient of u ,
Inequalities for an Eigenvalue Problem
229
and 101 denotes the n-dimensional measure of 0. We also mention that if R is bounded and has a continuous boundary, then the imbedding of W1*2(R) into L2(R) is compact (cf. Burenkov [l].) As is well known, even for bounded open sets, the spectrum of the Neumann Laplacian may be quite wild (see for example Hempel, Seco and Simon [3].) Thus we choose to impose some restrictions on our 'admissible domain deformations'. Our deformed domains will be in the form q5(0),where q5 is a homeomorphism of 0 onto q5(0).We now introduce our conditions on the 'admissible 4's'. That is, we set
@(n) =
{
q5 E (L1)"(R))n : the continuous representative of
I
4 is injective, essinf I det 041 > 0 , n
(3)
where L"(0) denotes the space of (equivalence classes of) real-valued essentially bounded functions in 0. Our class of admissible homeomorphisms will If q5 E @(a), then one can easily show that q5(0>is open and satisfies be @(n). the set of conditions (w). Thus problem (1) has a sequence of eigenvalues as in (2). We now introduce the pseudometric S on @(a). We set
for all 4,
4 E @(Q),where Dq5 denotes the gradient matrix of q5 E (q5,.),,=1,...,~,
(Dq5)-l denotes the inverse matrix of Dq5, and (Dq5)-t denotes the transpose matrix of (Dq5)-l. Then we have the following result. Theorem 1 Let n 2 2. There exist two functions 10, + O O [ ~ X[0,+ o o [ ~to [0,+oo[, such that
A1
and
A2
of the set
whenever (71,. . . , 7 5 ) , and ([I,. . . ,[5) E 10,+00[~x[O,+ 0 0 [ ~ satisfy the inequalities "/I 2 ( 1 f o r 1 = 1,2, and 71 5 [l for 1 = 3,4,5,and such that
Pier Domenico Lamberti and Massimo Lanza de Cristoforis
230
4
f o r all subsets 0 of R" as in ( w ) , and f o r all 4, E @(0), and f o r all j E N \ ( 0 ) . In particular, the functions A1 and A, depend only o n n and the right-hand sides of (5) do not depend o n j E N \ (0). For a proof of Theorem 1, we refer to [5]. Here, we just outline the main ideas. We 'pull back' or 'transplant' the eigenvalue equation (1) by means of the map 4, and we observe that the resulting equation can be viewed as an eigenvalue equation for a self adjoint operator T+ acting in the quotient space W1~2(0)/R endowed with an 'energy' scalar product Q+ depending on 4. Then we estimate the distance between the eigenvalues of T4 and of TJ by means of the distance between Qb and QJ, and of the distance between T4 and TJ in the operator norm. Finally, we estimate the distance between Q 4 and QJ, and the distance between T+and TJ in terms of 6(+,
4).
We now make a few comments on Theorem 1: 0
0
0
The right-hand sides of inequalities ( 5 ) are independent of j E N \ (0). It would be a different story if we tried to estimate IXj[$] - Xj[$]l. If 4 , $ E @(a) differ by a rigid motion, i.e. if 4 = R$ c, where R is an n x n-orthogonal matrix, and c E R", then 6(4,4)= 0, and Xj[4] = Aj[4] for all j E N \ (0}, as expected. Let r > 0. Let 0 satisfy condition ( w ) . Let
+
G T ( 0 )=
{
1
4 E G ( 0 ) : - 5 essinf IdetD41, r
R
esssup 1041 5 7 R
I
.
Then the maps 4 H XY1[4] are all Lipschitz continuous from the pseudometric space ( G T ( 0 ) 6) , to R, with a Lipschitz constant independent of j E 0
N \ (0).
Theorem 1 holds also under weaker assumptions on 0. Namely, in the set of conditions ( w ) , we could replace the condition of compact imbedding of W1i2(0) into L 2 ( 0 ) ,with the weaker condition that .[a] < 00. In this case, however, the numbers Xj[4] should be interpreted as the variational eigenvalues delivered by the Rayleigh quotient, via the Poincar6 formula.
Inequalities for an Eigenvalue Problem
231
Comparison with the Results of Burenkov and Davies We now comment on the result of Burenkov and Davies [2, Thm. 211. For brevity, we shall expose their work in a simplified setting. The authors consider a uniform class of bounded domains with Lipschitz continuous boundary, which for brevity we do not introduce here. Then Burenkov and Davies have proved that if 521 is a domain in such a class, and if j E W \ {0}, then there , 0 2 is another domain in the exist ~j > 0, bj > 0 such that if 0 < E < ~ j and 522 01, where d denotes same class, such that {z E 01 : d(z, 8521) > E } the Euclidean distance, then
c
c
Actually, Burenkov and Davies [2] have also considered domains with Holder continuous boundary. For brevity, we do not report their work here. The result of Burenkov and Davies, and our result are independent from one another and have been obtained by different methods. We now point out some advantages of each result. 0
0
0
0
0
Burenkov and Davies express the closeness of the domains 521, 522 in a very visible way. Burenkov and Davies do not assume that Q1 and 0 2 are homeomorphic images of some ‘base’ domain. Our result has a global nature, but the closeness of the ‘deformed domains’, is expressed in terms of the more-difficult-to-handle pseudodistance 6, and our ‘deformed domains’ must be homeomorphic images of some ‘base’ domain. Our result holds for the more general set of conditions ( w ) . Our result yields an estimate independent of the eigenvalue index j.
An Inequality for the Projections onto the Eigenspaces We now turn our attention to eigenfunctions. We first make two considerations: 1. If 211 and 212 are eigenfunctions for our problem on q51(52), 42(0), respectively, then we compare 211 and 2r2 by comparing the corresponding ‘pull-backs’ 211 o 41 and 212 o 4 2 in W1!2(0). 2. When we consider multiple eigenvalues, we shall compare the projections onto the eigenspaces. In W1t2(q5(0)),we shall use its standard scalar product, which we ‘pull-back’ on W’~2((n)to the following scalar product
232
Pier Domenico Lamberti and Massimo Lanza de Cristoforis
for all
u1,u2 E
W1yz((n).
We now confine our attention to a certain set of eigenvalues, indexed by the integers contained in a certain nonempty subset F of N \ {0}, and we introduce the set
of those domain deformations $ for which the eigenvalues indexed by the integers in F may coincide, but must not touch the remaining eigenvalues. If 4E F),then we can consider the space generated by all &pull-backs of some eigenfunction corresponding to eigenvalues indexed by some integer of F. That is, we can consider the space @[4,F ] generated by the subset
@(a,
u E W1i2(f2) : u o q5(-') is a Neumann eigenfunction of - A on $(O) for some X E {Xj[4]: j E F } of W'>'((n).Then we denote by &[$I the orthogonal projection of W1i2(f2) with the scalar product Qb onto @[4,F ] . Our goal is now to estimate the distance between &[4], &[$I for different 'domain deformations' 4, $ in @((n,F)in the standard operator norm 1) . ) ) . c ( w I , ~ ( ~ ) , w Iof , ~ the ( Q )linear ) and continuous operators in W'i2((n). To do so we introduce some notation. We set F* = { I E N \(0) : (1 - l , l , Z + l } n F # 8) , and
d'41
min Iq"41 - XL"41I j€F,l€F'\F
7
for all $ E *(a, F ) , and for all finite nonempty subsets F of W \ ( 0 ) . The significance of 441 is that of distance of the set of the reciprocals of the eigenvalues indexed by integers of F from the set of the reciprocals of the remaining nonzero eigenvalues. Then we can state our main result for eigenfunctions, which has been proved in [5] by exploiting the same ideas as in the proof of Theorem 1, and the formula of Kato for the projection onto an eigenspace.
Theorem 2 Let n 2 2. Let F be a finite nonempty subset of N \ { O } . Then there ezist functions r1 and r2of 10, + w [ ~ x[0, to [0,+m[ such that C(71,.. . ,7 7 ) 5
r&,. . . ,&),
r = 1,2
whenever (71,. . . ,y7) and ([I,. . . ,<7) in 10, +00[~x[O,+ 0 0 [ ~ satisfy the inequalities 71 2 C for 1 = 1,.. . ,4,and 71 5 C f o r 1 = 5,6,7, and such that
Inequalities for an Eigenvalue Problem
233
ID4 - D$l for all subsets 0 of R" satisfying ( w ) , and for all
4, $ E @(0, F).
We remark that if F is a finite subset of N \ {0}, then the continuity of the dependence of the eigenvalues upon 4 E Q ( 0 , F ) (see Theorem l), and Theorem 2, ensure that &[.I satisfies a Lipschitz condition on the compact F). subsets of Q(0, Finally, we mention that Lipschitz continuity results of the type presented in this paper can be proved also for the Dirichlet problem (see [4].)
References 1. V.I. Burenkov, Soboleu spaces on domains, B.G. Teubner, Stuttgart, 1998. 2. V.I. Burenkov, E.B. Davies, Spectral stability of the Neumann Laplacian, J. Differential Equations, 186, (2002), pp. 485-508. 3. R. Hempel, L.A. Seco, B. Simon, The essential spectrum of Neumann Laplacians on some bounded singular domains, J. F'unct. Anal., 102, (1991), pp. 448-483. 4. P.D. Lamberti, M. Lanza de Cristoforis, A global Lipschitz continuity result for a domain dependent Dirichlet eigenvalue problem for the Laplace operator, to appear, (2003). 5. P.D. Lamberti, M. Lanza de Cristoforis, A global Lipschitz continuity result for a domain dependent Neumann eigenvalue problem for the Laplace operator, to appear, (2003).
Half Robin Problems for the Dirac Operator in the Unit Ball of R" ( m 2 3) Zhenyuan Xu Department of Mathematics, Physics and Comuter Science Ryerson University 350 Victoria Street Toronto, ON M5B 2K3 Canada zxuQryerson.ca
Summary. In this paper, half Robin problems are considered for the operator D in the unit ball B of R" (rn 2 3). D is the Dirac operator in Euclidean space.
1 Introduction Let C, be the complex Clifford algebra over Rosm ( m 1 3) . Its basis consists of the elements eA = e,lcuz...ah - ealeaz - . . e c u h ,where A = {al,a2,-.. ,ah} C {1,2,..- ,m} with a1 < a2 < < ah. For A = 0, eg = eo is the identity element of C,. An arbitrary element a of C, may then be written as a = C A a A e A with U A E c. The multiplication in is governed by the basic rules ejek ekej = -26jk, j , k = 1 , 2 , . . . ,m, with b j k the Kronecker delta. A bar map in C, is defined by ii = C Ai i ~ i ?where ~, EA is the conjugate of a~ , CA = Eah .. .b,,E,, and E j = - e j , J' = 1 , 2 , . . . ,m. The norm of a is defined by la( = (CA1 ~ ~ 1 ~ )Note ~ / that, ~ . in order t o obtain an algebra-norm in C,, we define (a(o = 2m/21a(, a E C,. Then we have lablo i lalolblo.
c,
+
In the sequel, we identify each point x = (x1,x2,. . . ,xm) E IW" with x= xjej E Rojm. We have x2 = -(xI2,xy = -x.y+xAy for x,y E Rotm, where x . y is the inner product of x and y, x A y is the wedge product of x and Y, x A Y = &
cj"=,
Cj"=,
Let D stand for the Dirac operator, D = ejtI/tIxj. Then D2 = -A, A being the Laplacian over R". The fundamental solution E of D is given by E ( x ) = -x/(A,JxJ"). In what follows, we denote a unit ball and a unit sphere in R" by B and
236
Zhenyuan Xu
Sm-', respectively. In [3], the following boundary value problems have been solved for the Dirac operator: Dw(2)= 0
,2EB ,
(1 + iw)(w(w)- f ( w ) ) = 0 , w E S"-l , or
Dw(2) = 0 , z E B (1 - iw)(w(w)- f ( w ) ) = 0
,
,wE
sm-l ,
+
Notice that (1+ i w ) ( l - i w ) = 0, for w E Sm-', which means that 1 i w and 1 - iw are zero divisors, thus inspiring us to call the above boundary value problems half Dirichlet problems for the Dirac operator in the unit ball. It has been shown that under some assumptions on f, these problems have a unique solution. In this paper we study half Robin problems for the Dirac operator in the unit ball of R" ( m 2 3). We look for a C,-valued function ~ ( 2satisfying )
Dw(2)= 0 , 2 E B
(1 + Z w ) ( W dn
,
+ Xw(w) - f(w)) = 0 , w E S"-'
,
or
Dw(2)= 0 , 2 E B
,
(1 - i w ) ( W + Xw(w) - f ( w ) ) = 0 , w E S"-l
dn where X is a constant in C and f ( w ) is a given C,-valued d / d n stands for the outer normal derivative on S"-l.
,
function on Sm-',
If a C,-valued function ~ ( zE)C'(B;C,) satisfies Dw(2) = 0 in B, then W(X) is said to be left monogenic in B. For the sake of simplicity, we call it monogenic in B. A monogenic function in B automatically belongs to C"(B; C,).
2 Terminology and Notations Definition The restriction s k ( W ) for a homogeneous harmonic of degree k to the unit sphere S"-l is said to be a spherical harmonic of degree k , i.e., (1)
Spherical harmonics are used to construct an orthogonal basis for the Hilbert space L2(Sm-l), provided with the standard inner product and norm
Half Robin Problems for the Dirac Operator
ds(w) being the Lebesgue measure on
237
S"-l and A, the area of S"-l.
The Poisson kernel is
and the Poisson integral is defined as
As is well known, for f ( w ) E C ( S m - l ) , h ( z ) is harmonic in B extended to Sm-', and h(z)lsm-l= f ( w ) .
, h ( z ) can be
For f ( w ) E L2(Sm-'), h ( z )is harmonic in B and lim h(rw) = lim P f ( r w ) = f ( w ) rfl
rfl
(5)
in the L2(Sm-l) sense. The Poisson kernel can be decomposed in spherical harmonics 00
where
(7) are spherical harmonics of degree k in z and Ck ( m - 2 ) / 2 ( t ) are Gegenbauer polynomials which are given by the generating function (1- 2 t r + ~ ~ ) - ( " - ' ) / ~ expanded in r , i.e.
We may then decompose f E L2(Sm-') in spherical harmonic by means of the formula 00
f(u) = where
Skf("J) 9
k=O
238
Zhenyuan Xu
Furthermore, the operators S k : f on L2(S"-l) satisfying
f are orthogonal projection operators
HSk
00
1= C S k . k=O
Definition The restriction Pk(w) of a homogeneous monogenic polynomial of degree k in W" to the unit sphere Sm-l is said to be inner spherical monogenic of degree k, i.e., D(T"k(W))
= 0 for 2 = TW E
R",
w =
X
1x1
+
The restriction Q k ( W ) of a homogeneous monogenic function of degree - ( m k - 1) in R"\{O} to Smelis said to be outer spherical monogenic of degree k, i.e., D(r-(m+k-l)
Q ~ ( w )= ) 0 for
2
= TW E W"\{O},
w=
X
-
I4 .
square integrable function space L2 (S"-l; Cm)with
Consider the C,-valued
We denote the C,-valued function space of inner spherical monogenics of degree k and outer spherical monogenics of degree k by M+(m;k ) and M - ( m ;k ) , respectively. Then we have
M+(m;k ) l M - ( m ; k ) , M+(m;k ) l M + ( m j;) , M - ( m ; k ) l M - ( m ; j ) ,k # j We denote the C,-valued H(m;k). We have
function space of spherical harmonic of degree k by
H ( m : k ) = M + ( m ;k ) @ M - ( m ; k i.e., for each
sk
.
- 1)
,
E H(m;k ) , we have
Moreover,
where r is the spherical Dirac operator, which may be represented in the form
Half Robin Problems for the Dirac Operator
239
in rectangular coordinates. In particular, we get
The first and second terms on the right-hand side of the above formula are inner spherical monogenic of degree k and outer spherical monogenic of degree k - 1 respectively. Thus we get, for each f E L2(Sm-'; C,),
where
are inner spherical and outer spherical monogenic of degree k respectively. We also have M
P f ( x )=
rk(pkf(w)
+Qk-if(W))-
(12)
k=O
Here Q - l f ( w ) is regarded as 0.
3 Half Robin Problems Now we consider the half Robin problems. We look for a C,-valued function w(2) satisfying Dw(z)= 0 , 2 E B , (13) (1
+ h)(* an + Xw(w) - f(w))= 0 , w E sm-l,
(14)
or
Dw(z)=0 , z E B , (1 - iW)(*
dn
+ Xw(w) - f ( w ) ) = 0 , w E s"-l
,
(16)
240
Zhenyuan Xu
where X is a constant in C and f ( w ) is a given C,-valued d / d n stands for the outer normal derivative on S"-'.
function on Srn-l,
By (13) we may suppose that the solution to the problem (13), (14) is in the form 00 = C P k ( 2 ), 2 E
W(.)
B ,
(17)
k=O
where Pk(z) is homogeneous monogenic polynomial of degree k to be determined. Let r = 1x1,z = T W , w = z/lxI, then Pk(z) = rkPk(w),where P,(w)is an inner spherical monogenic function of degree k . If the series
c
c
k=O
k=O
00
00
w(z) =
Pk(Z)=
TkPk(W),
2
E
B ,
(18)
and its derivative are uniformly convergent in B and the series (9) is uniformly convergent on Sm-', it follows from (13) that
k=O
k=O
k=O
Expanding the above gives 00
+
C [ k P k ( w ) XPk(w) - Pkf(w)- Q
+i ~ k f ' k ( ~ )
k f ( ~ )
k=O
+ i ~ X P k ( w) i w P k f ( w ) - i w Q k f ( ~ )= ]0. Since P~(w), Pkf(w)are inner spherical monogenic of degree k and Q k f ( w ) is outer spherical monogenic of degree k, thus w P , ( w ) , w P k f ( ~ are ) outer spherical monogenic of degree k and w Q k f ( w ) is inner spherical monogenic of degree k. We then have
+
+
- Q k f ( ~ ) i w k P k ( ~ ) ZWXP~(W) - i ~ P k f ( w= )0. Notice that the above two equations are actually equivalent, so we have
Theorem 1 Given f E Lz(S"-l; C,), if X # -ko, ko = 0 , 1 , 2 , . . . , the problem (13), (14) has a unique solution and the solution is given b y an explicit formula
Half Robin Problems for the Dirac Operator
241
where P k f ( w ) and Q k f ( w ) are the functions expressed in the formula (10) and (11) respectively. The boundary condition (14) is satisfied as lim,.tl in the sense ofL2(Srn-l;Cm).
A necessary condition for the solvability of the problem (13), (14), if X = -ko, ko =0,1,2,...1is
P,,f ( w ) + i u Q k o f
(w) = 0
,
i.e.
If the condition (21) is satisfied, then the solution has the f o r m m
1
W(z)=
x+[pkf
(w)
+ iwQkf ( W ) ] r k + r k o p k o ( w ) ,
(22)
k=O,k#ko
where P k o ( w ) is an arbitrary inner spherical monogenic function of degree ko. Proof If X # -ko, ko = 0,1,2,. . . , let ~ ( zbe) the function defined by (20), where P k f ( W ) and Qk f ( w ) are defined by (10) and (11) respectively. Let Pf (z) be the function defined by (12).Then
k=O m
1t i w
k=O
+[krk-'iw 1 iw oc)
+
+ Xr'iw krk-l(l
- (A
+ k)r"']Qk
f ( w ) } + (1 + i w ) ( P f (z) - f(w)).
- r)Pkf ( w )
k=l +
g{ +
k=O
(1 i w ) ( i w ) [krk-l X+k
+ X r k ] Q k f ( w ) - (1 + iw)r"'Qrcf
(w)}
242
Zhenyuan Xu
1
+ iw [kr"' + Ark - (A + k)rk+']Qkf(w) + ( 1 + i w ) ( P f ( z )- f(w))
k=O
It is clear that the condition ( 1 4 ) is satisfied in the sense of L2(Sm-l; Cm) as lim,Tl. Similarly the theorem holds for the problem ( 1 5 ) , (16).
Theorem 2 Given f E L2(Sm-l; Cm),i f X # -ko, ko = 0 , 1 , 2 , . . . , the problem ( 1 5 ) ,( 1 6 ) has a unique solution and the solution has a n explicit formula
where Pk f ( w ) and Q k f ( w ) are the functions expressed in the formula ( 1 0 ) and ( 1 1 ) respectively. The boundary condition ( 1 6 ) is satisfied as lim,T1 in the sense of L2(Sm-l; Cm).
A necessary condition for the solvability of the problem ( 1 5 ) ,( 1 6 ) , if X k o = O , 1 , 2 ,..., is
= -ko,
If the condition ( 2 4 ) is satisfied, then the solution has the f o r m 00
W(Z) =
k=O,k#ko
1 -[Pkf(w) X+k
+ rkoPko(w)
- iwQrcf(w)]rk
(25)
where Pko( w ) is a n arbitrary inner spherical monogenic function of degree ko. Remark For X = 0, the problem (13),(14)becomes the half Neumann Problem
Half Robin Problems for the Dirac Operator
Dw(z)= 0, x E B (1
+ iW)(*
,
- f ( w ) )= 0 , w
an By Stokes' theorem we should have
A[(l
Since A[(l reduced to
km-l s,
=
-[(l
E
sm-l.
+iz)w(x)]ds #
+ iz)w(rc)] = 2iDw(z)+ (1+ i z ) Q w ( x )= 0, the above equation is a
0=
a
+ i z ) ~ ( ~ ) ]=d V
243
-[(l an
+ i z ) w ( x ) ] d s=
iDw(z)dV
+
+ + iw)-]ds W W an
[ ( i w ) w ( w ) (1 L
-
1
)
+
(1 i w ) f ( w ) d s =
Since C,"L'2(w. <) = 1 , and C_"j2((w. [) is regarded as 0, the above condition coincides with the condition (21) for Ica = 0.
References 1. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Pitman, London, 1982. 2. Delanghe, R., Sommen, F., Soucek, V.: Clifford Algebra and Spinor-Valed Functions, A Function Theory for the Dirac operators, Kluwer Academic Publishers,
1992. 3. Delanghe, R., Sommen, F., Xu, Z.: Half Dirichlet Problems for Power of the Dirac Opertor in the Unit Ball of R"(m 2 3), Bull. SOC.Math. Belg-Tijdschr. Belg. Wisk Gen. 42(1990) 3. Ser. B pp. 409-429. 4. Begehr, H., Xu, Z.: Nonlinear Half-Dirichlet problems for first order elliptic equations in the unit Ball of R" (rn 2 3 ) , Applicable Analysis, 45 (1992), pp. 3-18. 5 . Wawrzynczyk, A.: Group Representation and Special Functions, Reidel, Dordrecht, 1984.
Problems Related to the Analytic Representation of Tempered Distributions C. Carton-Lebrun Department of Mathematics University of Mons-Hainaut B-7000 Mons Belgium christiane.lebrunQumh.ac.be Summary. Using results of previous studies concerning Hilbert transforms and Cauchy transforms in weighted spaces of integrable distributions, we propose a constructive real variable approach to several problems related to the analytic representation of tempered distributions. To that purpose, we show that the space of real valued tempered distributions is the inductive limit of an increasing sequence of subspaces defined in terms of Hilbert transforms. This method provides extensions of classical results and alternative proofs of known results.
1 Introduction The purpose of this article is to propose a constructive real variable approach t o several problems related t o the analytic representation of tempered distributions. We adopt the usual notation and definitions concerning spaces of functions and distributions. For given f in the space S' of all tempered distributions, we say that F : C \ R + C is an analytic representation of f , if F is analytic in C \ R and
+
lim [ F ( . i y ) - F ( - - i y ) ] = f y-O+
in the topology of S'. In [8, Section 11,Tillmann proved the following fundamental results concerning analytic representations of tempered distributions:
Theorem 1.1 [8]
(i) Suppose f E S'. Then, there exists analytic representations of f. These can be chosen in the class
246 (aa)
C. Carton-Lebrun
If F E 7, then the boundary values
exist in the topology of S'. The mapping @ = F --f f = F' - F- is thus surjective from 'T onto S'. (zii) Suppose f E S'. If F1 E 7, FZ E 'T are two analytic representations off , then F'1 and F2 diger by a polynomial. The space S' is thus isomorphic t o the quotient space 'TIP, where P as the space of all polynomials. In the present paper, our aim is to obtain explicit Plemelj relations, and thus explicit analytic representations, for any given f E S'. The method we propose is based on results of previous studies [l],[2] in which we defined and investigated Hilbert transforms [l]and Cauchy transforms [2], in the context of the scale of spaces
v,l=Wk++lVL, = { f E S' : w-k-1 f E VLl } ,k E N (1.2) where ~ ( z = ) (1 + z2)1/2and VL, is the space of integrable distributions. If k = 0, Vd is the space considered in [6], by Schwartz. In [6], Schwartz proved that the Hilbert transform H f of f E wV;, defined by S' Wf, +)s = - v;, V f , W ( H + ) ) B , 7 E 5 6.
+
can be (1.3)
+.
where H+ denotes the classical Hilbert transform of Also, the operator H : f H Hf is continuous frow wDL, into S' and coincides with the usual Hilbert transform on the subspaces &' and V i p(1 5 p < cm)of wVL, [6]. In [l],we utilized the above definition of H to define more general Hilbert transforms in the context of the scale of spaces V,l, k E N. For given k E N and index m 2 k, the Hilbert transform Sn,,f o f f E V,l is defined in [l]by
where ( E D ,0 _< ((z) _< 1, ((z) = 1 in a neighborhood of the origin and 77 = 1- (. In case k = m = 0, S,,o coincides on Vd = wVLl with the operator H defined by (1.3). S,,O is thus independent of 7. If f = b E B M O , then S,,lb coincides with the Hilbert transform defined by Fefferman in [4].Also, S,,l(l)= HE z H ( f ) is a constant function depending on the partition of unity (E, 7 ) . Note also that the difference (S,,,,,f - S,,,,, f ) of two Hilbert transforms of f E V,l with 0 < k 1 5 inf(m1, m2), is a polynomial that can be determined explicitly in terms of 71,q2,ml,mz,k and f ([1, p.1521). Main results in [l]are concerned with mapping properties, continuity, pseudoinversion and composability of the Hilbert transforms defined by (1.4). In [2], we defined, in the same context of spaces as above, the Cauchy transform of index m, o f f E V,l, 0 _< k _< m, by
+
+
Analytic Representation of Tempered Distributions
where Co : g
H
247
Cog is defined on Vd by
+
for z = 2 iy, y # 0. The main results in [2] are concerned with boundary values of Cauchy transforms, Plemelj relations and mapping properties of related operators in weighted topologies. We mention here that two types of results are given in [l]and [2]. For the sake of clarity, we will say that a result is of homogeneous type if it holds in a subspace of the form
Gb,k = {f E
v; : Sq,kf E VL},k EN.
(1.6)
On the other hand, results that are obtained in the context of the scale of spaces k E N,e.g. continuity of Sq,k from V; into V;+', will be referred to as results of non-homogeneous type. In the present article, our purpose is to show that in a very specific sense, all results obtained in [l] [2], are applicable to any given f E S'. Actually, we show that for each given f E S', it is possible to associate with that f , an index k = k[f] E N such that all results of non-homogeneous type obtained in [l],[2] will be applicable to that f . Furthermore, to obtain results of homogeneous type, for that f also, we show that the scale V;, k E N,can be refined as follows: Gb C v d C Gb,l C C Gb,k C v; C Gb,k+1 , k E N, (1.7) with continuous embedding of each space into the subsequent one. This paper comprises 3 Sections: In Section 2, we give a brief survey of results obtained in [l],[2]. In Section 3, we give extensions of these results to distributions in S'.
v;,
-
a
*
2 Hilbert Transforms and Cauchy Transforms in
Weighted Spaces of Integrable Distributions We use the notation given in [l]and [2] for function and distribution spaces. We refer to [5],[7] for the study of ,130,a, DL-,D L ~D, i p , 1 5 p , q 5 00 and to [3] for other characterizations of Dil. As in [2], we consider, for w(z) = (1 + z2)lI2and k E N,the space
,- {+ = W - k - l
v k = w-k-1,13
cp : cp E
a,}.
(2.1)
+
This space is endowed with the topology induced by the map + wk+'+ : vk 23,. The dual of v k is V; = wk+'DLl. Unless stated otherwise, DLl and V,lare supposed to carry the strong topology ,8(DLl;BC),,8(Vk);Vk).For the .--$
248
C. Carton-Lebrun
definitions of these topologies and weaker ones, we refer to [9], for instance. The following notation from [l],[2] will be used too : The integral off E Dkl is (f?1). Its “normalized moment” is denoted M(f ) = (1/7r) (f,1). A sequence <j is an approximate unit, for short .$ E AU, if Cj E D for each j , & converges to 1 in Bc and if for every compact K E R, there exists J E N such that (j(x) = 1 for all x E K , j 2 J . A distribution f belongs to DLl if and only if, for any <j E AU, limj+m( f , &) exists. In the latter case, lirnj+m( f , (j) = ( f , 1). We mention also that we write E E U ( 0 ) if E E D,0 5 c(x) 5 1, E(x) = 1 in a neighborhood of the origin. We write q E U(o0)if q = 1 for some ( E U ( 0 ) .
-<
Results of Non-Homogeneous Type for Hilbert Transforms The following results of non-homogeneoustype are concerned with the Hilbert transforms defined by (1.4). We refer to [17pp.152-1551 for the proofs of these results and related corollaries. Lemma 2.1 Let f E V,. Then, f o r m 2 k + 1, and q E U(m),
<
Theorem 2.2 (Continuity. Duality). Let k E N, E U(O),q = 1 - E. Define Tq,k O n v k + 1 = W - k - l a c , by
+ %H(tk’$), X
Tq,k’$= t(H’d’)
’$ E
vk+1-
Then, 2(‘)
for every f E V , vL+,(Sq,kf,@)Vk+i
(ii)
sq,k
= -vL(f,Tq,k’$)Vk
v’d’ E v k + 1 *
is continuous from V, into VL+l.
Theorem 2.3 (Pseudo-inversion). Suppose E E U(O), q = 1 - E.
(i) Iff E Vd, then
(ii)
If k 2 1, f
E V,, then
(2.3)
Analytic Representation of Tempered Distributions
249
in VL+l, where
Theorem 2.4 (Associativity). Iff E V,l, then for any ( E U(O), 11 = 1 - E,
Sv,k+l (Sv,k+lSn,kf)= ( s v , k + l s v , k + l > ( S v , k f ) Results of Non-Homogeneous Type for Cauchy Transforms The following theorem is concerned with the main results of non-homogeneous type, concerning the Cauchy transforms defined by (1.5). These results are based on a preliminary study given in [2, Sections 3,4] concerning Poisson and conjugate Poisson operators, and their boundary values in weighted topologies. We refer to [2, Section 51, for the proofs of the results given hereafter in Theorem 2.5. In the following statement, Sq,of and C,,O stands for Hf and Cof respectively, when f E Vd. Theorem 2.5 (Analyticity, Boundary Values of Cauchy transforms, and Plemelj relations). Let f E VL, [ E U(O), 17 = 1 - (. Then, (i) c q , k f E E(@ \ R), (ii) for each y # 0 , c q , k : f H Cq,kf ( . i y ) is continuous from V,l into Wk++2DL 1; (iii) Cq,kf + i y ) admits the boundary values
+
(a
in the topology of VL+l. (iw) cTk 7 . : f -+ ( c q , kf ) + and v k znto V,l+l.
C-%k : f
-+ ( c q , kf ) -
are continuous from
We mention here that a number of algebraic relations of non-homogeneous type for products of operators S,,,, C&, C& can be deduced from Theorem 2.5. For instance, for k 2 1, 77 E U ( c o ) ,f E VL, f real valued, the following relation holds in VL,, :
where
250
C. Carton-Lebrun
and where Gjr is defined as in Theorem 2.3 (ii). We refer to [2, p.1071 for more details about these explicit relations.
Results of Homogeneous Type for Hilbert Transforms The following results hold for the restriction s 7 , k IG&,kof the Hilbert transform S7,k.f o f f E Gh,k : Theorem 2.6 [l,Theorem 3, p.1591 (Inversion in G&,k). 1. Suppose f E Gb. Then,
HWf) = -f and for every u E Go,
where Go = {u E VO: Hu E Vo}. 2. Let E u(O),7 = 1 - (, k 2 1. Iff E GQ,klthen
and for every $ E G7,kl
Results of Homogeneous Type for Cauchy Transforms The following results holds for the restriction C,,klGb,k of the Cauchy transform Cq,kf o f f € Gh,k,k € N.This result shows that Plemelj type relations quite similar to the classical Plemelj relations in LP and DLp (1 < p < 00) hold in Gb,k. Theorem 2.7 [2, Theorem 6.21 (Plemelj relations in Gh,k.) Let k
2 1, 7 E
w4. (2)
For y # 0, C7,k,Y(Gh,k: f Wk+lVLI.
(ii) I f f E Gh,k, then
H
c 7 , k f (. -k Zy) iS C O n t i n U O U S
f’Om Gh,k into
Analytic Representation of Tempered Distributions
251
From Theorem 2.7, one deduces a number of algebraic relations for products of operators. These hold in G;,k for distributions in G;,k. Typical examples are : H(C$ f ) = -iC$f ; H(C; f ) = iC{ f Vf E Gb.
For k 2 1, q E U(Oo), f E
G;,k,
S,,kCq&., f) = -
q k f
c;,(q,f)= q
k f
- i R 4 . f ;77;
1
+ fk-l(f;
k ;4
7 ; k;4
Vf
Vf
E G;,k
E G;,k
where P k - 1 ( f ;q; k ; x) is defined by (2.4). Other related relations can be found in [2, Corollaries 6.3, 6.41.
3 Plemelj Relations for Tempered Distributions We first prove two results concerning the scale of spaces VL, k E N . Proposition 3.1 The space S' is the inductive limit of the spaces VL, k E N. Proof Suppose f E S' and f j converges to f in the topology of S'. We have to show that there exists k = k [ f ] E N, such that gj = f - fj converges to zero in the topology of V'k[f1 * By [7, p.2401, there exists r E R such that w+gj converges to zero in the topology of Dim.Also, we deduce from [7, p.2031 that for any u > 1, w-r-agjconverges to zero in the topology of DLl. Choosing u > 1 such that r + u E N, we see that gj converges to zero in the topology of V, = wk+lDL1, for k = r u.This ends the proof.
+
Theorem 3.2 For each given q E U(oo), the space S' is the inductive limit of the refined scale Gb C Vd C G;,1 C
- . *
C G;,k c VL c G;,k+l,
k E N,
with continuous inclusion of each space into the subsequent one. Proof Let k E N. It is clear that G$,kis continuously embedded in VL. To show that V,l is continuously embedded in G;,,+,, we first note that i f f E V,l, then f E VL+l. Furthermore, it follows from Lemma 2.1, that if f E V,, then
where S q , k f E vL+land zk E vL+l.As a consequence, Sv,k+lf belongs to vL+l and thus f E G;,,+,. Furthermore, if f m E V, converge to f in the topology of Vi, then Sq,k f m converges to Sq,kf in the topology of V,+l. Furthermore, since
(%)converges to
in D i l , the constant functions M
252 to M
C. Carton-Lebrun
(&)
in the topology of w2D>, , and as a consequence xkM
{
>>
>>
(@
x k M & in the topology of VL+l. From this and what { ( precedes, we deduce that V,l is continuously included in Gk,k+!. converges t o
In view of the above theorems, we can extend all results given in Section 2, as well as related results given in 111, [2] to any given f E S’.As an example, the following theorem, concerning analytic representations of homogeneous type holds :
Theorem 3.3 Let
E E U(O), q = 1 - E . Let f E S’. Then,
(i) there exists k = k [ f ] such that f E G;,,[,]. (ii) f admits analytic representations of the f o r m :
($) ( z ) , z E
(Cq,mf>(z= ) ~o(~f)(z) +z m ~ o
\
f o r m 2 k [ f ]2 0. (iii) These analytic representations belong t o the class I of Tillmann and the following Plemelj relations hold in Gb,m, f o r each m 2 k [ f ] :
(iu) If F l ( z ) = (C,,,,, f ) (2) and F2(z) = ( C q 2 , m z(fz)) are two analytic representations off E VL with 0 < Ic 1 5 inf(m1, m2) t h e n Fl(z),F2(z) differ by a polynomial P ( z ) that can be determined explicitly in t e r m s of 71, 7 2 , m1, m2.
+
Proof Assertion (i) follows from Theorem 3.2. Assertions (ii), (iii) follow from Theorem 2.7. Assertion (iv) follows from [2, Lemmas 2,3]. Note here that similar results can be obtained for analytic representations of non-homogeneous type. In the latter case, the results can be deduced from Proposition 3.1 and Theorem 2.5. Also, results concerning mapping properties and algebraic properties of operators S,,,, C$+, C$ can be stated in the context of S’, and proved, by using Proposition 3.1 and Theorem 3.2.
References [l] C. Carton-Lebrun (1995). Hilbert transforms in weighted distribution spaces. In : J. Garcia-Cuerva et al. (Eds.), Fourier Analysis and Partial Differential Equations, pp. 149-162. Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, USA. [2] C. Carton-Lebrun and F. Colacito (2003). Boundary Values of Cauchy Transform in Weighted Spaces of Integrable Distributions. Complex Variables, 48, 95-110.
Analytic Representation of Tempered Distributions
253
[3] P. Dierolf and J. Voigt (1978). Convolution and S'-convolution of distributions. Collectanea Mathernatica, 29, 185-196. [4]C. Fefferman (1971). Characterization of bounded mean oscillation. BuZ1. Amer. Math. Soc., 77,587-588. 151 J. Horvath (1966). Topological vector spaces and distributions, Vol. I. Add'isonWesley, Reading, MA. [6] L. Schwartz (1962). Causaliti. and analyticiti.. A n . Acad. Brasil. Cien., 34,13-21. [7] L. Schwartz (1978). The'orie des distributions. Hermann, Paris. [8] H.G. Tillmann (1961). Darstellung der Schwartzschen Distributionen durch analytische Funktionen. Math. Zeitschr., 77,106-124. (91 F. Trsves (1967). Topological Vector Spaces, Distributions and Kernels. Academic Press, New-York.
Strong Unique Continuation for Generalized Baouendi-Grushin Operators Nicola Garofalo’* and Dimiter Vassilev2 Department of Mathematics Purdue University West Lafayette IN 47907 USA garofaloQmath.purdue.edu CRM/ISM and Universite du Quebec Q Montrkal
Montrkal, Quebec Canada vassilevQmath.uqam.ca Summary. We present a strong unique continuation property for a class of subelliptic “variable coefficient”operator whose “constant coefficient” model at one point is the so-called Baouendi-Grushin operator.
1 Introduction and Statement of the Results We say that a given partial differential operator IC in RN has the strong unique continuation property (SUCP) if every weak solution u of the equation Cu = 0, which vanishes t o infinite order at some z, E RN, i.e.,
must vanish identically in some neighborhood of 2,. In other words non-trivial solutions can have at most finite order of vanishing. In this paper we study the strong unique continuation property for a class of “variable coefficient” operators whose “constant coefficient” model at one point is the so called Baouendi-Grushin operator [B], [Grl], [Gr2]. We recall that the latter is the following operator on RN = Rn x R”, N = n m,
+
N
c, = c x i x a u , a= 1
where the vector fields are given by *The first author was supported in part by NSF Grant No. DMS-9706892.
256
Nicola Garofalo and Dimiter Vassilev x k =
a
-, k = l , ..., n, dxk
Xn+j
=I
a
,
.
3 = 1,...,m.
(2)
Here a > 0 is a fixed parameter, x = ( 5 1 , . . . ,x") E R" and y = ( y i , . . . , y m ) E R". When a = 0, C, is just the standard Laplacian in RN.For a > 0 the ellipticity of the operator Lo becomes degenerate on the characteristic submanifold M = R" x (0) of RN. The SUCP for the operator C, was proved in [G2]. In the same paper --t
this is also proved for the operator Cu+ +
assumptions on
V1
< V1,Du > +V,u
= 0 with suitable
and V,. To give an idea, for example
+ C -$ and I < V1,Du > I 5 C ~ X U ~ $ ~ / ~ P is enough (note the use of an a priori estimate on the gradient Du in the above conditions). Here Du is the gradient of u,X u is the horizontal gradient (7) of u,and p and $ are defined correspondingly in ( 5 ) and ( 6 ) . With a completely different method, based on a subtle two-weighted Carleman estimate, the sucp was established in [GSl] for zero order perturbations Lo - V, when a = 1 and y E R (i.e. m = l), where the potential V, is allowed to belong to some appropriate LP spaces,
IV,l
I
vo E L;,, p > Q - 2 if n-even, p > if n-odd. The operator for which we prove the SUCP is the following
c N
cu =
Xj(Uij(X,Y ) X i U ) = 0.
t,j=l
(3)
We assume that A = ( a i j ( x ,y)), i , j = 1,...,N , is a N x N matrix-valued function on RN which, for simplicity, we take such that A ( 0 ) =Id. Furthermore, we assume A is symmetric and uniformly elliptic matrix. Thus aij = ajiand there exists X > 0 such that for any 77 E RN
Xl77I2 I < A(X,Y)77,7 > IX-llr1I2.
(4)
Our main concern is whether, under suitable assumptions on the matrix A , the sucp continues to hold for the operator C.To put our result in perspective we mention that when a = 0 in (2), so that Lois the standard Laplacian, we have the following well-known SUCP due to Aronszajn, Krzywicki and Szarski [AKSI
Theorem 1.1 Suppose 0 i s a connected open set in R". Let u E W,1,',2( 0 ) be a weak solution of Lu = C a i ( a i j ( z ) a j u )= 0 , where A ( z ) = ( a i j ( z ) ) is a symmetric, positive definite matrix with Lipschitz continuous entries. If there i s point zo E 0 at which u has a zero of infinite order in, then u = 0 a.e..
SUC for Bmuendi-Grushin Operators
257
Furthermore, it was shown in [MI that such an assumption is optimal. The case n = 2 is exceptional since, according to a result of Bers and Nirenberg, the SUCP holds for bounded measurable coefficients. Our results, Theorems 1.3 and 1.4 can be seen as a generalization of Theorem 1.1, in the sense that, in the limit as Q --t 0 we recapture both the assumptions and the conclusion of the elliptic case, see Remark 1.3. The approach, however, is different from that in [AKS], which is based on Carleman inequalities along with results from Riemannian geometry that do not seem to be adaptable to our context due to the lack of ellipticity. Instead, we have used the ideas developed in [GLl], [GL2], [G2], and simplified in [K]. Our main result is Theorem 1.3, which gives a quantitative control of the order of zero of a weak solution to (3). Such result is proved under some hypothesis on the matrix A which are listed as assumptions (H) below. The latter should be interpreted as a sort of Lipschitz continuity with respect to a suitable pseudo-distance associated to the system of vector fields (2). In what follows we let ( = (z,y). To state our main result, we recall the following gauge from [G2] associated to the operator C, p = p(()
dsf( I Z p + l ) + ( a + 1)21y1”-.
(5)
Let B,. = { p < r } be the pseudo-balls with respect to p centered at the origin in EN with radius r. It is worth stressing that if Q is an even positive integer, then the Carnot-Carathdodory distance associated to the system of vector fields in (2) is comparable to p ( ( ) . We will also need the angle function $J defined as follows [G2]
Hereafter, given a function f, we denote the gradient along the system of vector fields in (2) (called also horizontal gradient)
Xf
= (Xlf, ...,X N f )
(7)
and let lXfI2 = CF1(Xjf)z. The function $ vanishes at every point of the characteristic manifold M , and clearly satisfies 0 5 $ 5 1. Definition 1.2 A weak solution to C u = 0 in a n open set 0 i s a function u E C ( 0 ) such that the (distributional) horizontal gradient Xu E Lfoc(f2), and the equation C u = 0 is satisfied in the variational sense in 0 , i.e.,
LdV = 0 f o r every
4 E CF (Q) .
For convenience, we have required that a weak solution be a continuous function since we will take traces on hypersurfaces. We note however that such
258
Nicola Garofalo and Dimiter Vassilev
an assumption could be considerably relaxed if one assumes the existence of sub-unit curve joining any two points. Under this additional hypothesis, the assumption u , X u E L;,,(R) would suffice to apply the results in [FL], [FS], and conclude that a weak solution u is (after modification on a set of measure zero) Holder continuous with respect to the Carnot-Carathdodory distance, and therefore (with a different exponent) also with respect to the Euclidean distance. Of course, when a is an even positive integer the system of vector fields is smooth and satisfies the Hormander finite rank condition. In this case, the existence of a sub-unit curve joining any two points follows from the theorem of Chow-Rashevsky.
Theorem 1.3 Let A be a symmetric matrix satisfying (4)and the hypothesis (H) below with relative constant A . Suppose u is a weak solution of (3) in a neighborhood of the origin R. Under these assumptions, there exist positive constants C = C(u,a , A, A , N ) and r , = r,(u, a , A, A , N ) , such that, for any 2r 5 r,, we have u2$ dV
IC
L,
u2$ dV.
The dependence of the constant C on u is quite explicit. As it is well known [GLl], Theorem 1.3 implies the following sucp.
Theorem 1.4 With the assumptions of Theorem 1.3, the operator C has the SUCP. We have stated the above theorem when the point of vanishing is the origin. Obviously the result is true for any other point with the appropriate modification of the hypothesis (H). In order to state our main assumptions (H) on the matrix A it will be useful to think of the latter in the following block form,
All A12 A = (Ax Az2) . Here the entries are respectively n x n, n x m, m x n and rn x m matrices, and we assume that At2 = A21. The proof of Theorem 1.3 relies crucially on the following assumptions on the matrix A. These will be our main hypothesis and will be assumed to hold throughout the paper.
HYPOTHESIS 1.5 There exists a positive constant A such that one has an B, for some E > 0 the following estimates (Ap,
for 1 < a , j I n
SUC for Baouendi-Grushin Operators
259
A simple, yet interesting example of a matrix satisfying the conditions (H) is
where f,g and h are functions which are Lipschitz continuous at the origin of Iw2 with respect to the Euclidean metric. Here, n = m = 1.
2 Monotonicity of the Generalized Frequency We begin by introducing the relevant quantities. Since our results are local in nature, from now on, we focus our attention on a pseudo-ball BR,, centered at the origin and such that u is a weak solution of C u = 0 in BR,, in which (4) and the hypothesis (H) hold. For 0 < r < R, we define correspondingly the height H ( r ) and the Dirichlet integral D ( r ) of u on the pseudo-ball B,
H(r) =
D(r)
lBT
u2 < A X p , X p >
=
k.
do
lDPl
< A X u , X u > dV.
Consider further the frequency function
The following two lemmas are the key to proving the monotonicity of the modified frequency R ( r ) = N ( r )e2Mr, where M > 0 will be suitably chosen. In the sequel we shall briefly sketch the main steps in their proofs, referring to [GV] for full details. Lemma 2.1 a) There exists a positive constant C1 = C l ( a , A , A , N ) such that for a.e. r E (0, R,) one has
b) There exists a positive number r, = r,(a, A, A, N ) 5 R, such that either H ( r ) = 0 on (0, r,) or H ( r ) > 0 on (0, r,).
260
Nicola Garofalo and Dimiter Vassilev
The proof involves a lengthy computation with the use of integration by parts, the co-area formula and the fact that, up to a constant, p2-Q is the fundamental solution of C, with singularity at (0,O). Here, Q = n (a l ) m is the homogeneous dimension corresponding to dilations with infinitesimal generator (radial vector field)
+ +
Lemma 2.2 There exists a constant a.e. r E (0,R,) one has
where p
C2 =
C2(a,A, A, N )
> 0 such that for
def< A X p , X p >.
To prove this lemma we use the co-area formula and Rellich's identity, which imply
< A X U , X ~ >d a~ P
1
JDpJ +
< A X u , [ X ,Flu > +;
1
Lr
( d i v F ) < A X u , Xu >
< ( F A ) X u ,X u >,
with
We show that F can be extended continuously to a vector field near the origin. In fact, such extension is sufficiently smooth. The vector field F should be thought of as a small perturbation of the radial vector field. A computation shows that we can estimate the terms in the right-hand side of the above identity as required. The above two lemmas imply the following monotonicity theorem
Theorem 2.3 Suppose u is a solution of Cu = 0 in a neighborhood of the origin in which (4) and the hypothesis [H) hold true. Under these conditions there exist positive constants r, = r,(u,a,A,A,N) and A4 = M ( a , A , A , N ) such that &(r) = ~ ( r l e x p ( 2 ~ r ) is a continuous monotone nondecreasing function for r E (0, r,).
SUC for Baouendi-Grushin Operators
261
Proof The proof of Theorem 2.3 follows from lemmas 2.1 and 2.2. Let Cz}, where C1 and C2 are the constants from Lemmas 2.1 and 2.2. With r, as defined in Lemma 2.1 we have that either u = 0 in By,or H ( r ) > 0 for 0 < r < r,. In the first case the frequency is identically zero on (O,r,) so let us consider the second case for which H(r) > 0. The continuity of R(r)follows from the continuity of each of the functions involved in its definition. Furthermore, for a.e. r E (O,r,) we have
M
= max(C1,
(In
rD(r)e2Mr H (r) ) I
1 D’ H’ = ; + - -D - + 2 M H
2D ->0,
-I - -Q r
H-
where we have applied first lemmas 2.1 and 2.2, and then the Cauchy-Schwarz inequality. The reader should keep in mind the following formula
3 Proof of the Doubling Property In this section we shall prove Theorem 1.3. If the solution vanishes in some neighborhood of the origin then the doubling for all sufficiently small balls is trivially satisfied. Let us consider next the case of a non-trivial solution. Let r, be the number defined in Lemma 2.1 and 2r 5 r,. By the co-area formula
From the ellipticity of A we have
iR
H ( r )d r x
LR
u21C,dV,
which shows it is enough to prove the doubling property for the height function H. Now we compute
5
12r <
e-2Mt
2r
2N(t)-
{2%
+ M } dt
2fi(r0)
IzT + dt
T
=
M
=
t
dt
2fi(~,)ln2
+Mr
+ M.
262
Nicola Garofalo and Dimiter Vassilev
In the above inequalities we have used the monotonicity of the modified frequency. Hence ~ ( 2 r5 ) 2Q-le2fl(ro)In2 + M H (4 i.e., the doubling property holds.
Remark 3.1 W e observe that f o r non-trivial solution we have the doubling property f o r all balls B2r c 0 and 2r 5 R, where R > 0 i s a fixed number, since f o r ”big” balls, i.e., 2r 2 r, we have
Of course, in this case the constant C in the doubling property depends o n
N(R).
4 Proof of the SUCP In this section we shall prove Theorem 1.4. Suppose u is a solution which vanishes to infinite order at the origin. Let lBTl= w,rQ. Fix a number IC > 0 such that C02-Qn = 1. For any r sufficiently small and p E N the doubling property applied p times gives
k,
when p
3 00.
u 2 $ d V 5 C,P
/
u2$dV
&/2p
This ends the proof.
References [AKS] N. Aronszajn, A. Krzywicki, J. Szarski, A unique continuation theorem f o r exterior differential f o r m s o n Riemannian manifolds, Arkiv for Matematik, 4 (1962), 417-435. [B] M. Baouendi, Sur une classe d’opkrateurs elliptiques degknkrks, Bull. SOC.Math. Fkance, 95 (1967), 45-87. [Ba] H. Bahouri, N o n prolongement unique des solutions d’op6mteurs “Somme de camks”, Ann. Inst. Fourier, Grenoble, 36-4 (1986), 137-155.
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263
[Bo] J. M. Bony, Princape du maximum, ine'galite' de Harnack et unicite' d u probltme de Cauchy pour les ope'rateurs elliptique de'ge'ne're's,Ann. Inst. Fourier, Grenoble, 19-1 (1969), 277-304. [FL] B. Franchi, E. Lanconelli, Holder regularity theorem for a class of linear non uniformly elliptic operators with measurable coefficients, Ann. Sc. Norm. Sup. Pisa, 4 (1983), 523-541. [FS] B. Franchi, R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: A geometrical approach, Ann. Sc. Norm. Sup. Pisa, 1 4 (1987), 527-568. [Gl] N. Garofalo, Lecture notes on unique continuation, Summer School on Elliptic Equations, Cortona, 1991. [G2] N. Garofalo, Unique continuation for a class of ellipic operators which degenerate on a manifold of arbitrary codimension, J. Diff. Eq., 104-1 (1993), 117-146. [GLa] N. Garofalo, E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier, Grenoble, 442 (1990), 313-356. [GLl] N. Garofalo, F. H. Lin, Monotonicity properties of variational integrals, A, weights and unique continuation, Indiana Univ. Math J., 35-2 (1986), 245-268. [GL2] N. Garofalo, F. H. Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366. [GSl] N. Garofalo, Z. W. Shen, A Carleman estimate for a subelliptic operator and unique continuation, Ann. Inst. Fourier, Grenoble, 4 4 (1994), 129-166. [GS2] N. Garofalo, Z. W. Shen, Absence of positive eigenvalues for a class of subelliptic operators, Math. Annalen, 304 (1996), 701-715. [Grl] V. Grushin, O n a class of hypoelliptic operators, Math. USSR Sbornik, 123 (1970), 458-476. [Gr2] V. Grushin, O n a class of hypoelliptic pseudodifferential operators degenerate on a submanifold, Math. USSR Sbornik, 13-2 (1971), 155-186. [GV] N. Garofalo, D. Vassilev, Strong unique continuation of solutions of equations involving generalized Baouendi- Grushin operators, preprint 2003 [K] I. Kukavica, Quantitative uniqueness for second order elliptic operators, Duke Math. J. 91 (1998), no. 2, 225-240 [MI K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Holder continuous coefficients,Arch. Rational Mech. Anal., 54 (1974), 105-117. [RS] L. P. Rothschild, E. M. Stein, Hypoelliptic diflerential operators and nilpotent groups, Acta Math., 37 (1976), 247-320.
Schoenberg's Theorem for Positive Definite Functions on Heisenberg Groups Jinman Kim' and M. W. Wong2 Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada jinman0mathstat.yorku.ca
Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada
.
mwwong0mathstat yorku.ca
Summary. We give a characterization of all continuous functions F on R+ x R such that the function fn on the Heisenberg group MI" defined by f p z ( zt ,) = F(I.1, t ) , ( I ,t ) E W", is positive definite on W", n = 1 , 2 , .. , .
I Introduction The following characterization of radial and positive definite functions on R" is known as Schoenberg's theorem and can be found in the book [3] by Donoghue. The background and applications of Schoenberg's theorem can be found in the papers [lo]and [ll]by Schoenberg.
Theorem 1.1. Let F be a continuous function on [0,CQ). Then the function fn : R" + C defined by fn(2) = F(IxI),
2
E R",
is positive definite on R", n = 1 , 2 , . . . , i f and only if there exists a positive and finite Radon measure u o n [O,oo) such that
Jinman Kim and M. W. Wong
266
We give in this paper a version of Schoenberg's theorem for positive definite functions on the Heisenberg groups W",n = 1 , 2 , . . . . Positive definite functions on the Heisenberg group have been studied by Fischer in [4],and Jorgensen in [8] and [9]among others. The main result is stated and proved in Section 3 after a recollection of some analysis on the Heisenberg group in this section and some preliminary results in Section 2. We identify any point ( q , p ) in R2" with the point z = q + ip in C". We define the symplectic form [ , ] on C" by [ z ,w]= 2 Im(z . w), z , w E C",
(1.1)
where = ( Z l , z 2 , * * . 7 zn),
w = (w1, w2,.. . ,wn),
and
n j= 1
We define the binary operation . on C" x R by
+
( z ,t ) * (w, s) = (2 w,t
+ s + [ z ,w])
(1.2)
for all ( z , t ) and (w,s) in C" x R, where [ , ] is the symplectic form on C" defined by (1.1). With respect to the binary operation defined by (1.2), C" x R is a non-abelian group in which (0,O)is the identity element and the inverse of ( z ,t ) is (-z, -t) for all ( z ,t ) in @" x R. The group C" x R with respect to the binary operation defined by (1.2) is called the Heisenberg group and is denoted by W". It is a locally compact and Hausdorff group on which the left (and right) Haar measure is the Lebesgue measure dzdt on C" x R. For j = 1 , 2 , . . . , we let Zj and
zj
be the partial differential operators
defined by
and
where
- a= L ( L - i L L ) azj
and
2
axj
-='(".i&). a azj 2 axi
267 Schoenberg's Theorem on Heisenberg Groups - Let T = Then { & , 2 2 , . . . ,Z,, 21,Z2,. . . , Z,, T } is a basis for the Lie algebra lj" of the Heisenberg group W", and we have the commutation relations [zjZ,] , = 2i6j,T, j,k = 1 , 2 , . . . ,
6.
and all other commutators vanish. More information on the Heisenberg group can be found in [5] by Folland, [7] by Howe, [12] by Stein, [13] and [14] by Thangavelu, and [15] by Wong among others. Let U ( n )be the group of all n x n unitary matrices. Then the group action of U ( n ) on the Heisenberg group W" is given by
k
*
(2,t)=
(kz,t)
for all k in U ( n ) and ( z , t ) in W". It is easy to prove that ( U ( n ) , W " ) is a Gelfand pair in the sense that Lh(,,(W") given by
L,!j(,)(W")
= {f E
L1(Wn): f ( k z , t )= f ( z , t ) ,( 2 , t ) E
W",k
E
U(n)}
is a commutative algebra with respect to the convolution on W". Let us recall that the convolution f * g of the functions f and g in L1(IHI") is defined by
(f * g ) ( z ,t ) =
Ln
f ( w ,s)g((w,4-1 *
( 2 , t>>dw ds,
(274
E
W".
Functions in L;(") (W")are integrable and U(n)-invariant on W". Let $ : W" -+ C be a smooth and U(n)-invariant function on W" such that $(O, 0) = 1 and $ : W" -+ C is an eigenfunction of D for all D in lj". Then we call $ : MI" --f C a U(n)-spherical function on H".In fact, a smooth function $ : W" -+ C is U(n)-spherical if and only if
s,,
$ ( ( z ,t ) *
(kw,s))dk = $+, t>$(w,s),
(2,
t ) ,( w ,4 E W",
where dk is the Haar measure on the group U ( n ) with JUcn,d k = 1. See, for instance, Chapter 3 of the book [13] by Thangavelu in this connection. A measurable function f : W" + C is said to be positive definite if for all every positive integer m,
c m
f((%
*
( Z j , tj))(iCj
20
i,j=l
for all (z1,t l ) ,( 2 2 , t 2 ) , . . . , (zm, t m ) in W" and all C1, C2, . . . , proved that f : W" -+ C is positive definite if and only if
L
f ( z , t ) ( V * ' p * ) ( z ,t ) d z dt 2 0,
where S(Wn) is the Schwartz space on
W" and
'p E
cm in C. It can be
S(W"),
268
Jinman Kim and M. W. Wong
(p*(z,t ) = (p(-.,
-t),
( z ,t ) E
W".
We denote by A ( U ( n ) , W " ) the set of all bounded, U(n)-spherical and positive definite functions on W". The U(n)-spherical functions q5;.r of type 1 on W" can be written as
where X E Itx,T = 0,1,2,. . . , LF-I is the Laguerre polynomial of order n - 1 and degree T with LF-'(O) = 1. The U(n)-spherical functions 7, of type 2 on W" are given by T J O ( Z , t ) = 1, ( Z , t > E W",
where w E C" and Jn-l is the Bessel function of order n - 1 of the first kind. It is proved in the paper [l]by Benson, Jenkins and Ratcliff that every bounded and U(n)-spherical function on W" is either of type 1 or type 2, and rlW(Z,t)
=
L(")
eiRe(w.kz)dk =
L,,,
eiRe(z.kw)dk 7
(Z,t)
E
W",
for all w in C". Furthermore, in the paper [l]by Benson, Jenkins and Ratcliff, it is proved that every bounded and U(n)-spherical function on W" is positive definite. Thus, A ( U ( n ) , W " ) is the same as the set of all bounded U(n)-spherical functions. We endow the set A(U(n),Wn)with the compact open topology, i.e., the topology of uniform convergence on compact sets. See the paper [2] by Benson, Jenkins, Ratcliff and Worku for details. Let f E LhC,, (W").Then the U(n)-spherical transform of f is the function
f : A ( U ( n ) , W " ) -+ C defined by It can be shown that
f * g($) and
= f($>6($>, $ E
(FW)GT,
A(U(n>, H">,
n(wwn>,
$E for all f and g in L:(,,(W"). Now, using the fact that L,&(n,(W")is a Banach *-algebra with respect to the involution =
Schoenberg's Theorem on Heisenberg Groups
269
and the fact that the compact open topology on A(U(n),W") is the smallest topology that makes all the mappings in the set
{f:W ( n ) , l W
-+
:
f E L&(,)(Wn)I
continuous, we can conclude that f" is in the space Co(A(U(n),W")) of all continuous functions on A(U(n),H") that vanish at infinity. Moreover, we get I IlfllL1(Wn), f E ~l(fl"), SUP +€A
lml
where A = A(U(n),H"). Indeed, for all .Ic, in A ( U ( n ) , H " ) , we know that II, : W" CC is positive definite and hence --f
Using Plancherel's theorem in the paper [6] by Godement for Gelfand pairs, there exists a unique Borel measure p on the space A(U(n),lHI")such that p ( A z ( U ( n ) W") , = 0 and
L"
If(.,
t)12dzd t = L1(U("),W") lf^(+)I2dP($)
for all continuous functions f in Lt(,,(W")
n L;(,,(W"),
where
Al(U(n),H") = {&!,r : X E R x , r = O,1,2,. . .} and
A,(U(n),W") = {vw : w E C"}.
If f is a continuous function in Lh(,)(W") n Lc(,)(W") such that f" E L1(A(U(n), W"),then we have the following inversion formula, which gives
f ( 4=J
m r n d P ( . I c , ) , ( z ,t ) E W".
A(U(n),H")
In particular, the inversion formula is valid for every continuous, positive definite and U(n)-invariant function f : W" -+ C. We end this section with one more notation. Let F : A,(U(n),W") -+ CC be any Borel function. Then we sometimes write F(X,r ) in place of F($+), X € W X a n d r = 0 , 1 , 2 ,....
270
Jinman Kim and M. W. Wong
2 Preliminary Results We collect in this section the technical results, which we need for the proof of the main result in the next section.
Lemma 2.1 Let X be a separable normed vector space and let sequence of continuous linear functionals o n X such that SUP
l
IlVkll
{vk}&
be a
< 00,
where 11 1) denotes the norm in the dual space of X . Then we can find a continuous linear functional v o n X and a subsequence { u ~ of ~{ u }~ }~such ~ ~? ~ that lim V k j ( Z ) = u(x), x E X. j+m
A proof of Lemma 2.1 can be found on pages 29 and 30 of the book [3] by Donoghue. Lemma 2.2 The set C o ( A ( U ( n ) , W " ) ) is a separable Banach space with respect to the norm 11 llw given by
IlFllW = SUP l ~ ( $ ) l , F
ECo(A(wvn)).
+€A
Proof We need only prove that the Banach space Co(A(U(n),IHIn))is separable with respect to the norm 11 Ilm. To this end, we use a result on page 327 of the paper [2] by Benson, Jenkins, Ratcliff and Worku to conclude that the range of L&(,)(W") under the U(n)-spherical transform is a dense subspace of C o ( A ( U ( n ) W)). , Furthermore, the space S,TJ(,)(W") defined by SLl(n)(W") = {f E S ( W : f ( W= )f ( z , t ) , (.,t) E W",Ic E U ( n ) )
is dense in Lb(,,(Wn). The proof then follows from the fact that S,(,,(W") is topologically isomorphic to the separable space S,TJ(") (P@) S(R), where S,TJ(,)(@") = {f E S(Cn) : f ( h ) = f(z), z E C", k E
u(n)},
and the fact that the U(n)-spherical transform is a continuous linear mapping on &(n) (Hn).
Corollary 2.3 Let No be the set of all nonnegative integers. Then the set Co(R x No) of all continuous functions on R x No vanishing at infinity is a separable Banach space with respect to the supremum norm.
Schoenberg's Theorem on Heisenberg Groups
271
3 The Main Result The main result in this paper is the following version of Schoenberg's theorem for positive definite functions on the Heisenberg groups W",n = 1,2,. . . . For the sake of simplicity, we denote by R+ the interval [0,m).
Theorem 3.1. Let F be a continuous function on R+ x R. Then the function fn : W" + C defined by fn(z,t> = F(IzI,t>,
(4 E W",
is positive definite on W",n = I, 2,. . . , i f and only i f there exists a positive and finite Radon measure v on Rx x No such that
F(r,t)=
s
e-+plr2--iAt
w x x No
Proof Since
/
w x xNo
dv(X,m),
( ~ , tE)RX x No.
dv(X,m) < 00,
it follows that for n = 1,2,. .. , and for every cp in S(Hn),we get, by Fubini's
theorem,
s,. {L
e-flAlla12-iAt(cp * cp*)(z,t ) d z d t &(A, m).
=
xNo
Since L $ - ~ ( Z = ) 1, x
E R+,
n = 1,2,.
(3.1)
. . , we get
for X E Rx and n = 1 , 2 , . . . . Therefore, by (3.1) and (3.2), fn(.,t)((P
* cp*)(z,t)dzdt 2 0,
'p
S(W"),
for n = 1 , 2 , . . . . Conversely, suppose that fn : W" -+ C is positive definite for n = 1,2,. . . . Then, as has been shown in the paper [2] by Benson, Jenkins, Ratcliff and Worku,
272
Jinman Kim and M. W. Wong
where u, is a positive and finite Radon measure on R x x NO.Now, we extend v, on R x x No to fin on R x No defined by fin(9) =
J
c p ( ~m , )dvn(~ m,) , cp E
x NO).
w x xNo
Then fin is a continuous linear functional on Co(R x NO),which is a separable Banach space by Corollary 2.3. Furthermore, we have
Thus, by Lemma 2.1, we can find a continuous linear functional fi on Co(R x No) and a subsequence fink of fin such that fin,(cp>
as k
--t 00
+
fi(cp>
for all 'p in Co(R x NO). Since
it follows that for m = 0 , 1 , 2 , .. . , LK-' -+ 1 uniformly on compact subsets of 00. Also, using the fact that for every (A, m ) in Rx x No, q5F,m is a positive definite function on IHI", we get
R+ as n -+
l4Y,m(z,t)l I 14Y,m(O,O)I = 1, ( z , t >E HI". Therefore
e-;lXlr2-iXt
= s,.,No
for all ( z , t ) in the proof.
IHI",
d 4 A ,m)
where W n , E Cnk is such that Iun,l = 1. This completes
0
Remark 1. In view of Theorem 3.1, we can easily see that the function F :
R3 --+ C defined by
is positive definite on R3.
Acknowledgement This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0008562.
Schoenberg’s Theorem on Heisenberg Groups
273
References 1. C. Benson, J. Jenkins and G. Ratcliff, The spherical transform of a Schwartz function on the Heisenberg group, J. Funct. Anal. 154 (1998), 379-423. 2. C. Benson, J. Jenkins, G. Ratcliff and T. Worku, Spectra for Gelfand pairs associated with the Heisenberg group, Colloq. Math. 71 (1996), 305-328. 3. W. F. Donoghue, Jr., Distributions and Fourier Transforms, Academic Press, 1969. 4. D. R. Fischer, Functions positive-definite on R3 and the Heisenberg group, J. Funct. Anal. 42 (1981), 338-346. 5. G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989. 6. R. Godement, A theory of spherical functions I, Trans. Amer. Math. SOC.73 (1962), 496-556. 7. R. Howe, On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. SOC.3 (1980), 821-843. 8. P. E. T. Jorgensen, Positive definite functions on the Heisenberg group, Math. Z. 201 (1989), 455-476. 9. P. E. T . Jorgensen, Extensions of positive definite integral kernels on the Heisenberg group, J. Funct. Anal. 92 (1990), 474-508. 10. I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. SOC.44 (1938), 522-536. 11. I. J. Schoenberg, Metric spaces and completely monotone functions, Ann. Math. 39 (1938), 811-841. 12. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , Princeton University Press, 1993. 13. S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Birkhauser, 1998. 14. S. Thangavelu, An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups, Birkhauser, 2003. 15. M. W. Wong, Weyl Transforms, Springer-Verlag, 1998.
Weak and Strong Solutions for Pseudo-Differential Operators M. W. Wong Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada mwwong(0pascal.math.yorku.ca
Abstract We give an account of weak and strong solutions for pseudodifferential operators in LP(Rn),1 < p < m. We use Gbrding's inequalities for pseudo-differential operators to study the existence and sometimes uniqueness of weak and strong solutions in L2(R"). We give sufficient conditions on the existence of strong solutions in LP(R"), 1 < p < 00, for elliptic pseudodifferential operators. Positivity of the symbols and boundedness from below of the corresponding pseudo-differential operators are the overarching conditions that give the existence and sometimes uniqueness of solutions.
1 Pseudo-Differential Operators in a Nutshell Let m E R. Then we define S" to be the set of all C" functions cr on R" x R" such that for all multi-indices Q and p, there exists a positive constant Ca,p for which
I(D,"Dp)(z,I)II Ca,p(l
+ IEI)"-Y
2,E
E R".
We call any function in S" a symbol of order m. Let cr E S". Then the pseudo-differential operator T, is defined on the Schwartz space S by
(T,cp)(X) = (27r)-742
eZ"'b(2,
[)$(() dE,
2
E R",
for all functions cp in S, where
@(() = (27r)-"/2
e-i".Ecp(z) dz,
< E R".
276
M. W. Wong
It is easy to prove that T, maps S into S. It is well-known that the formal adjoint T,* of T, exists and is also a pseudo-differential operator with symbol in S". Using the formal adjoint, we can extend T, : S 4 S to a continuous linear mapping from S' into S', where S' is the space of all tempered distributions, bY (Tuu)(cp) = cp E S,
~(rn),
for all u in S'. It is also well-known that T, : H S ~ P Hs-"?P is a bounded linear operator for -00 < s < 00 and 1 < p < 00. Let us recall that for -00 < s c 00 and 1 < p < 00, H"J' is the LP-Sobolev space of order s defined bY HSJ' = {u E S' : J-,U E LP(Rn)}, --f
where J , is the pseudo-differential operator with symbol IS,given by = (1
IS&)
+ 1E12)-s'2,
E
E R".
It can be shown easily that HS+' is a Banach space in which the norm
11
is given by IIulls,p
= II J - s 4 l p ,
7J
E Hs7p,
where 11 [ I p is the norm in LP(R"). Let CY E S", m > 0. The viewpoint adopted in this paper is to look at the pseudo-differential operator T, as a linear operator from LP(Rn) into LP(R"), 1 < p < 00, with dense domain S. Then we can prove that T, is closable and hence the minimal operator T,,o of T, exists. The minimal operator T,,o is the smallest closed extension of T,. A function u in LP(Rn) lies in the domain d(T,,o) if and only if there exists a sequence { ( ~ k of} functions ~ ~ in S for which ( ~ k u in LP(Rn)and T,cpk -+ f for some f in LP(Rn) as k -+ 00; and if this is the case, then T,,ou = f. Let f E LP(Rn). Then a function u in LP(Rn) is said to be a weak solution of the pseudo-differential equation T,u = f on R" if (.,T,*cp) = ( f 4 , cp E S, --f
where (F,G) is defined by
(F,G)=
1
F(x)G(5)dx
IWn
for all measurable functions F and G on R" provided that the integral exists. Theorem 15.3 in the book [13] by Wong tells us that a weak solution exists if and only if there exists a positive constant C such that
Let CY E S", m > 0. Then we say that constants C and R such that
IS
is elliptic if there exist positive
Weak and Strong Solutions for Pseudo-DifferentialOperators
277
Let CT E S", rn > 0, be an elliptic symbol. Then the symbol of the formal adjoint T,*of T, is also elliptic. For every elliptic symbol in S", rn > 0, there exist positive constants C1 and Cz such that Clll'pIl%P
I IITUPIIP + Il'pllP I C2II'pllm,p,
'p
E 25-
This is the analogue of the celebrated Agmon-Douglis-Nirenberg inequality for pseudo-differential operators. The origin of the inequality can be traced back to the study of partial differential equations in the paper [l].It follows from 1 < p < 00, the Agmon-Douglis-Nirenberg inequality that for all f in LP(RWn), every weak solution u in LP(R") of the pseudo-differential equation T,u = f on R" is in Hm*P. All definitions and results in this section are fleshed out in the book [13] by Wong. The aim of this paper is to study the existence and sometimes uniqueness of weak and strong solutions for pseudo-differential operators in LP(RWn), 1< p < 00. Strong solutions are defined in Section 2. Weak and strong solutions of partial differential equations appear to have been first studied by Friedrichs in [3]. Weak and strong solutions for elliptic pseudo-differential operators in LP(Rn), 1 < p < 00,are first studied in the paper [la] by Wong and explained in detail in the book [13]. Weak and strong solutions are known, respectively, as maximal and minimal operators in the works [12, 131. The operator-theoretic terminology is to some extent in deference to the pioneering works [4]of Hormander and Schechter [lo]. In retrospect, however, it is fair to say that in the study of partial differential equations, the concepts of weak and strong solutions are more intuitive and hence more appealing than the concepts of maximal and minimal operators.
2 Strong Solutions in L2(Rn) To introduce the notion of a strong solution, let u E S", rn > 0, and let f E LP(W"), 1 < p < 00. Then a function u in LP(Rn) is said to be a strong solution of the pseudo-differential equation T,u = f on R" if u E d(T,,o) and T,,Ou = f . It can be shown easily that every strong solution is a weak solution of T,u = f on R". Theorem 2.1 Let u E S 2 m , m > 0 , be an elliptic symbol such that there exists a positive constant C such that
Then f o r every function f in L2(Rn),the pseudo-differential equation T,u = f o n R" has a unique strong solution u.
We give two proofs of Theorem 2.1. The first proof depends on the LaxMilgram lemma given, for instance, on page 57 of the book [7] by Lax. The
M. W. Wong
278
second proof is based on Theorem 15.3 in the book [13] by Wong. Both proofs depend on the following lemma.
Lemma 2.2 Under the hypotheses of the preceding theorem, there exists a positive constant C such that Re (T,u, u)2 C112~11%,~,u E HmY2.
Proof Let u E H m > 2Then . there exists a sequence {p};fO=,of functions in S such that (Pk -+ u in Hm>2as k -+ 00. Using the fact that T, : Hm>2-+ H - " T ~ is a bounded linear operator, we get a positive constant C' such that
as k
-+
00.
Therefore
First Proof of Theorem 2.1 Define the bilinear mapping B : Hm!2 x Hmi2 --t C by B(zL,v) = (u,T,v), U,WE Hmy2. Then for all u and v in H m , 2 ,
and, by Lemma 2.2,
p(W)l2 I(T0W)I 2 CIIu111,2,
E
Hrnt2.
Let f E L2(RWn). Then we define the linear functional F :
Hml2
+C
by
F ( w ) = ( ~ , f ) , w E Hm12. It is a bounded linear functional because
lW41 = I ( W ? f ) l
5
llWllm,2llfIl-m,2
I Ilfll21lWIlm,2,
W
E
So, by the Lax-Milgram lemma, we get a unique function u in Hm!2such that F ( w )= B ( w , u ) , w
E
Hmy2,
Weak and Strong Solutions for Pseudo-DifferentialOperators
279
or equivalently,
(w, f) = (w, T,u), w E Hrnt2. So, u is a weak solution in L2(Rn) of the pseudo-differential equation on R". Since (T is elliptic, it follows that u is a strong solution. To prove uniqueness, we let v be another strong solution. Then, by Lemma 2.2, llu
1
- V I I & , ~ 5 -Re (T,(u
c
- v ) ,u - v ) = 0.
Therefore u = v and the proof is complete.
Second Proof of Theorem 2.1 By (2.1) and Schwarz' inequality, we get 'p in S,
for all functions
and hence
1
ll'p112
L ~IIT,t'pl12.
Let f E L2(Wn).Then
I(f,'p)I5
llf11211'p112
5
1 ~llfll2llT,*'pll2,
'p
E S.
So, the pseudo-differential equation T,u = f on R" has a weak solution u in L2(Rn).That u is the unique strong solution follows from the same argument used at the end of the first proof. A major theme in the modern theory of general partial differential equations is to learn the properties of the partial and pseudo-differential operators from the properties of the corresponding symbols. Thus, it is important to have good conditions on the symbols (T to ensure that the inequality in the hypotheses of Theorem 2.1 is fulfilled. To this end, the notion of a strongly elliptic symbol is in order. A symbol in S", m E R, is said to be strongly elliptic if there exist positive constants C and R such that
It is clear that strongly elliptic symbols are elliptic. We need Ggrding's inequality for pseudo-differential operators with strongly elliptic symbols, which follows from the pseudo-differential calculus developed in the book [13] by Wong. A proof can be found on page 57 of the book [9]by Saint Raymond. Theorem 2.3 (Gbrding's Inequality) Let (T E S2rn,m > 0, be a strongly elliptic symbol. T h e n there exists a positive constant A and a constant C, f o r every nonnegative real number s such that Re (T,v7
'p)
2 Allvll:,2
- csII'pII&-s,z,
'p
E S.
280
M. W. Wong
We can now give sufficient conditions on the existence and sometimes uniqueness of strong solutions in L2(Rn) for pseudo-differential operators T, in terms of the symbols CT.
Theorem 2.4 Let CT E S2m,m > 0 , be a strongly elliptic symbol. Then there exists a real number Xo such that for all f in L2(R") and X 2 XO, the pseudodifferential equation (T, + XI)u = f on Rn, where I i s the identity operator on L2(Wn),has a unique strong solution u. Proof By Girding's inequality in Theorem 2.3, there exist constants A and Xo such that A > 0 and Re (Tocp,cp) 2 AIIvllt,2 - ~ o I l y o l l % cp E s. Therefore for X
2 XO,
Re ((Tu+ X ~ ) c p l c p >2 Allvllt,2 +
2 AllcpllE,21 cp E s*
- X0)llcpII;
0
Thus, by Theorem 2.1, the proof is complete.
The Girding inequality in Theorem 2.3 implies that if CT E S2m,m > 0, is strongly elliptic, then for every nonnegative real number s, there exists a positive constant C, such that Re (Tucp,cp) 2 - ~ s l l P l l ~ - - s , 2 ,
'p
E
s.
The following theorem contains the well-known sharp Girding inequality, which is valid without ellipticity. As such, the Girding inequality in Theorem 2.3 is often known as the classical Girding inequality.
Theorem 2.5 Let
ISE
S2m,m > 0, be such that
+,t)
2 0,
x , E E R".
Then for all s in [0,2], there exists a positive constant C, such that
RePucp,yo) 2
-CsllcpIlt-$,2,
cp E
s.
The case for s = 1 is in the work [5] of Hormander and the much more delicate case for s = 2 is due to Fefferman and Phong [2]. See also the paper [8] by Melin and Section 18.6 of the book [6] by Hormander on these matters. The sharp GBrding inequality of Fefferman and Phong can be used to prove the existence of weak solutions for pseudo-differential operators with nonnegative symbols.
Theorem 2.6 Let
CT E
S2m, 0 < m 5 1, be such that
+,E)
2 0,
z,E E R".
Then there exists a positive constant A0 such that for all f in L2(Rn) and X > Xo, the pseudo-digerential equation (T, X1)u = f o n R" has a weak solution u in L2(Rn).
+
Weak and Strong Solutions for Pseudo-Differential Operators
281
Proof By the sharp Gkding inequality in Theorem 2.5, there exists a positive constant XO such that for X > XO,
Re((T0 + ~ ~ ) c p , c p L )
- Xo)llcpIIL-1,2
2 (A
- Xo)llcpII;,
cp E
s.
Thus, by Schwarz' inequality, we get for all cp in S, ,
which is the same as
Let f E L2(Rn).Then, by Schwarz' inequality,
I(f,cp)l 5
Ilfll2llcpll2
5
1
Ilfll2ll(T0 - W*cplln,
cp E
s.
By Theorem 15.3 in the book [13] by Wong, the proof is complete.
0
Using the fact that weak solutions are strong solutions for pseudo-differential operators with elliptic symbols, we have the following theorem. Theorem 2.7 Let o E SZm, m > 0 , be a n elliptic symbol such that a ( s , t ) L 0,
Z,t
E
R".
T h e n there exists a positive number A0 such that f o r all f in L2(Rn) and X > XO, the pseudo-differential equation (To XI)u = f o n R" has a unique strong solution.
+
Theorem 2.7 follows from the fact that the symbol in question is strongly elliptic and hence Theorem 2.4 applies.
3 Strong Solutions in LP(R"), 1
< p < 00
We give sufficient conditions in this section for the existence and sometimes uniqueness of Lp-strong solutions for pseudo-differential operators, 1 < p < 00.
Theorem 3.1 Let o E S", m > 0 , be elliptic and such that there exists a positive constant C such that ll'pllp)
I ClI~,*(Pllp5 'p E s.
T h e n the pseudo-differential equation T,u = f o n R" has a strong solution u every function f in LP(Rn).
for
282
M. W. Wong
Proof For all functions 'p in S, we use the Agmon-Douglis-Nirenberg inequality and the hypothesis of the theorem to obtain
Thus, by Theorem 15.3 in the book [13] by Wong, the pseudo-differential equation T,u = f on R" has a weak solution u in LP(Rn). Since CT is elliptic, it follows that u E d(T,,o) and T,,ou = f. Therefore u is a strong solution of T,u = f on R". 0 We give an LP-analogue of Theorem 2.7 as an application of Theorem 3.1.
Theorem 3.2 Let u E Sm, m pendent of x and
> 0 , be a n elliptic symbol such that u i s inde-
.(E> # 0 , E
E R".
T h e n for every function f in LP(R"), the pseudo-differential equation T,u = f o n R" has a unique strong solution u.
We see that Theorem 3.2 is very similar to Theorem 2.7 except that in the Lp-case, we are dealing with "constant coefficient" pseudo-differential operators. Theorem 3.2 follows directly from Theorem 2.3 in the paper [ll]by Wong, which states that { a ( ( ): ( E R"} is the spectrum of T,,o. See Theorem 2.3 in the paper [ll]by Wong in this connection. Hewever, the following direct proof is of some interest.
Proof of Theorem 3.2 By Theorem 3.1, it is sufficient to prove that there exists a positive constant C such that
llvllpt I CIITF'pllP5 To do this, let
T
'p
E
s.
be the function on Iw" defined by 1
Then for all multi-indices a , we can use Leibniz' formula to get
where Ca(t),.,,,,(k) is a constant depending on a(+'), . . . , a ( k )and the sum is taken over all multi-indices a(1), . . . , a ( k )that partition a. Thus, we can find positive constants CL'), . . . ,C,( k ) such that
Weak and Strong Solutions for Pseudo-Differential Operators
283
for all E in R". Since u is elliptic, we can find positive constants C and R such that
b(E)I L c(1+ k%", 161 2 R-
Since
R"
:
& is a continuous and positive function on the compact set { E E
1J1 5 R}, it follows that there exists a positive number S such that
Therefore we can find a positive constant C' such that
2 c'(1+ IEI)",
E E R".
So, there exists a positive constat C" such that I(aaT)(J)I
L C"(1+ l E I ) - " - ' ~ l ,
J E R".
Therefore 7 E 5'-". Using the fact that TT : H-"J'' positive constant C"' such that
llcpllp~= I l G ~ Z c p I l p5~ cl//II%Pll-m,p~ L
-+
IIT*lIP',
cl/l
Lp'(R"), we get a
'p
E S.
It remains to prove uniqueness. To do this, let u and v be strong solutions of T,u = f onRn, where f E LP(Rn), 1 < p < 00. Let w = u-v. Then w E HmJ' and T,w = 0 on R". Taking the Fourier transform in the distribution sense, we get a$ = 0. So, qucp) = (u.;r)(cp)= 0,
cp E
S(R").
(3.1)
But, for all $ in S(Rn), we can find a function cp in S(R") such that up = $. Hence, by (3.1), 'Li)(+) = 0, 7 ) E S(R"). So, 6 = 0 and we get w = 0. Thus, u = v and the proof is complete.
Acknowledgement This research has been partially supported by the NSERC under Grant OPG0008562. The author is grateful to the referee for constructive comments on the first version of the paper.
References 1. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959), 623-727.
284
M. W. Wong
2. C. Fefferman and D. H. Phong, On positivity of pseudo-differential operators, Proc. Nat. Acad. Sci. USA 75 (1978), 4673-4674. 3. K. 0. F'riedrichs, The identity of weak and strong extensions of differential operators, Trans. Amer. Math. SOC.55 (1944), 132-151. 4. L. Hormander, On the theory of general partial differential operators, Acta Math. 94 (1955), 161-248. 5. L. Hormander, Pseudo-differential operators and non-elliptic boundary problems, Ann. Math. 83 (1966), 129-209. 6. L. Hormander, The Analysis of Linear Partial Differential Operators 111,SpringerVerlag, 1985. 7. P. D. Lax, Functional Analysis, Wiley-Interscience, 2002. 8. A. Melin, Lower bounds for pseudc-differential operators, Ark. Mat. 9 (1971), 117-140. 9. X. Saint Raymond, Elementary Introduction to the Theory of Pseudo-Differential Operators, CRC Press, 1991. 10. M. Schechter, Spectra of Partial Differential Operators, Second Edition, NorthHolland, 1986. 11. M. W. Wong, Spectra of pseudo-differential operators on Lp(Rn.), Comm. Partial Differential Equations 4 (1979), 1389-1401. 12. M. W. Wong, On some spectral properties of elliptic pseudo-differential operators, Proc. Amer. Math. SOC.99 (1987), 683-689. 13. M. W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999.
Comparison Results for Quasilinear Elliptic Hemivariational Inequalities Siegfried Carl Institut fiir Analysis Fachbereich Mathematik und Informatik Martin-Luther-Universitat Halle-Wittenberg 06099 Halle Germany [email protected]
Summary. We generalize the sub-supersolution method known for weak solutions of single and multivalued equations to quasilinear elliptic hemivariational inequalities. To this end we first introduce our basic notion of sub- and supersolutions on the basis of which we then prove existence, comparison, compactness and extremality results for the hemivariational inequalities under considerations.
1 Introduction Let 0
c RN
be a bounded domain with Lipschitz boundary 8 0 , and let
V = W1ip(0) and V, = W i ' p ( 0 ) 1 , < p < 00, denote the usual Sobolev spaces with their dual spaces V* and V:, respectively. In this paper we deal with the following quasilinear hemivariational inequality:
u E Vo, 7 E M ( u ): (Au - f,w - U )
+
L
j " ( ~w ;- U ) d~
2 (7,TJ - u ) , (1.1)
for all w E VO,where j"(s; r ) denotes the generalized directional derivative of the locally Lipschitz function j : R -+ R at s in the direction r given by j o ( s ; r ) = limsup Y+S,
tL0
j(Y
+ t r >- j ( Y ) t
7
(1.2)
cf., e.g., [6, Chap. 21, f E V:, and a@(u)is the subdifferential of the continuous and convex functional @ ! : L P ( 0 ) -+ R given by
where h : R -+ R is some monotone nondecreasing (not necessarily continuous) function satisfying a certain growth condition specified later. The operator
286
Siegfried Carl
A : V --t V: is assumed to be a second order quasilinear differential operator in divergence form Au(z)= -
CN - -au ~ ( x , Vu(z)), i=l
(&.
with Vu = . . , a) alN . Let gradient of j defined by a j ( s ) := { C E
dXi
aj : R
--t
2'\{0}
denote Clarke's generalized
R I j o ( s ; r ) 2 C T , Vr E R}.
A method of super-subsolutions has been established recently in [3] for quasilinear elliptic differential inclusions of hemivariational type in the form
Au+ aj(u)- a$(u) 3 f , in Q, where $ : R
-+
u = 0, on 8 0 ,
(1.6)
R is given by
which is continuous (but not necessarily smooth) and convex due to the monotonicity of h. One can show that any solution of (1.6) is a solution of the hemivariational inequality (1.1).The reverse is true only if the function j is regular in the sense of Clarke which means that the one-sided directional derivative and the generalized directional derivative coincide, cf. [6, Chap. 2.31. The main goal of this paper is to provide existence and comparison results of (1.1) in terms of an appropriately generalized notion of super- and subsolutions. Moreover, we are going to prove the existence of extremal solutions and the compactness of the solution set within the order interval of super- and subsolutions. A generalization of the super-subsolution method of hemivariational inequalities in the form u E Vo :
(Au - f,w - U )
+
L
j 0 ( u ;w - U ) dz
2 0 , V w E V,,
(1.S)
has been developed recently in a joint paper with D. Motreanu and Vy Le, It should be noted that the extension of the super-subsolution method see [4]. for (1.1)is by no means a straightforward generalization of the theory developed for the multivalued problems (1.6) because of the intrinsic asymmetry of hemivariational inequalities.
2 Notations, Hypotheses and Main Result We impose the following hypotheses of Leray-Lions type on the coefficient functions ai, i = 1,...,N , of the operator A:
Comparison Results for Hemivariational Inequalities
(Al) Each ai : R x lRN
287
R satisfies the Carathhodory conditions, i.e., ( x , c ) H aa(x,c) is measurable in 2 E R for all E E RN and continuous in for almost all x E R.There exist a constant co > 0 and a function ICO E L*(R), l / p l / q = 1, such that --f
+
I ICo(x)+ co J d P - l
I.i(.,t)l for a.e. z E f2 and for all
7
E RN.
(A2)CL&i(X,0 - sib, t"i
- ti) > 0 for a.e. x E 0 , and for all t,6' E RN with E # t'. ( A 3 ) z E lai(z,E)& 2 vltlP - b ( X ) for a.e. z E R,and for all E E RN with some constant v function kl E L 1 ( R ) .
> 0 and some
As a consequence of (Al), (A2) the semilinear form a associated with the operator A by
is well-defined for any u E V, and the operator A : Vo -+ V: is continuous, bounded, and monotone. For functions w, z : R -+ R and sets W and 2 of functions defined on 0 we use the notations: w A z = min{w, z } , w V z = max{w, z } , WAZ = {wAz I w E W, z E WVZ = {wVz I w E W, z E Z } , and w A Z = {w} A 2, w V 2 = {w} V 2. By using (1.7) the subdifferential of 1c, is given by WJ(4= Jl(41, (2.1) where h and 6 denote the left-sided and right-sided limits of h. Denote by H the Nemytskij operators associated with h and 6,respectively, then and we introduce our fundamental notion of sub- and supersolution as follows.
z},
[W,
Definition 2.1 21 E V is called a subsolution of (1.1) i f the following holds: (i) g < 0 o n a R , (ii) ( A g - f , v - g ) + f , j o ( g ; v - g ) d x 2
(E(g),v-g), V v E g A V o .
Definition 2.2 ii E V is a supersolution of (1.1) i f the following holds:
an,
(i) ii 2 0 o n (ii) (Afi - f , v - U )
+ Jszj o ( U ; v - U ) dx 1 (n(fi), v -U),
V v E U V Vo.
We assume the following hypothesis for j and h: (H1)The function j : R -+ lR is locally Lipschitz and its Clarke's generalized gradient aj satisfies: (i) there exists a constant c1 2 0 such that El
for all
I t + c1 (s2 - s1)"-l
E aj(si), i = 1,2, and for all s1,
s2
with s1
< s2.
Siegfried Carl
288
(ii) there is a constant
(H2) The function h : R
c2
2 0 such that
+R
is monotone nondecreasing and satisfies:
Ih(s)l I c3 (1
+ Isy-l),
v s E R.
Let LP(L?) be equipped with the natural partial ordering of functions defined by u 5 w if and only if w - u belongs to the positive cone L?(L') of all nonnegative elements of LP( L?). This induces a corresponding partial ordering also in the subspace V of LP(L?),and if u,w E V with u 5 w then [u,w] = { z E
v I u 5 z 5 w}
denotes the order interval formed by u and w.
Definition 2.3 A solution u* is the greatest solution of (1.1) within [g,ii]if f o r any solution u E [g,ii] we have u 5 u*.Similarly, u* is the least solution within [g,ii] i f f o r any solution u E [g,ii] we have u, 5 u. The least and greatest solutions are the extremal ones. Our main result is given by the following theorem.
Theorem 2.4 Let hypotheses (Al)-(A3) and (Hl)-(H2) be satisfied, and let u and ii be sub- and supersolutions of (1.1) with 2 5 ii. Then the hemivariational inequality (1.1) possesses extremal solutions within the order interval and the solution set of all solutions of (1.1) within [g,U] is a compact subset in Vo.
[u, 211,
The proof of Theorem 2.4 relies on an extremality result which has been obtained recently in [4]for the following special case of (1.1): U E V O :( A u - f , v - u ) +
I.
jo(u;~-u)d~>OV , V E ~ .
(2.2)
Hemivariational inequality (2.2) is a special case of (1.1) if @(u)G 0. Here we recall the extremality result for (2.2) obtained in [4].
Lemma 2.5 Let hypotheses (Al)-(A3) and ( H l ) be satisfied and assume the existence of an ordered pair of sub-supersolutions, g and 20, of (2.2) according t o Definition 2.1 and Definition 2.2 (with H = 0 , H = 0). Then there exist extremal solutions of (2.2) within [a,el.
3 Proof of Theorem 2.1 In this section we are going to prove our main result. Therefore we assume that all the hypotheses of Theorem 2.4 are satisfied with g , ii being an ordered pair
Comparison Results for Hemivariational Inequalities
289
of sub-supersolutions of (1.1). The proof is inspired by an idea of the author used in [2]to treat elliptic problems with flux boundary inclusions of d.c.-type. The proof will be given in two steps.
Proof (a) Existence of extremal solutions of (1.1) The hypothesis (H2) ensures that the functional 9 : LP(0) -+ R given by (1.3) is well defined, convex and locally Lipschitz continuous, and so is the restriction of 9 to V, denoted 9)vo.Since VOC LP(0) is densely embedded, we get the following characterization of the subgradients of a ( @ I v , ) (due ~ ) to [ 5 , Theorem 2.31:
y E a(!P.(v0)(u)
y E L q ( 0 ) with y(x) E a+(u(x)) for a.e. x E 0. (3.1)
Due to (2.1) the inclusion on the right-hand side is equivalent to y E [H(u),I?(.)]. Therefore u is a solution of (1.1)if the following holds: u E VO and there exists a y E L q ( 0 ) with y E [H(u),H(u)]such that
(Au - f , - ~U ) +
~ O ( U ; V
- U ) dz
2 ( y , -~u ) , V v
E
VO.
(34
Let us consider the following hemivariational inequality related with (3.2):
u E VO: (Au - f,v - U )
+ J, j " ( ~v;-
U)
dx _> ( H ( u ) ,w - u ) ,
(3.3)
for all ZI E VO.Note that in view of (H2) the Nemytskij operator g : LP(f2) t L'J(0)is well defined, but not necessarily continuous, which makes the treatment of (3.3) more difficult. We are going to show that (3.3) has the greatest solution U* within the interval [%,ti], and that U* is at the same time the greatest solution of the original problem (1.1) within [g,ti].To this end we consider first the following hemivariational inequality with given right-hand side R(3)E Lq(f2) c V$ : u E Vo : (Au - f,w
-
U)
+
s,
j " ( ~w;- U ) dx 2
(R(ti), ZI - u ) ,
(3.4)
for all ZI E G. By hypothesis ii is a supersolution of (l.l),and thus, in particular a supersolution of (3.4). Because of H(u) 5 R(g)5 R(ii)one readily can see that the given subsolution of (1.1) is also a subsolution of (3.4). Therefore we may apply Lemma 2.5 with f replaced by f R(ii)E V$ which ensures the existence of extremal solutions of (3.4) within [g,ti]. We denote by u1 the greatest solution of (3.4) within [g,ii] and consider next the hemivariational ), inequality with B ( G ) replaced by B ( u ~i.e.,
+
u E Vo : (Au - f,w - U )
+
w - U ) da: 2
~ O ( U ;
( R ( u ~w )-, u ) ,
(3.5)
Siegfried Carl
290
a(ti),
for all v E Vo. Since u1 5 ti, we get A(u1) 5 which shows that u1 is a supersolution of (3.5). Furthermore, in view of 2 5 u1 we have H ( g ) 5 A(g)5 ~ ( u I and ) , this implies that g is also a subsolution of (3.5). Again by applying Lemma 2.5 there exist extremal solutions of (3.5) within [ g ,u l ] ,and we denote the greatest one by 1-42. Continuing this process we get by induction the following iteration: uo = ti, and u,+~E [g,u,] is the greatest solution of u E VO : (Au - f,Y - U )
+
s,
j O ( ' ~ 1 ;Y
- U ) d~
2 (R(u~ Y) ,u ) ,
(3.6)
for all Y E VO,which yields a monotone nonincreasing sequence (u,)satisfying
and (3.6) with u replaced by u,+l, i.e., we have
2 (A(%),21 - %+l>
(3.8)
E Vo. Due to (3.7) the sequence (u,)is LP(Q)-bounded, which implies that the sequence (B(un)) is LQ(0)-bounded, and thus from (3.8) we get by taking Y = 0 the following estimate: for all
Y
+,+I,
U,+1)
5 ( n ( U n > + f,un+l)+
JR jo(un+l;-un+l>dx.
(3.9)
Since j 0 ( s ; r )= max{Cr : C E a j ( s ) } , from (3.9) we get by using (A3) and (Hl) (ii) the boundedness of (u,)in VO,i.e., I1~nllVo5 c,
if 12.
(3.10)
The boundedness (3.10) and monotonicity (3.7) of the sequence (u,) as well as the compact embedding Vo c LP(0) imply the following convergence properties: (i) u,(z) + u*(z)a.e. in 0, (ii) u, 4 u*in Lp(O), (iii) u, 3 u*in Vo. Replacing
Y
in (3.8) by u*we get (Au,+1, ~ , + 1 - u*>5
Since ( s ,T-)
H
s,+(f +
jo(un+l;u* - u n + ~ dx )
A(%J,%+1
- u*>.
(3.11)
j o ( s ; T - ) is upper semicontinuous Fatou's lemma yields
jo(un;u*-u,)dx<
-u,)dx=O.
(3.12)
Comparison Results for Hemivariational Inequalities
291
From (3.11), (3.12), the boundedness of (R(u,))in Lq(L') and the convergence properties (i)-(iii) we obtain (3.13)
Hypotheses (A1)-(A3) imply that the operator A enjoys the (S+)-property (see, e.g., [l, Theorem D.2.1]), which in view of (iii) and (3.13) yields the strong convergence (iv) u,
--f
u*in Vo.
Furthermore, because the function s H E(s) is monotone nondecreasing and right-sided continuous, we get by means of Lebesgue's dominated convergence theorem and the a.e. monotone pointwise convergence of the sequence (u,)
that is
s(u,)
-
R(u*)Y dz, v 21 E LP(R),
(3.14)
H(u*) in Lq(R) which due to the compact embedding
LQ(f2)c V t results in (v) H(u,) ---f H(u*)in V:. Thus taking into account the convergencies (i)-(v) and passing to the lim sup in (3.8) we arrive at U*
+
E Vo : (Au*- f , Y - u*)
1.0
j0(u*; Y - u*)dz 2
(R(u*), v - u*),
(3.15)
for all Y E Vo,which shows that u*E [g,4 is a solution of (3.3). Moreover, u* is also the greatest solution of (3.3) within [g, 4. To see this, let u E [g,U ] be any solution of (3.3). Because (3.3) holds, in particular, for all Y E uAV0, i.e., Y is of the form Y = u - (u- .I)+ with UJ E fi, and since H ( u ) 5 R(U)we infer that u is a subsolution of (3.4). Replacing g by u in the iteration above and noticing that the iterates u, are defined as the greatest solutions, the same iterates as before satisfy u 5 u, 5 ii for all n, and thus u 5 u*,which proves that u*is greatest solution of (3.3) within [ZL, 211. If we set y* = H(u*), then, in particular, y* E [H(u*), B(u*)]and we have U*
E
V, : (Au* - f , Y - u*)+
L
j0(u*; - u*)d~ 2 (y*,Y - u*), (3.16)
for all v E VO,which proves that u* is a solution of (1.1). Finally, we shall show that u* is the greatest solution of (1.1) in [g,U]as well. To this end which means that there is an E let ii be any solution of (1.1) in [%,a], a!P(ii) = [ff(ii),H(ii)] such that (1.1) holds with u and y replaced by 21 and 7. Since 7 5 I H ( E ) , similar arguments as above imply that ii is a subsolution of (3.4), which by interchanging the role of g and ii yields the
a(ii)
Siegfried Carl
292
following inequality for the iterates (u,) : ii 5 u, 5 .ii, and thus we get U. 5 u*, which proves that u*is the greatest solution of (1.1) within [g,?i].By similar reasoning the existence of the least solution u* can be proved.
(b) Compactness of the solution set of (1.1) Let S c [g,.ii] denote the set of all solutions of (1.1) within [g,'u],and let (un)c S be any sequence, i.e., we have: there are y, E 6'P(un) = [H(u,), H(un)]such that
u, E
VO
:
for all w E bounded in implies the and (yk) of
(Au, - f,21 - un)
+
jo(u,; 21 - u,) dx 2 (y,, 21 - u,), (3.17)
VO.The sequence (un)is bounded in LP(L'), and hence (7,) is Lq(L') in view of (Hl), which by similar reasoning as in step (a) boundedness of (u,)in VO.Thus there exist subsequences (uk) (u,)and (y,), respectively, satisfying
(1) uk(x) + u(x)a.e. in 0, (2) u k + u in LP(O), (3) u k u in VO, (4)Y k y in Lq(O),
--
where yk E aP(uk), and y E a#(u) due t o the compact embedding Lq(O) C V,.. By means of the convergence properties (1)-(4) we get similarly as in step (a) that the following holds: lim sup(Auk, u k - u)5 0 , k
which by the (S+)-property of A implies the strong convergence
(5) Uk + u in Vo. Passing to the limit in (3.17) with u, replaced by (uk)as k -+00 shows that the limit u belongs to S. This completes the proof of Theorem 2.4.
References 1. S. Carl and S. Heikkila, Nonlinear Differential Equations in Ordered Spaces, Chapman & Hall/CRC, Boca Raton, FL, 2000. 2. S. Carl, Existence of extremal solutions of boundary hemivariational inequalities, J. Differential Equations 171 (20011, 370-396. 3. S. Carl and D. Motreanu, Quasilinear elliptic inclusions of hemivariational type: extremality and compactness of the solution set, J. Math. Anal. Appl. 286 (2003), 147-159. 4. S. Carl, Vy K. Le, and D. Motreanu, The sub-supersohtion method and extremal solutions for quasilinear hemivariational inequalities, Differential Integral Equa-
tions (to appear).
Comparison Results for Hemivariational Inequalities
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5. K.C. Chang, Variational methods f o r non-differentiable funtionals and their applications to partial differentialequations, J. Math. Anal. Appl. 80 (19Sl), 102129. 6. F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990.
Hemivariat ional Inequalities Modeling Dynamic Viscoelastic Contact Problems with Friction Stanislaw Mig6rski* Faculty of Mathematics and Computer Science Institute of Computer Science Jagiellonian University ul. Nawojki 11 PL-30072 Krakow Poland migorskiQsoftlab.ii.uj.edu.pl
Summary. We consider a mathematical model which describes the dynamic viscoelastic frictional contact between a deformable body and an obstacle. The contact process is modeled by a general normal damped response condition and the dependence of the normal stress on the normal velocity is assumed to have nonmonotone character of the subdifferential form. The problem is formulated as a hyperbolic hemivariational inequality with the nonmonotone multidimensional and multivalued boundary conditions. We establish the existence of solutions by using a surjectivity result for multivalued pseudomonotone operators. Under a stronger hypothesis the uniqueness of a solution is obtained.
Introduction In this paper we examine the dynamic process of frictional contact between a deformable body and an obstacle. The body is assumed to be viscoelastic and obeys the Kelvin-Voigt constitutive law with a linear elasticity operator and a nonlinear viscosity operator. The contact is modeled with a general normal damped response condition. The dependence of the normal stress on the normal velocity is assumed to have nonmonotone character of the subdifferential form. We model the friction assuming that the tangential shear on the contact surface is given as a nonmonotone and possibly multivalued function of the tangential velocity. Due to the nonmonotone character of the multivalued boundary conditions, the contact problem is formulated as a hemivariational inequality. The aim of the paper is to deliver existence and uniqueness results *Supported in part by the State Committee for Scientific Research of the Republic of Poland (KBN) under Grants no. 2 P03A 003 25 and 4 T07A 027 26
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Stanislaw Mig6rski
for a hyperbolic hemivariational inequality with the nonmonotone multidimensional and multivalued boundary conditions. We formulate the hemivariational inequality under consideration as a second order evolution inclusion. For the latter, we establish the existence of weak solutions by employing a surjectivity result for multivalued pseudomonotone operators. The uniqueness of a solution is obtained in a case when the damping operator is strongly monotone and the subdifferential of the superpotential satisfies a relaxed monotonocity condition. We mention that the notion of hemivariational inequality was introduced and studied by P.D. Panagiotopoulos in the early eighties as variational formulation for certain classes of mechanical problems with the nonconvex and nonsmooth energy functionals, cf. Panagiotopoulos [16, 171. In the case the superpotentials are convex functionals the hemivariational inequalities reduce to variational inequalities, cf. Duvaut and Lions [7]. The recent mathematical results on the stationary hemivariational inequalities can be found in Naniewicz and Panagiotopoulos [15], Goeleven et al. [8] and the references therein. We refer to Migorski [ll] and the references therein for the results on the first order evolution and parabolic hemivariational inequalities. The hyperbolic hemivariational inequalities with a multivalued relation depending on the first order derivative of the unknown function were treated by Goeleven et al. [9] and Migorski [14]. The contact problems for viscoelastic bodies have been recently investigated in several papers, see e.g. Awbi et al. [l],Chau et al. [3], Rochdi et al. [19], Sofonea and Shillor [20] and the literature therein. In the subsequent parts of this note we describe the model for the process, present its variational formulation and state our main existence and uniqueness results.
1 Problem Statement We consider a deformable viscoelastic body which occupies the reference configuration 0 c R d with d = 2 , 3 . We assume that 0 is a bounded domain with Lipschitz boundary and r is divided into three mutually disjoint measursuch that rneas(rD) > 0. We set Q = 0 x (0,T) able parts TO, r N and for T > 0. The body is clamped on so the displacement field vanishes there. Volume forces of density f1 act in 0 and surface tractions of density fi are applied on r N . The body may come in contact with an obstacle, the so-called foundation, over the potential contact surface rc. We denote by u = (u1,. . . ,ud) the displacement vector, by u = ( a i j ) the stress tensor and by E(U) = (~ij(u)) the linearized (small) strain tensor, where i, j = 1 , . . . , d. We suppose that the material is viscoelastic and obeys a Kelvin-Voigt type constitutive relation u(u,u’)= C(E(U’)) Q(E(u)),where C and Q are given nonlinear and linear constitutive functions, respectively. We remark that in linear viscoelasticity the above law takes the form aij = cijki~ki(u’) +gijkzEkl(u),where C = { c i j k l } and Q = {gijlci}, i , j , Ic, 1 = 1 , . . . ,d
r
rc
+
Hemivariational Inequalities in Viscoelasticity
297
are the viscosity and elasticity tensors, respectively (cf. also Duvaut and Lions [7] for the description of short memory materials). As concerns the laws on the contact surface rc,we consider subdifferential boundary conditions. We assume that the normal stress UN and the normal velocity uh satisfy the nonmonotone normal damped response condition of the form --ON E aj~(x,t,t&) on rc x (O,T), (1) j ~I‘c: x ( 0 , T )x R --t R is locally Lipschitz in its last variable and a j represents the Clarke subdifferential. The friction relation is given by
and describes the multivalued law between the tangential force (TTon rc and the tangential velocity u$,where j ~rc: x (0, T )x R d + R is locally Lipschitz in the third variable. Let uo and u1 denote the initial displacement and the initial velocity, respectively. We denote by Sd the space of second order symmetric tensors on R d . The classical formulation of the contact problem may be stated as follows: find a displacement field u:Q + R and a stress field 0:Q -+ S d such that
u” - diva fs = C(+’))
= fl
in Q in Q u=o on r D x ( 0 , T ) on r N x ( 0 , T ) O n = f2 -ON E a j ~ ( x , t , ~ -OT k ) , E a j ~ ( x , t , ~ $on ) rc x (O,T) , u(0) = uo, u’(0) = u1 in 0. ’
+
Q(E(U))
(3)
The boundary conditions (1) and (2) include as special cases the classical boundary conditions of mechanics (cf. Panagiotopoulos [17],Chapter 2.3). As for the normal damped response condition (l),we consider, as an example, a function p ~R :+ R satisfying
H ( ~ N: ) p ~R :+ R is a function such that ). (i) p~ E Lzc(R) and I ~ N ( s ) ( 5 p ~ ( l 1+ (ii) there exist lim p ~ ( 7 for ) sER
for s E R with p l
> 0;
T+SfO
and (for simplicity we drop the (x,t)-dependence) we define
It is well known (see Chang [2], Goeleven et al. [S]) that if p~ satisfies H(pN)(i), then a j ~ ( c~p )5 ( s ) for s E R,where the multivalued function p 3 : R -+ 2’ is given by p 3 ( s ) = [PN(1)( s ) , p(2) N (s)] denotes the interval in R) and ([a,.]
~
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Stanislaw Mig6rski p (1) , (s) = lim essinf 6+0+
p (2) , (s) = lim esssuppN(7).
~N(T),
6+o+
lT-S1<6
1T-.91<6
If p~ satisfies H ( p , ) , then aj,(s) = s ( s ) for s E R. In this case j, defined by (4) satisfies the hypotheses H ( ~ Nbelow ) and (1) takes the form -UN E p%(u/N) on rc x ( 0 , T )(see Sections 2.4 and 7.2 of Panagiotopoulos [17] for particular cases). Furthermore, if additionally p~ is a continuous function, then (1)reduces to -ON = p ~ ( u / Non ) I'c x ( 0 , T ) which states the relation considered by Awbi et al. [l]. If p ~ ( s =) k l s with Icl 2 0, then we have -UN = k1u/N on rc x (0, T ) ,i.e. the resistance of the foundation to penetration is proportional to the normal velocity. This type of boundary condition, studied by Sofonea and Shillor [20], models the motion of a deformable body on a support of granular material. If p ~ ( s = ) k2s+ k 3 , where s+ = max{O,s}, we have -ON = k2(u/N)+ k3 on I'c x (0,T).This behavior was studied by Rochdi et al. [19], where the contact surface rc was assumed to be covered with a lubricant that contains solid particles, such as one of the new smart lubricants, or with worn metallic particles. Here k2 > 0 represents the damping resistance constant and k3 2 0 is the prescribed oil pressure. This contact condition models the fact that the oil layer presents damping or resistance, only when the surface moves towards the foundation. Concerning the relation (2), we mention that a simple case of the nonmonotone onedimensional friction law which holds on the part rc of the boundary of a plane body 0 c R2 was considered by Panagiotopoulos in Section 7.2 of [17]. In this case the law of the form
+
+
r
where p~ is a function satisfying the same conditions as in H ( ~ N )appears , in the tangential direction of the adhesive interface and describes the partial cracking and crushing of the adhesive bonding material. Here the function j ~R :--+ R is defined by j ~ ( s = ) ~ T ( Td~ ) and it satisfies the conditions H ( ~ Tstated ) below. We refer to Section 2.4 of Panagiotopoulos [17] for several examples of the zig-zag friction laws which can be put in the form ( 5 ) . If 0 c R3,then the friction law is two dimensional in a local coordinate system on the tangential plane to each point t o rc,or three dimensional in the global orthogonal Cartesian system. In the first case it relates {uT,} with { u ; h } , where a = 1, 2 denotes the local coordinates, and in the second case it relates { a ~with ~ } where i = 1, 2, 3 denotes the global coordinates. The nonconvex superpotential j~ for 0 c R3 is formulated by extending to R2 or to R3 certain onedimensional nonmonotone multivalued laws, e.g. by considering minimum type and maximum type functions (cf. Section 4.6.1 of Naniewicz and Panagiotopoulos [15] for concrete examples). In the case, when p ~ ( s=) ps ( p > 0 represents the friction coefficient) the law ( 5 ) reduces to - - a ~ = pu$ on rc x (0,T)which simply means that the tangential shear is proportional to the tangential velocity. This is the case in Sofonea and
Ji
Hemivariational Inequalities in Viscoelasticity
299
Shillor [20] when the contact surface is lubricated with a thin layer of nonNewtonian fluid. The description of other friction laws can be found in Han and Sofonea [lo], and Denkowski and Migorski [5].
2 Hemivariational Inequality Model In this section we set the problem (3) in a variational form. We use the following spaces H = ~ ~ R~), ( 0 3~ ; = {r = {rij) : rij = ~ j iE L 2 ( ~ )=} L2(O;&), H1 = {U E H : E ( U ) E 3.1) = H1(Q;Rd),'F11 = { T E 3.1 : div7 E H } , where &:H1(Q;Rd)+ L2(Q;&) and div:3.11 -+ L 2 ( Q ; R d )denote the deformation and the divergence operators, respectively, given by E ( U ) = { ~ i j ( u ) } , ~ i j ( u= ) 1/2(ui,j uj,i), divo = { o i j , j } and the index following a comma indicates a partial derivative. The spaces H , 'F1, H I and 3Cl are Hilbert spaces equipped with the inner products
+
(% V ) H 1 = ( U , V ) H
+ (+),
E(V))W,
(0, 7)X1
= (a,T ) X
+ (diva, diVT)H.
H1 we denote by v its trace yv on r, where y: H 1 ( Q ;R d ) H1/2(r;Rd)c L 2 ( r ; R d )is the trace map. Given v E H1/2(r;Rd)we
For every v E +
denote by V N and VT the usual normal and the tangential components of v on the boundary r,i.e. V N = v .n and VT = v - vNn. Similarly, for a regular (say Cl)tensor field u:L' S d , we define its normal and tangential components by CN = (on). n and CTT = an - aNn. We recall also (see Clarke [4])that given a locally Lipschitz function h: E ---t IR,where E is a Banach space, the generalized directional derivative of h at x E E in the direction v E E , denoted by ho(z;v), is defined by --f
The generalized gradient of h at z, denoted by d h ( z ) , is a subset of a dual space E* given by d h ( z ) = { C E E* : h0(z;v)2 ( C , V ) ~ for . ~ ~all v E E } . We denote by V the closed subspace of H1 defined by V = {v E H I : v = 0 on To}. On V we consider the inner product and the corresponding norm given by ( U , V ) V = ( E ( U ) , E ( V ) ) N , 1(w((= I/ . E (V)(~. H for U , V E V . Identifying H with its dual, we have an evolution triple V c H c V* (see e.g. Zeidler [21], Denkowski et al. [ 6 ] )with dense, continuous and compact embeddings. We denote by (., .)vlxv the duality of V and its dual V * ,by I ( . IIv* the norm in V*. In the following we also need the spaces V = L2(0,T ;V ) ,'f?= L2(0,T ;H ) and W = {w E V : w' E V * } , where the time derivative involved in the definition of W is understood in the sense of vector valued distributions. We
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Stanislaw Mig6rski
have W C V c c V * . The duality for the pair ( V , V * ) is denoted by ( ( z ,w))v*xv= s,’(z(s), W ( S ) ) V * ~ Vds. It is well known that the embeddings W C C(0,T ;H ) and {w E V : w‘ E W } c C(0,T ;V) are continuous. We suppose the following regularity for the body forces and surface tractions f i E L 2 ( 0 , T ; H ) f2 , E L 2 ( 0 , T ; L 2 ( r ~ ; R and d ) ) we define f E V* by (f(t),~)v*xv = ( f i ( t ) , ~ )(f2(t),v)LZ(rN;Wd) ~ for v E V and a.e. t E (O,T). F’rom the equilibrium equation in (3) and the Green formula (assuming the regularity of the functions involved) and we have
+
+
+
L
v d. r ( ~ ” ( ~ ) , v ) v * x v( ~ ( ~ > , E ( V ) ) .=H ( f i ( t ) , ~ ) ~ ~ ( t ) n for v E V and t E ( 0 , T ) .Taking into account the boundary condition on we obtain
( u ” ( ~ ) , v ) v *+~ (v~ ( ~ ) , E ( v ) ) . H
r N
a ( t ) n - ~ d=r(f(t),~)v*xv.
On the other hand, we have
By the definition of the subdifferential, the relations (1) and (2) on rc x (0, T ) can be written as follows ON^ 5 jk(x, t ,uh;7 ) for 77 E R and -OT . t 2 j$(x,t,u&;J)for E E R d , respectively. Hence we obtain the variational formulation of (3): find a displacement field u:( 0 , T )+ V and a stress field a: ( 0 , T )+ ‘Ft such that (U”(t),
V)V*XV
+Lc
+ (a(t),E(V))w+
(&(x, t , uh; V N ) -k j $ ( x ~t,u&; , V T ) ) d r ( x )2
2 ( f ( t ) , ~ ) v * for ~ v all v E V and a.e. t E ( 0 , T )
(6)
+
a ( t )= C ( ~ ( u ’ ( t ) )G ) ( ~ ( u ( t )for ) ) a.e. t E ( 0 ,T ) u(0) = uo, u’(0) = u1.
3 Main Results In the study of the problem (6) we need the following hypotheses.
H ( C ) : The viscosity operator C:Q x S d --f S d satisfies the Carathhodory condition (i.e. C ( . , . , E ) is measurable on Q for all E E s d and C(x,t,.) is continuous on S d for 8.e. (IC, t ) E Q ) and for a.e. (x,t ) E Q we have
+
0) IIC(x,t,&)llsd5 ci ( b ( x , t ) Il&llsd)for E E b E L 2 ( Q ) ,ci (ii) ( C ( Z , ~ , E-~ C(x, ) t , ~ 2 ): )( E I - ~ 2 2) 0 for all E I , ~2 E s d ;
> 0;
Hemivariational Inequalities in Viscoelasticity (iii) C(z, t ,E ) : E
301
2 c ~ \ \ E \ \ &for all E E Sd with c2 > 0.
H ( B ) : The elasticity operator G: Q x S d S d is of the form G(z, t , E ) = E(Z)E with a symmetric and positive elasticity tensor E E L"(O), i.e. IE = (gijkz), i , j , k , l = 1,.. . , d with gijkZ = g j i k i = gikij and g i j k i 1 0. H ( ~ N: ) j ~rc: x ( 0 , T )x R -+ R is a function such that --f
(i) (ii) (iii) (iv)
j ~ ( .,() . , is measurable for all ( E R and j ~ ( .,O) . , E L1(rc x ( 0 , T ) ) ; j ~ ( z , .)t ,is locally Lipschitz for all z E rc, t E ( 0 , T ) ; 171 5 C N (1 \(I) for all 7 E aj,v(z,t,t) with C N > 0; j g ( z , t , ( ;-€J 5 d~ (1+ \el) for all ( E R with d~ 2 0.
+
H ( ~ T: ) j ~rc: x ( 0 , T )x R d (i) (ii) (iii) (iv)
--f
R is a function such that
j ~ ( . , . , (is) measurable for all ( E Rd, j ~ ( ..,O) , E L1(rc x ( 0 , T ) ) ; j ~ ( zt ,,.) is locally Lipschitz for all z E fc,t E (0, T ) ; 11711 5 CT (1 11t11)for all q E aj~(x,t,t) with CT > 0; & ( z , t , ( ;-0 5 dT (1 Il
+
H ( f ):
fl
E
+
L 2 ( 0 , T ; H )f,2 E L2(O,T;L2(rAr;Rd)), uo E
v
and u1 E H . In the hypotheses H ( ~ Nand ) H ( ~ Tthe ) symbol a j denotes the Clarke subdifferential of j with respect to the variable E. Next, we introduce the operators A: (0, T ) x V + V * ,B: V
t
V * defined
by
( A ( t , u ) , 4 v * x= v (C(z,t,E(U>),&(4)x (% 4 V " X V = ( G b , t , E ( 4 , +))H for u,2, E V and t E (O,T),and the functional J : ( 0 , T )x L 2 ( I ' c ; R d+ ) R given by
J ( t ,4 = for t E (0, T ) and
2)
1 (m,
E
t , w ( z ) )+ jT(& t , V T ( 4 ) ) d r ( 4
rc
L 2 ( r C ;Rd).
We associate with the problem (6) the following evolution inclusion:
(
find u E V with u' E W such that u"(t) + A @ ,u l ( t ) ) Bu(t)+?* (aJ(t,Tu'(t))) 3 f ( t ) a.e. t u(0) = UO, u'(0) = u1,
+
(i,l),
(7)
where 2 = H6(R;Rd) with a fixed 6 E T:Z t L2(T;Rd) is the trace operator and T * denotes its adjoint. The reason to introduce the problem (7) is the following. Under the above assumptions, every solution t o problem (7) is also a solution t o the problem (6). Hence, in order to prove the existence of solutions to (6) it is enough to establish the existence of solutions to (7).
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Stanislaw Mig6rski
Theorem 1 Under the hypotheses H(C), H(G), H(jnr), H ( ~ Tand ) H ( f ) , the problem (7) has a solution. Proof The proof is based on a surjectivity result for pseudomonotone operators. Full details can be found in Mig6rski [13]. Let us consider the operator K : U + C(0,T; V ) given by K v ( t ) = s,” w ( s ) d s + uo for w E U.The problem (7) can be formulated as follows
{
find z E W such that z’(t) A(t,z ( t ) ) B ( K z ( t ) ) 7* ( a J ( t , V z ( t ) ) )3 f ( t ) a.e. t
+
+
z ( 0 ) = 211
+
(8)
We notice that z is a solution to (8) if and only if u = K z satisfies (7). In what follows the proof proceeds in two steps, first suppose that 211 E V and then we pass to the more general case 211 E H . For u1 E V and define the following operators d 1 : U + U*,B 1 : U + U* and NI:U -,2’’ by (dlw) ( t )= A(t,v(t) ul), (Blv) ( t ) = B ( K ( v ( t ) 211)) and Nlw = {w E 2* : w ( t ) E 7 * (aJ(t,Y(w(t) ~ 1 ) ) )a.e. t E ( 0 , T ) )for w E V . Using these operators, from (8) we get
+
+
+
{
Z’+dlZ+BlZ+N1Z
3
f
(9)
z(0)= 0
and observe that z E W solves (8) if and only if z - u 1 E W is a solution to (9). Let the operator L: D ( L ) c V + U* be defined by Lv = v’ with D ( L ) = { u E W : v(0) = 0). Recall that L is a linear, densely defined and maximal monotone operator. Now the problem (9) can be written as find z E D ( L ) such that ( L
+ 7). 3 f,
+ +
where 7:U + 2’’ is given by 7 v = (dl B1 N1)v for w E U. We are able to prove that the operator 7 is bounded, coercive and pseudomonotone with respect to D ( L ) , and therefore it is surjective, see Papageorgiou et al. [18] and Denkowski et al. [6]. Under some additional assumptions on the data we can show that the hemivariational inequality (7) has a unique solution.
Theorem 2 Under the following additional hypotheses (1) A is strongly monotone, i.e. for all u, w E V and a.e. t we have
(A(t,u)- A(t,w), u - v ) ~x, v 2 rnlllu - w1I2 with some ml > 0 , (2) J satisfies the relaxed monotonicity condition (21 - Z 2 , W l - w 2 ) L z ( r ; W d ) 2 - m 2 l l w 1
2
- W211L2(r;Wd)
for all zi E a J ( t , w i ) , wi E L 2 ( r c ; R d )i, = 1 , 2 , a.e. t E ( 0 , T ) with m2 > 0,
Hemivariational Inequalities in Viscoelasticity
E2,
303
f?
(3) ml > m2 where > 0 is the constant of the continuity of the trace operator, i.e. I / Y V I J ~ ( ~ ;5R E11~11 ~) for v E V , the solution to the problem (7) is unique.
The examples of functionals J which satisfies the relaxed monotonicity condition are provided in Mig6rski ([13]).
References 1. Awbi, B. and Essoufi, El H. and Sofonea, M., A viscoelastic contact problem with normal damped response and friction, Ann. Polon. Math., 75 (2000), 233246. 2. Chang, K.C., Variational methods for nondifferentiable functionals and applications to partial differential equations, J. Math. Anal. Appl., 80, (1981), 102-129. 3. Chau, 0. and Han, W and Sofonea, M., A Dynamic Frictional Contact Problem with Normal Damped Response, Acta Appl. Math., 71 (2002), 159-178. 4. Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. 5. Denkowski, Z. and Migbski. S, Existence of Solutions to Evolution Second Order Hemivariational Inequalities with Multivalued Damping, Proceedings of the 21st IFIP TC7 Conference on System Modelling and Optimization, Sophia Antipolis, France, July 21-25, 2003, 203-216. 6. Denkowski, Z. and MigQski, S. and Papageorgiou, N.S., An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003 7. Duvaut, G. and Lions, J.L., Les Inequations en MBcanique et en Physique, Dunod, Paris, 1972. 8. Goeleven, D. and Motreanu, D. and Dumont, Y. and Rochdi, M, Variational and Hemivariational Inequalities: Theory, Methods and Applications, vol. I and 11, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003. 9. Goeleven, D. and Miettinen, M. and Panagiotopoulos, P.D., Dynamic hemivariational inequalities and their applications, J. Optimiz. Theory and Appl., 103, (1999), 567-601. 10. Han, W. and Sofonea, M., Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS, International Press, Providence, Rhode Island, 2002 11. Migbrski, S., Existence and convergence results for evolution hemivariational inequalities, Topological Methods Nonlinear Anal., 16,(2000), 125-144, 12. Migbrski, S., Evolution hemivariational inequalities in infinite dimension and their control, Nonlinear Analysis, 47, (2001), 101-112, 13. Migbrski, S., Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Applicable Anal., in press, 14. Migbrski, S., Boundary Hemivariational Inequalities of Hyperbolic Type and Applications, J. Global Optim., submitted, 15. Naniewicz, Z. and Panagiotopoulos, P.D., Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995 16. Panagiotopoulos, P.D., Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhauser, Basel, 1985
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17. Panagiotopoulos, P.D., Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993 18. Papageorgiou, N.S. and Papalini, F. and Renzacci, F., Existence of Solutions and Periodic Solutions for Nonlinear Evolution Inclusions, Rend. Circolo Mat. di Palermo, 48, (1999), 341-364, 19. Rochdi, M. and Shillor, M. and Sofonea, M., A quasistatic contact problem with directional friction and damped response, Applicable Anal., 68, (1998), 409-422, 20. Sofonea, M. and Shillor, M., A quasistatic viscoelastic contact problem with friction, Comm. Appl. Anal., 5 , (2001), 135-151. 21. Zeidler, E., Nonlinear Functional Analysis and Applications IT A/B, Springer, New York, 1990
Sensitivity Analysis for Generalized Variational and Hemivariational Inequalities Boris S. Mordukhovich* Department of Mathematics Wayne State University Detroit MI 48202 USA borisQmath.wayne .edu
Summary. This paper mainly concerns applications of the generalized differentiation theory in variational analysis to sensitivity analysis/robust Lipschitzian stability for parametric systems governed by variational hemivariational inequalities and their extensions in infinite-dimensional spaces. The basic tools of our analysis involve coderivatives of set-valued mappings and second-order subdifferentials of extended-real-valuedfunctions. Using these tools, we establish new sufficient as well as necessary and sufficient conditions for robust Lipschitzian stability of generalize hemivariational inequalities evaluating also the exact Lipschitzian bounds. Most results are obtained for the class of Asplund spaces, which particularly includes all reflexive spaces, but some of them require the finite-dimensional setting which some other hold in general Banach spaces.
1 Introduction The paper is devoted t o applications of advanced tools of variational analysis and generalized differentiation t o sensitivity analysis/robust Lipschitzian stability for parametric systems in infinite dimensions that are viewed as extensions of the classical variational and hemivariational inequalities. The class of variational systems under consideration can be conveniently described by
perturbed generalized equations
0E
fk,Y> + Qh Y)
(1)
in the sense of [8], where f:X x Y -+ 2 is a single-valued mapping while Q : X x Y =t 2 is a set-valued mapping between Banach spaces. In this note we use the terms base and field referring to the single-valued and set-valued parts of (l),respectively, with the decision variable y and the parameter z. It *This research has been supported by the National Science Foundation under grants DMS-0072179 and DMS-0304989.
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Boris S. Mordukhovich
has been well recognized that (1)provides an appropriate model for sensitivity analysis in a broad framework of constrained optimization and equilibria. In particular, generalized equations (1) cover classical variational inequalities find y E R with ( f ( x , y ) , v - y ) 2 0 for all v E R and hence complementarity problems corresponding to the normal cone field Q(y) = N(y;R) in (1). Note that, in contrast to the standard framework, model (1)includes the case when the field Q may depend o n the perturbation parameter x. Generalizing variational inequalities in the framework of (1),we consider this model in the case of subdifferential fields Q = d’p, where d stands for a subdifferential operator on the class of lower semicontinuous (1.s.c.) functions ‘p: X t := (-00,00], which may be nonconvex and extended-real-valued. When d stands for the Clarke generalized gradient of Lipschitz continuous functions, such systems are known as hemivariational inequalities. In what follows we pay the main attention to variational systems of this type generated by the smaller limiting/basic subdifferential of 1.s.c. functions (see Section 2), which is appropriate to describe certain mechanical and economic equilibria. In fact, we mostly consider two classes of variational systems involving subdifferential operators. The first one called (borrowing the mechanical terminology) by variational systems with composite potentials is described as
z
where g: X x Y -+ W and ‘p: W t are mappings between Banach spaces. The second class labelled as variational systems with composite fields is given by
+ (89 9>(s,Y), + z, and f : X x Y
0 E f(z,Y)
0
(3)
with g : X x Y W , ‘p: W t W*. Both classes are particularly important in composite optimization. Our primary interest is robust Lipschitzian stability of parametric variational systems (2) and (3). The basic concept is a kind of robust Lipschitzian behavior introduced by [ 11 under the name of “pseudo-Lipschitz” multifunctions. In our opinion, it would be better to use the term of Lipschitz-like multifunctions referring to this kind of Lipschitzian behavior, which is probably the most proper extension of the classical Lipschitz continuity to set-valued mappings (while “pseudo” means “false”; cf. [9], where this property of multifunctions is called the Aubin property without specifying its Lipschitzian nature). It is well known that Aubin’s Lipschitz-like property of an arbitrary set-valued mapping F : X =t Y between Banach spaces is equivalent to metric regularity as well as to linear openness of its inverse F-’: Y 3 X . These properties play a fundamental role in nonlinear analysis, optimization, and their applications. Note that both Lipschitz-like and classical Lipschitz properties are robust (stable) with respect to perturbations of initial data, which is important for sensitivity analysis. ---f
Generalized Variational and Hemivariational Inequalities
307
Throughout the paper we use standard notation. Unless otherwise stated, all spaces are Banach whose norms are always denoted by 11 .I). For multifunctions F : X 3 X * the expression
I
Limsup F ( x ) := {x* E X * 3 sequences z k + 5 and X-b?
with
5;
E F ( Q ) for all
f x* k E N} 2;
signifies the sequential Painlev6-Kuratowski upper/outer limit with respect to the norm topology in X and the weak* topology in X*;N := {1,2,. . .}. Recall that F : X =f Y is positively homogeneous if F(CYZ) = a F ( x ) for all x E X and a! > 0. The n o r m a positively homogeneous multifunction is defined by
IlFll := SUP {IIYII I Y E F ( z )
and
11415 1).
2 Lipschitzian Properties and Generalized Differentiation Let us first define the Lipschitz-like property of multifunctions following [l]. Given F : X =! Y and (5,i j ) E gph F , we say that F is Lipschitz-like around (5,ij) with modulus f2 >_ 0 if there are neighborhood U of 5 and V of y such that
F ( x )n V
c F ( u )+ lllx - ullBy
for all x,u E U,
(1)
where B y stands for the closed unit ball in Y . The infimum of all such moduli { l }is called the exact Lipschitzian bound of F around ( 5 , i j ) and is denoted by lip F ( 3 ,y). If V = Y in (l),the above property reduces to the local Lipschitz continuity of F around with respect to the Pompieu-Hausdorff distance on 2 y ;for single-valued mappings F = f:X 4Y it agrees with the classical local Lipschitz continuity. In general the Lipschitz-like property can be viewed as a localization of Lipschitzian behavior not only relative to a point of the domain but also relative to a particular point of the image i j E F ( 5 ) . We are able to provide complete dual characterizations of the Lipschitzlike property (and hence the classical local Lipschitzian property) using appropriate constructions of generalized differentiation. Let us recall the basic definitions referring the reader to [6], [9], and [5] for more details, history, and discussions. Given a nonempty subset R of a Banach space X and a number E 1 0, we first define the collection of E-normals to fl by
and by fie(x;f2) := 8 for x at 5 E R is defined by
4 R. Then the basic/limiting
normal cone to 0
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Boris S. Mordukhovich
When the space X is Asplund (i.e., its every separable subspace has a separable dual) and the set 0 is closed around 3,one can equivalently replace fie(.;0 ) in (2) with fi(z;a):= fio(z;0). Given a set-valued mapping F : X =iY , the (normal) coderivative of F at (5,jj) E gph F is defined is a set-valued mapping D*F(Z,jj): Y* =f X* with the values
D * F ( Z , j j ) ( y * ):= {x* E X*l ( ~ * , - y * )E N((3,jj);gphF)). When F = f:X+ Y is single-valued and strictly differentiable at 3, the coderivative D * f ( ? ) ( y * ) reduces to the adjoint derivative operator
D * f ( ~ ) ( y *=) { V f ( ~ ) * y * for } all y* E Y*.
-
Given an extended-real-valued function p:X + R := [-w, m] finite at 3, we define its (first-order) basic subdifferential acp and singular subdifferential at Z by ap(Z) := D*E,(Z, p(3))(1), amp@):= D*E,(Z, p(Z))(O),
where E,(z) := {v E R I v 2 p(z)} is the corresponding epigraphical multifunction; see [6] for equivalent representations. Note that the basic subdifferential dp(Z) is always smaller than the generalized gradient of Clarke, while the singular subdifferential amp(?)reduces to zero for locally Lipschitzian functions. Recall the scalarization formula
D*f(%)(y*) = a(y*,f)(~) # 8 for all y* E Y*
(3)
held when X is Asplund and dimY < 00; see [6] for more general cases. The second-order subdifferential of p at 5 relative to jj E acp(Z) is defined as the coderivative of the first-order subdifferential mapping by
02p(5,ij)(u) := D*(Op)(z,?j)(u), u E
x**.
If p is twice continuously differentiable around 5 , then a2p(s)(u> = { ~ ~ p ( . ) * u }for all u E
x**,
where V2p(z) stands for the classical second-order derivative operator. To conclude this section, we recall some “normal compactness” properties used in the paper; it is needed only in infinite dimensions. A mapping F : X 3 Y is sequentially normally compact (SNC) at (Z,?j) E gph F if for any sequences ( ~ kz,k ,yk, xi,y z ) E [0,co) x (gphF) x X* x Y* with Ek
L 0,
(%Yk>
+
(w>, ( 4 , Y ; ) E fiEk((%Yk);gPhF)
(4)
Generalized Variational and Hemivariational Inequalities
309
one has (xz,y;) (0,O) + //(xz, yi)II --+ 0 as k + 00. One may equivalently put ~k = 0 in (4) if both X and Y are Asplund while F is closed-graph around ( 3 ,jj). Finally, a set 0 c X is SNC at 5 E 0 if the constant mapping F ( x ) G 0 satisfies this property. We refer the reader to the recent papers [2] and [7] for more information about these and related properties and their calculus.
5
3 Coderivatives of Generalized HVIs In this section we obtain efficient upper estimates as well as exact formulas for coderivatives of solution maps to the variational systems described in (2) and (3), which we call generalized hemivariational inequalities (HVIs). This is an essential part of our approach to the sensitivity analysis for such systems based on coderivative characterizations of Lipschitzian stability. First consider the solution maps S ( 4 := { Y E YI 0 E
f (x,Y ) + a(v O g > ( x Y, ) }
(1)
to the subdifferential systems (2) with composite potentials.
Theorem 3.1 (computing coderivatives of solutions maps to generalized HVIs with composite potentials). Let X , Y , and W be Banach spaces, and let (3,jj) E g p h S for S defined in (1) with g: Y + W and p: W -+E . Put 4 := -f ( 3 ,jj) E a((po g)(jj) and assume the following: (a) f : X x Y --f Y * is strictly differentiable at ( Z , j j ) with the surjective partial derivative V, f (3,y):X + Y * . (b) g E C1 around jj with the surjective derivative V g ( j j ) : Y + W , and the mapping Vg:Y L(Y,W ) is strictly differentiable at jj. --f
Let
v
E W * be a unique functional satisfying the relations
4 = V g ( j j ) * B and
.is E ap(W)
with 3 := g(jj).
T h e n one has the inclusion
which becomes a n equality i f the kernel of V g ( j j ) is complemented in Y , i e . , there is a closed subspace L c Y with L @ ker V g ( j j ) = Y . Proof Using first Theorem 4.l(b) from [4] and the definition of a 2 ( p o g ) ,we get D * s ( ~ , g ) ( y *=) {x* E X*I 3u E Y** with x* = Vzf(Z,jj)*u,
310
Boris S. Mordukhovich
under assumption (a). Applying the second-order chain rule for a2(po g ) from Theorem 4.1 in [3], we arrive at the desired result in both inclusion and 0 equality forms. Our next result concerns generalized HVI with parameter-dependent potentials governed by (strongly) amenable functions that play a major role in finite-dimensional variational analysis and optimization; see [Rockafeller and Wets(1998)l. Recall that $: 2 -+ is strongly amenable at E if there is a neighborhood U of E on which $ can be represented in the form $ = 'pog with a C2 mapping g: U -+ IR" and a proper 1.s.c. convex function p: IR" -+ Z satisfying the qualification condition
amp(g(z))n ker V g ( Z ) * = (0).
Theorem 3.2 (coderivative estimate for solution maps to generalized HVIs with amenable potentials). Let ( 3 ,ij) E gph S for S defined in (1) by mappings betweenjnite-dimensional spaces. A s s u m e that f is con:= 'p o g i s strongly amenable at tinuous around (*, y), that the potential (3,y), and that
+
a 2 9 ( W , f i ) ( O ) n ker V g ( 3 ,y)* = ( 0 ) f o r all 8 E M ( 3 ,y),
(2)
where W := g ( 3 , i j ) and the set M ( Z , g ) i s given by
M ( s ,y) := (i7 E
w*( v E d'p(W),
V g ( 3 ,y)*. = -f (2,y)}
T h e n one has the inclusion
D*S(s,y)(y*) c
+
u
*.{
E
x*I 3u E x x Y
with (.*,
-y*) E
D*f ( 2 ,ij)(.)
[ V 2 ( W m 4+ V9(~,y)*a2'p(~,O)(Vg(~,8).)]}.
G € M(5,&)
Proof. Similar to the one in Theorem 3.1 with the use of Theorem 4.1(a) from [4] and the second-order subdifferential chain rule for strongly amenable functions that follows from Corollary 4.3 in [Mordukhovich (2002)l. 0 Finally in this section we present an upper estimate for coderivatives of solutions maps to generalized HVIs governed by variational systems (3) with composite fields S(.)
:= {Y E
YI 0 E f(.,
Y) + (a'p0 d . 7
Y)}.
(3)
For simplicity we consider only the case of locally Lipschitzian mappings f ; see [5] for further extensions. Theorem 3.3 (coderivatives of solutions to generalized HVIs with composite fields). Let ( 5 , i j ) E g p h S for S defined in (3) with some mappings g: X x Y + W , f: X x Y -+ W* and 'p: W -+ between Banach
Generalized Variational and Hemivariational Inequalities
31 1
spaces. Denote fi := g ( 3 , y) and ij := -f ( 3 ,y) E acp(8). The following assertions hold: (i) Assume that f is strictly differentiable at (3,jj) with the surjective partial derivative V,f(Z,jj) and that g = g ( y ) i s strictly differentiable at jj with the surjective derivative Vg(jj).Then one has D * S ( Z , y ) ( y * )= {z* E X*l 3u E W** with z* = V,f(Z,ij)*u, -?4* E
V,f(.,
y>*.
+ Vm*a2cp(fi,m)}.
(ii) Assume that X and Y are Asplund, that W = R", that g : X x Y
+
R" i s continuous while f : X t Y -+ R" are Lipschitz continuous around (3,g), that the graph of 89:R" 3 R" is closed around ( G ,q) (which is automatics for continuous and for amenable functions cp), that the qualification conditions a2q(G,q)(O)
n ker D*g(ji,jj) = { 0 } ,
[o E a@,f)(&y) + D*g(3,y) 0 @cp(fi, q)(u)] ==+21 = 0 are fulfilled, and that either g is locally Lipschitzian around (3,y) or X and
Y are finite-dimensional. Then one has the inclusion D * S ( 3 , y ) ( y * )c {z* E X*l 3u E R" with
+ O*g(.,Y)
(z*, -y*) E a(%f)(3,a)
0 a2cp(.ISr,B)(4}.
Proof. Proceeding similarly to the proof of Theorems 3.1 and 3.2 and using the scalarization formula (3), we have the inclusion D * S ( Z , y ) ( y * )c {z* E X*l 3u E
R" with
, (z*,-Y*) E D*f(3,8)(4+ ~ * ( ~ c p o g ) ( ~ , Y , ~ ) ( ~ ) } ~
which becomes the equality under the assumptions in (i). Employing now appropriate chain rules for coderivatives of compositions, we arrive at the desired results of the theorem; see [5] for more details, as well as for specifications and generalizations.
4 Lipschitzian Stability of Generalized HVIs As mentioned above, our sensitivity analysis relies on coderivative characterizations of robust Lipschitzisn properties via the coderivative evaluations in Section 3. We are not going to provide in this paper such an analysis in
312
Boris S. Mordukhovich
full generality and refer the reader to [5] for more developments. Although complete coderivative characterizations of Lipschitzian properties are available in general infinite-dimensional settings as in Theorem 3.3 from [4], we use here only the following partial result; cf. also Theorem 9.40 from [9] and the references therein for the case of finite-dimensional spaces.
Theorem 4.1 (coderivative conditions for the Lipschitz-like property). Let F : X =$ Y be closed-graph around (3,y) and 5°C at this point, and let both spaces X and Y be Asplund. T h e n condition
D*F(Z,jj)(O)= (0)
(1)
is suficient f o r the Lipschitz-like property of F around ( 2 ,jj). Moreover,
when in addition dimX < 00. If both X and Y are finite-dimensional, then condition (1) is necessary and suficient for the Lipschitz-like property of F around ( 3 , j j ) and estimate (2) holds as equality. Now using the coderivative formulas from Section 3 together with SNC calculus for set-valued mappings in infinite dimensions, we derive from Theorem 4.1 efficient conditions for Lipschitzian stability of variational systems (1) and (3), respectively.
Theorem 4.2 (Lipschitzian stability for generalized HVIs with composite potentials). Let (3,jj) E g p h S f o r S defined in ( l ) ,where f:X X Y + X* x Y* with 4 := -f(z,jj), g:X x Y -+ W with tij := g(Z,g), and 'p: W -+ The following assertions hold: (i) Suppose that W is Banach, X is Asplund while Y = R", that g = g ( y ) , and that assumptions (a) and (b) of Theorem 3.1 are fulfilled with 5 defined therein. T h e n S is Lipschitz-like around (3,jj) i f and only if u = 0 E R" is the only vector satisfying
z.
0 E V,f(.,
+
+
g)*u 02(v,g)(jj)*u Og(~)*a~'p(w,21)(vg(y)u).
In in addition X i s finite-dimensional, then one has
{
lip S(3,j j ) = sup IIV, f (3,jj)*ull with - y* E V, f (3,jj)*u
+ V 2 ( W ) ( % )+ * uVg(Y)*~~Cp(W,~)(Vg(Y).21), IlY*ll 5 I}, where the maximum is attained when, in particular, W is finite-dimensional. (ii) Assume that the potential = 'p o g in (1) is strongly amenable at (3,jj) in finite dimensions, that f is locally Lipschitzian around (3,j j ) , and that the qualification conditions (2) and
Generalized Variational and Hemivariational Inequalities [(Z*,O)
E
a(U,f>(%a>
u
+
313
[V2(@,g>(.,!7)(4
G€M(l,G)
+Vg(z,y>*a2$9(*,@)(Vg(Z,y)u)]]
==+ 2* = u = 0
are fulfilled. Then S is Lipschitz-like around (5, fj) with lipS(2,ij)
{ I
5 s u p I(z*l( 3u E X x Y with (z*, -y*)
E
a(u, f)(Z,g)
73 EM ( 3,g)
Theorem 4.3 (Lipschitzian stability for generalized HVIs with composite fields). Let (Z,y) E g p h S with S defined in (3), where X and Y are Asplund, and where g : X x Y + R" and f : X x Y + R" are strictly differentiable at ( 3 , j j ) . Assume that the graph of 89 is closed around ( i i j , q ) , that
and that one has
Then S is Lipschitz-like around (Z, y). If in addition dim X lip S(Z, jj) 5 sup 2* - V,f(Z,
3u E Rm,y*
E V,g(Z,
< 00, then
g)*a2p(w,Q ) ( u )with
+
g)*u E V,g(Z, y)*d2p(w,Q)(u),IIVyf(2, Y)*u Y*II 5 I}.
References 1. Aubin, J.-P. (1984), Lipschitz behavior of solutions to convex minimization problems, Mathematics in Operations Research, Vol. 9, pp. 87-111. 2. Fabian, M. and Mordukhovich, B.S. (2003), Sequential normal compactness versus topological normal compactness in variational analysis, Nonlinear Analysis, Vol. 54, pp. 1057-1067. 3. Mordukhovich, B.S. (2002), Calculus of second-order subdifferentials in infinite dimensions, Control and Cybernetics, Vol. 31, pp. 557-573. 4. Mordukhovich, B.S. (20031, Coderivative analysis of variational systems, Journal of Global Optimization, Vol. 28, pp.347-362. 5. Mordukhovich, B.S. (2005), Variational Analysis and Generalized Differentiation, Vol. I Basic Theory, Vol. 11: Applications, Springer. 6. Mordukhovich, B.S. and Shao, Y. (1996), Nonsmooth sequential analysis in Asplund spaces, Transactions of the American Mathematical Society, Vol. 349, pp. 1235-1280.
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7. Mordukhovich, B.S. and Wang, Y . (2003), Calculus of sequential normal compactness in variational analysis, Journal of Mathematical Analysis and Applications, Vol. 282, pp. 63-84. 8. Robinson, S.M. (1979), Generalized equations and their solutions, part I: basic theory, Mathematical Programming Study, Vol. 10, pp. 128-141. 9. Rockafellar, R.T. and Wets, R.J.-B. (1998), Variational Analysis, Springer, Berlin.
A Smoothing Property of Schrodinger Equations and a Global Existence Result for Derivative Nonlinear Equations Mitsuru Sugimotol and Michael Ruzhansky2 Department of Mathematics Graduate School of Science Osaka University Toyonaka Osaka 560-0043
Japan sugimotoQmath.wani.osaka-u.ac
Department of Mathematics Imperial College 180 Queen's Gate London SW7 2BZ UK ruzhQic.ac.uk
Summary. We give a global (in time) existence result for a class of derivative nonlinear equations based on a global smoothing property of generalized Schrodinger equations.
1 Derivative Nonlinear Schrodinger Equation What is the condition on the initial data p for the equation
(a+ A,) u(t,z) =(Vu(t,z)lN u(0,z) = 'p(z), t E
R,z E R"
to have a time global solution? Here are some answers: 0
(Chihara 1996 . . .N 2 3). Smooth, rapidly decay, and sufficiently small. (Hayashi-Miao-Naumkin 1999 ... N 2 2). 'p E H["/21f5,rapidly decay, and sufficiently small.
In the condition above, can we weaken the smoothness assumption for The purpose of this article is t o answer it affirmatively. We can replace the regularity index [n/2] 5 by a smaller one if the non-linear term has a structure! We consider the equation 'p?
+
316
Mitsuru Sugimoto and Michael Ruzhansky
~ ( 0X ), = c p ( ~ ) , t E R, x E R". Here (x)= (1
+ lx12)1'z and
+
Theorem 1 Assume N 2 4, s > ( n 3 ) / 2 + ( N - 3 ) / ( 2 N - 2 ) . Suppose that (x)(DX)'cpE L2 and its L2-norm is suficiently small. Then equation ( 2 ) has a time global solution. But we still have questions: 0 0
Question 1 : What happens in the cases N = 2 , 3 ? Question 2 : What about the relation between the linear term and the structure of the nonlinear term?
To answer them, we generalize the linear term -Axto L,, where
A V, for -Ax, we By finding the structure o ( X ,Dx)for L p instead of answer the questions. Assume that the Gaussian curvature of the hypersurface
EP = {E;p(E>= 1) never vanishes (p(E) = eralized equation:
{
is the usual case!), and consider the following gen-
(28, - L p )u(t,x) =Io(X, Dx)uIN ~ ( 0 , x=) c p ( ~ ) , t E R,2 E R".
(3)
(In the case N = 2 , replace o ( X ,D,) by o ( X , We remark that the classical orbit { ( ~ ( tE )( t ,) ) ; t 2 0} associated to (3) is the solution of j.(t) = VEP2(E(t)), =0 (4) ~ ( 0= ) 0, E(0) = k.
{
i(t)
We define the set of the path of all classical orbits:
rp= { ( x ( t ) , t ( t ) )sol. ; of (4),t 1 0 , k E R" \ 0 } = { ( t W E ) ,0;E E R" \ 0, t 2 0).
Schrodinger Equations and Derivative Nonlinear Equations
317
By using this terminology, we assume the structure for the non-linear term:
(In the case N = 2, assume a(x,<) used the notation
N
*
a(z7t)
{
(x)o/(11/2instead.) Here we have
(4"Itlb
+, 0 E C" (R:
x (Ri"\ 0 ) ),
= A%(x,t) ; (A
.(Ax,()
o(x,A() = Abo(z,t ); (A
> 1, 1x1 >> 1)) > 0).
Under these assumptions, we have the following theorem which answers our questions:
Theorem 2 Suppose n 2 2, s > (n+3)/2. Assume that (x>(DZ)'pE L2 and its L2-norm is suficiently small. Then equation (3) has a time global solution. (In the case N = 2, assume ( Z ) ~ ( D ~E)L2 ~ (instead.) P Examples of nonlinear terms which satisfy ( 5 ) in the case L, = -Ax:
; 1 (1
a(x,t )=
2
A
1 t 1 1
for large x ( N 2 3),
4
a(x,t)2= - A t
1i:
I
for large x ( N = 2).
2 Smoothing effect of Schrodinger equations The proof of Theorem 2 is based on a global smoothing property of generalized Schrodinger equations. Let L, = P ( D , ) ~be as before, and consider the equation - Lp)u ( t ,x) = 0 (6) ~ ( 0 , x=) p(x) E L2(R").
Theorem 3 Suppose n 2 2. Assume
+,t) = o on r,,
+,c)
(x)-1/21t11/2.
Then the solution u to equation ( 6 ) satisfies
a ( X ,D z ) u ( t ,x) E L2(Rt x R:).
(7)
318
Mitsuru Sugimoto and Michael Ruzhansky
c)
There are some previous results without the structure assumption o(x, = 0 on rp.In fact, (7) is true for the following: 0 0
(Z)-'/~I~/~
(Ben-Artzi & Klainerman 1992) a(z,()= (Kato & Yajima 1989) a(z,E) = lxla-lIEla (0 < a
(s > 1/2) < 1/2)
Theorem 3 means that we can take the critical index s = 1/2 or a = 1/2 if we assume the structure condition. We have a similar result for inhomogeneous equations
Theorem 4 Suppose n 2 2 and k E N. Assume
Then the solution u to ( 8 ) satisfies the estimate
Combining Theorems 3 and 4, we have an a priori estimate for solutions of the Cauchy problem
Corollary 5 Suppose n 2 2, s,S 2 0, and k E N. Assume
+,E) 4x7 E )
1 0 , +,E)
- (4°1E11/k.
= 0 on r
p
Then the solution u to (9) satisfies the estimate
N
To prove Theorem 2, use Corollary 5 with f = Io(X,D,)'uI key point is that the space H,"(Hf) is an algebra if s > 1/2 and fact, we have
. Then, the > n/2. In
Schrodinger Equations and Derivative Nonlinear Equations
319
3 Mapping property of Fourier integral operators Finally, we mention the main tool for proving the results in Section 2. Let
be a Fourier integral operator with an amplitude function u = u ( z , y , < ) and a real phase function 4 = 4(z, y, (). Especially, we need the case
In fact, we have L, .T = T
*
(-Ax)
if u ( z ,y, () = 1. By this relation, the L2-property of -Ax can be interpreted as that of L,. To justify the argument above, we need the (weighted) global L2-boundedness of the operator T . Then, what is the condition for T to be L2(R")-bounded? About this question, we knew only the following sufficient condition: 0
(Asada-Fujiwara 1978) All the derivatives of u ( z , y , c ) and all the derivatives of each element of D ( 4 ) is bounded and (detD(q5)I 2 C > 0, where
But we cannot use the result of Asada-Fujiwara since q& is not bounded for 4 given by (10). Hence we need the following extended result. For m E R, let LL(R") be the set of functions f such that the norm
is finite.
320
Mitsuru Sugimoto and Michael Ruzhansky
References [l] K. Asada and D. F'ujiwara, O n some oscillatory integral transformations in L2(R"), Japan. J. Math. (N.S.) 4 (1978), 299-361. [2] M. Ben-Artzi and S. Klainerman, Decay and regularity f o r the Schrodinger equation, J. Analyse Math. 58 (1992), 25-37. [3] H. Chihara, The initial value problem f o r cubic semilinear Schrodinger equations, Publ. Res. Inst. Math. Sci. 32 (1996), 445-471. [4] N. Hayashi, C. Mia0 and P. I. Naumkin, Global existence of small solutions to the generalized derivative nonlinear Schrodinger equation, Asymptot. Anal. 21 (1999), 133-147. [5] T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), 481-496.
Existence and Blow up for a Wave Equation with a Cubic Convolution Kimitoshi Tsutaya* Department of Mathematics Hokkaido University Sapporo 060-0810 Japan tsutayaQmath.sci.hokudai.ac.jp
Summary. We consider a wave equation with a cubic convolution together with a potential in three space dimensions. We show the borderline value of the decay rate at infinity of the initial data between global existence and blow up.
1 Introduction This article is based on the paper [13] and is intended to present recent developments on the wave equation with a cubic convolution. We consider the wave equation @U - AU = V ~ ( X ) U (V2 * U')U (1)
+
for z E R3, where V l ( x ) = O(lxl-") as 1x1 + 00, Vz(x) = v21z1-YZ, u2 E R, and * denotes spatial convolution. The Schrodinger equation with the interaction term Vl(x)u (V2 * u2)uwas studied by Hayashi and Ozawa [6]. See also Coclite and Georgiev [2]. We study the global existence of smooth solutions of (1)for small initial data. Moreover, in this paper the potential V1 is assumed to be small since the solution may blow up in a finite time unless VI is small. See Strauss and Tsutaya [ll].In the case V l ( x )= 0, the initial data have small amplitude, and V2(x)satisfies some conditions, it is known that smooth solutions exist for all time. Hidano [4]proved not only the global existence but also the existence of the scattering operator for 2 < 79 < 5/2 using the Lorentz invariance method. For other dimensional cases and the Klein-Gordon equation with a cubic convolution, we refer to Menzala and Strauss [8],Mochizuki and Motai [9],and Mochizuki [lo].
+
*Supported by a JSPS research grant.
322
Kimitoshi Tsutaya
One aim of this article is to permit V ~ ( Z which ) is small and decays like IxI-71. We show that if y1 > 2, 2 < y 2 < 3, and the small initial data decay like lxl-l-k with k > 1 (3 - 7 2 ) / 2 , then smooth solutions exist for all time, which improves on the requirement 2 < 7 2 < 5 / 2 in Hidano [4]. On the other hand, we show that if any of these three conditions are relaxed, then there exist arbitrarily small initial data such that the corresponding solutions blow up in a finite time.
+
2 Global Existence In this section we state the global existence theorem. We consider the Cauchy problem 8,". - AU = Vi(z)U (V2 * u ~ ) ut ,E R,x E R3,
{
+
(2)
u ( 0 , z )= cp(.>,
& U ( O , 4 = $J(z), z
where V ~ ( X = )~ 2 1 ~ 1 - 7 2with v2 E R and we give some hypotheses : (Hl) The potential Vl(x)satisfies
72
E
R3,
> 0. Before stating our results,
with y1 > 0, where v1 > 0 is a small parameter. (H2)The data p(z) E C3(R3),+(z) E C2(R3)satisfy
with k > 0, where E
> 0 is a small parameter.
Theorem 1 Consider the problem (2). Assume the hypotheses ( H l ) and (H2). If 71 > 2, 2 < 7 2 < 3, k > 1+(3-72)/2, and if E and v1 are sufficiently small, depending on k , 71, 7 2 and v2, then there exists a unique global C2-solution u = u(t,z) of (2). Moreover, for the solution u of (2), there exist unique C2-solutions u&(2,t ) of d,"u - Au = 0 such that I(u(t)- uk(t)IIe
provided that k
Sketch of
> 716, where
Proof of Theorem 1.
+
0
(t
-+
*m),
(3)
Existence and Blow up for a Wave Equation with a Cubic Convolution
323
To prove Theorem 2.1 we consider the linear problem 6’:.
- AU= f ( t , z),
t E R , xER3,
~ ( 0z), = c~(z), &u(O, X) = $(z),2 E R3.
(4)
The solution u = u ( t , x ) of (4)is given by
is the solution of d ; u - A u = 0 with the initial data cp(z), $(z), where uO(t,z) and
is the solution of 6 ’ ; u - A ~ = f with zero data, where dw is the surface measure on the unit sphere Iw( = 1. We use the following lemma:
Lemma 2 (Asakura [l])Suppose that p(z), $(z) satisfy (H.2). Let k > 1. Then the solution uo of 6’:u0 - Au’ = 0 with the data cp(z), $(x) satisfies
where the constant
c k
depends only o n k .
We next introduce the weight function
4 t 7 4 = (1+ It1 + 14N + lltl - 141)m, where rn = min(1, k - l}, and define the norm for functions u = u(t,z) by
The following lemma is the basic estimate for the existence proof. Lemma 3 Let V l ( z )satisfy (HI). If 7 1 > 2, 2 < 7 2 < 3 and k > 1+(3-72)/2, then there exists a constant C > 0 depending only o n k, 71, 7 2 and u2 such that IIL[lV14+ I(& * U2)~1111I C ( ~ 1 I I U I+ l 11~113>.
Using Lemmas 2 and 3, we can prove Theorem 1 by the contraction mapping principle. The result (3) is obtained from IIVIU
+ (V2 *
U2)U(t)II
E L1(0,m).
324
Kimitoshi Tsutaya
3 Blow up In this section we state a blow up result. We consider (1) for t > 0. We can verify local existence and uniqueness for the problem, and existence of non-negative solutions, provided that
v,E L”, v, 2 0,
v2
> 0, 0
‘p f 0,
$ 2 0.
(5)
Let T be the existence time.
Theorem 4 Assume the hypothesis (5). Let one of the following conditions be satisfied: (i) 0 < y1 < 2 and V~(X) 2 C(1+ I z I ) - Y 1 for some constant C > 0 , (ii) 0 < y2 < 2, (iii) 1/2 < k < 1+(3-y2)/2 and $(x) 2 &(1+1 ~ 1 ) - ~ - for ~ some constant & > 0. Then T < 00. The result under the condition (ii) is due to Hidano [4].The proofs for the conditions (i) and (iii) are based on the method of Glassey [3] and that of John [ 5 ] , respectively.
References 1. F. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations 11, No.13 (1986), 1459-1487. 2. G.M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrodinger equations, preprint. 3. R.T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177 (1981), 323-340. 4. K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcialaj Ekvacioj 43 (2000), 559-588. 5. F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235-268. 6. N. Hayashi and T. Ozawa, Smoothing effect for some Schrodinger equations, J. Functional Analysis 85 (1989), 307-348. 7. J. B. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10 (1957), 523-530. 8. G. Perla Menzala and W. A. Strauss, On a wave equation with a cubic convolution, J. Diff. Eq. 43 (1982), 93-105. 9. K. Mochizuki and T. Motai, On small data scattering for some nonlinear wave equations, “Patterns and waves-qualitative analysis of nonlinear differential equations”, Stud. Math. Appl. N0.18, 543-560, North-Holland, Amsterdam, 1986. 10. K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. SOC.Japan 41 (1989), 143-160.
Existence and Blow up for a Wave Equation with a Cubic Convolution
325
11. W. A. Strauss and K. Tsutaya, Existence and blow up of small amplitude nonlinear waves with a negative potential, Discrete and continuous Dynamical Systems 3 (1997), 175-188. 12. K. Tsutaya, Global existence and the life span of solutions of semilinear wave equations with data of non compact support in three space dimensions, Funkcialaj Ekvacioj 37 (1994), 1-18. 13. K. Tsutaya, Global existence and blow up for a wave equation with a potential and a cubic convolution, to appear.
Zeros and Signs of Solutions for Some Reaction-Diffusion Systems Hiroshi Uesaka Department of Mathematics College of Science and Technology Nihon University Japan uesaka0math.cst.nihon-u.ac.jp
Summary. We prove two theorems on the change of sign of the components of the solutions of certain reaction-diffusion systems.
1 Introduction Let 0 be a bounded domain in R", n = 1 , 2 , 3 , ... with a smooth boundary In this paper we shall be concerned only with real-valued functions and real quantities. We consider the following problem,
an.
{
+ + + + + +
atu = a1Du blu ClW M ( u ,w), &w = a2Au b2u c2u N ( u ,w), in f2 x [0, oo), u(z,O)= uo(z) and w ( z , O ) = wo(z)in 0, B(u) = B ( u ) = 0 on 8 0 x [ O , o o ) ,
(1.1)
where A denotes the usual Laplace operator in R" and B(.) = 0 means some self-adjoint homogeneous boundary condition on d o . The examples of B ( - )= 0 are such as the Dirichlet, the Neumann or the Robin homogeneous boundary condition. B ( - )= 0 is given so that -A has the self-adjoint realization in L 2 ( n ) .The initial data U O , V O is assumed to satisfy suitable regularity and suitable compatibility conditions necessary to assure the global existence of smooth solutions and the uniqueness. M , N are semi-linear terms defined later. We set the assumptions as follows:
[A4
There exists a pair of a unique global solution of (1.1) and it has regularity u). enough to justify calculation in this paper. We denote its pair by (u, Refer to Section 2 of Chapter 9 in [3] for a simple example of (1.1)satisfying
[A. 13.
328
Hiroshi Uesaka
Our aim in this paper is to show that the component u and/or the component w of a solution (1.1) changes sign under suitable assumptions on the coefficients and the nonlinear terms M and N . We shall prove two theorems in Section 3. In Theorem 3.1 we shall show that u necessarily changes sign independent of the sign of v, but we can show nothing about the sign of v. In Theorem 3.2 we shall show that at least either u or w changes its sign. Acknowledgment. I express my sincere gratitude to the referee for carefully reading and correcting my manuscript, and for giving useful comments. I also thank Mr. J. Wirth of TU Bergakademie Freiberg so much for kindly helping me make my manuscript compatible with the style file and for correcting the manuscript.
2 Preliminaries We use a well-known fact concerning the first eigenvalue and its eigenfunction
of -A in 0 with a self-adjoint homogeneous boundary condition. FACTLet +(x) be an eigenfunction of a self-adjoint realization of -A associated with the first non-negative eigenwalue A. Then +(x) has always one sign in 0, which we may choose to be positive, and +(z) E Cm(f2) n C(n). For the proof of the fact we refer to Section 7 of Chapter 6 of [2]. Let ( u , ~are ) the solution of ( l . l ) ,and +(z) be the above introduced eigenfunction corresponding to the boundary condition of (u, v). We set u(z,t ) + ( z ) d z and
V ( t )=
Then we have the following lemma.
Lemma 2.1 Let U and V be the functions defined above, then
{ holds, where
+ +
U’(t) = (bl - a l A ) U ( t ) c l V ( t ) V’(t)= (b2 - a z A ) V ( t ) c2U(t)
+J +
M ( u ,v)+dz, N ( u ,v)+dx,
(2.1)
’ denotes differentiation with respect t o t .
Proof. Multiplying the first equation in (1.1) by the eigenfunction +(z) and applying the divergence formula to Ja Au4dx, the first equation of (2.1) is obtained. The second equation of (2.1) follows in the same way. We plan to deduce our results from a system of ordinary differential inequalities obtained from (2.1) by making suitable sign assumptions on the nonlinearities M and N . To prepare for the study of such inequalities we consider first an ordinary differential system of the form
{
+ +
W’ = a3W c3x, X’ = a4X c4w,
Zeros and Signs of Solutions for Some Reaction-Diffusion Systems
329
with real coefficients, U k , C k , k = 3 , 4 , and we assume that these coefficients satisfy the following assumption:
c3
> 0 and a4 + (a3 - a4)2 > 0. 44
For this situation we recall the following result from [5].
Lemma 2.2 Let W,X be a solution of (2.2) with coeficients satisfying [A.2]. Then W has infinitely many zeros in the interval [p, m ) , p E R, p arbitrary. We will obtain our results by applying comparison arguments linking the solution U,V of the resulting ordinary differential inequalities with a solution of an ordinary differential system of the form (2.2). The comparison will be done after applying the Priifer transform. In order to utilize the comparison theorem we assume for the coefficients occurring in (2.1) and (2.2) the following relation:
~ . 3 1 1. c1 1 c3 > 0 , 2. c2 2 c4, 3. ( C i - c3)(.2 - C 4 ) 2 +[(bi- b 2 ) -t(a2 - ai)A - (a3 - ~
4 ) ] ~ ,
3 Theorems Let u,v denote a solution of (1.1). We shall state two theorems on the behavior of the solution. In Theorem 3.1 we shall show that the sign of u necessarily changes independent of the sign of u,but we can show nothing about the sign of v. In Theorem 3.2 we shall show that u or v changes its sign. To formulate our results we make the following general assumptions concerning the non-linear terms M and N .
~4.41
v M ( u , v ) 2 0, and M ( 0 , v ) = 0 for all u , v E R. Remark. An example of M ( u ,v): M ( u ,u)= 3u2 e - ( u 2 + 2 )
~
5
1
u N ( u ,v) 5 0, and N ( 0 ,w) = 0 for all u,v E R. Remark. An example of N ( u ,v): N ( u ,v) = - ~ ~ v ~ e - ( ( " ~ + " ~ ) . We have the following theorems.
Theorem 3.1 Let M ( u , u ) = 0. Assume that [A.l], [A.2], [A.3] and [A.5]. Then the component u of a solution (1.1) changes its sign.
Theorem 3.2 Assume that [A.l], [A.2], [A.3], [A.4] and [A.5]. Then at least either u or v changes its sign.
Hiroshi Uesaka
330
4 Proofs of Theorems In order to prove Theorem 3.1 and Theorem 3.2, we shall apply the Prufer transform for the pair (U,V )and another pair (W,X). We use arguments and results by BGcher [l],Kamke [5], Kreith [6] and Uesaka [7].
Proof of Theorem 3.1. In this case we assume M ( u , v ) = 0. Thus the first equation of (1.1) is linear. Let u ( x , ~be) positive for some T 2 0 in O. We shall show that there exists (x,t ) E 0 x (7, m) such that u(x,t ) < 0. We omit the proof for u(x,T) < 0 because the argument of the proof is the same one as for U ( X , T ) > 0. Let T = sup{t Iu(x,s) > 0 for T 5 Vs < t , b’x E O}. By a contradiction argument we shall show that T < 00 and that there exists (20,t o ) E O x [T,00) such that u(x0,t o ) < 0 . So assume T = 00. Noting N ( u , v ) 5 0 for u > 0, we have from (2.1)
U f ( t )= ( b l - UlA)U(t)+ ClV(t> (b2 - uzX)V(t) c2U(t>.
{ V f ( t )I
+
(4.1)
If T = 00, U ( t ) > 0 always. We would like to show that U ( t ) has zeros. For that purpose we use the Prufer transform for U,V of (4.1). Let
U ( t ) = p ( t ) sin8 and V ( t )= p ( t ) cos8. We note that p and 8 are functions o f t , and p 2 0. If p ( i ) = 0 for some
i, U ( 0 = V ( i )= 0. Hence we assume that p ( t ) > 0. It follows from U ( t ) > 0 that 0
< 8 < 7r
{
and sin8 > 0. From (4.1) we obtain
+
pf sin8 p8’ cos 8 = (bl - ulX ) p( t ) sin 8 pf cos 8 - pBf sin 8 5 (b2 - ~ 2 X ) p ( tcos ) 8
+ clp(t)cos 8 , + c 2 p ( t )sin 8.
(4.2)
Compute the difference of the second equation of (4.2) multiplied by sin 8 and the first line of (4.2) multiplied by cost), and then we have
{
-pet
5
+
(b2 - u2X)p(t) cos 8 sin 8 czp(t>sin2 8 - u l X ) p ( t )sinecose - clp(t)C O S ~8.
-(bl
(4.3)
Dividing (4.3) by p, we have
O W ’ 2W W ,
+
+
(4.4)
where F(O(t>)= c1 cos28 [(bl - b 2 ) (u2 - ul)X] sin 8 cos 8 - c2 sin2 8. We also apply the Prufer transform for X and W of (2.2) in the same manner as for U and V . Setting
W ( t )= r ( t )sin8, X ( t ) = r ( t )cos8,
Zeros and Signs of Solutions for Some Reaction-Diffusion Systems
331
+
where f(6(t))= c3 cos26 (a3 - a4) sin6 cos 6 - c4 sin219. We would like to compare O ( t ) with 6(t).Subtracting f(cp) from F(cp), we have
F(cp) - f(cp) = (c1 - c3) cos2cp + [(bl - b2) -(c2
- cq) sin
2
+ (a2 - a l ) -~ (a3 -
ad)]
coscpsincp
cp.
(4.6)
By the assumption [A.3] we can compare F with f and we have F(cp) f(cp) 2 0 for any 9.We put d(0) = O(0). Then by virtue of the comparison theorem (see p. 27 of [4]) we can conclude that
W )I w .
(4-7)
[A.2],which concerns existence of infinitely many zeros, is assumed and hence Lemma 2.2 holds for (2.2). Consequently W has infinitely many zeros. Let a! and p be successive zeros of W . We assume that Q < ,B and 6(a!)= 0. Then we get S(p) = 7r because of 6(a)’= S(p)’ = c3 > 0. Hence there exists t o such that O ( t 0 ) = 7~ and a! < t o < p. Thus
Thus we can maintain that
I. There exist $1, z 2 E f2 such that u ( q ,to)u(zz,t o ) < 0 at t = t o , or 11. u(z,to)E 0 and u(x,t)2 0 for any z E f2 in the neighborhood of to.
I is contradictory to the assumption of T = 00. But we can show that I1 also does not hold as follows. If I1 holds, then we have Au(z,to) = 0 and t --t u(.,t)is positive in the neighborhood of t o except t # to. Hence u(.,t) has a minimum value at t = to. and so ut(z,t o ) = 0 for any E R. Then it follows from (1.1)that v(z,to)= 0 for any z E 0. Owing to the assumption of the uniqueness of a solution in [A.1] we obtain that u = w = 0 in f2 x [0, co). This is contradictory. Thus we can conclude that T < 00 and there exists (z, t ) with t < 00 such that u ( z ,t ) < 0 or u(z,t ) = 0 holds. We shall show further that there exists certainly (z, t ) E 0 x (0, co) such that u(z,t) < 0 holds. Assume that u ( z , t )2 0 for any ( z , t ) E R x [O,co). We consider the following two cases. (1).For any fixed T >_ 0 there exist t‘ > T such that U(t’) = 0 holds , or (2). U ( t ) > 0 holds if t
> 7 for some T > 0 . If case (1) holds, then we have the same fact as I or 11. On the other hand the case (2) does not hold because by the same argument as used to deduce
332
Hiroshi Uesaka
I and I1 owing to the Priifer transform and the comparison theorem there exists certainly t such that U ( t ) = 0. Hence there exists (x,t ) with T < t < 00 such that u(x,t)< 0 holds. Thus we have obtained the desired result and so 0 the proof of Theorem 3.1 is complete. Now we shall prove Theorem 3.2. In Theorem 3.2 a nonlinear term M ( u ,w) is actually put on the first line of (1.1). Proof of Theorem 3.2. A nonlinear term M ( u , v ) in Theorem 3.2 is actually put on the first equation of (1.1). We assume that u > 0 and w > 0 in 0 x (7,T ) with T 5 00 for some T 2 0. Other cases such as u < 0 and w > 0, or u > 0 and w < 0, or u < 0 and w < 0 can also be proved in the same argument as in u > 0 and w > 0 and therefore we omit the proof. Let
T
= sup{t Iu(x,s) > 0 and w(x,s) > 0 for T
5 Vs < t , Vx E 0).
We shall show T < 00. We assume that T = 00. By [A. 41, M ( u ,v ) d ( z ) d x2 0; it follows from (2.1) that uyt) 2 (bl - U l A ) U ( t ) ClV(t), (4.8) vyt) 5 (bz - azX)V(t) czU(t), for 0 5 t 2 T . We also apply the Priifer transform for U and V . Because of U,V > 0 we have 0 < 9 < 1r/2 and
+
{
+
U ( t ) = p(t) sin 9, V(t) = p(t) cos 9. In the same manner as in the proof of Theorem 3.1 we have
VW), +
9(tY 2 (4.9) where F(O(t))= c1 cos2 9 [(bl - b z ) (a2 - a l ) A ]sin Bcos 9 - c2 sin2 9. Also we have from the Priifer transform of W = r cos 6 and X = r sin 6,
+
W' 2 f(+(t)),
(4.10)
+
where f(d(t)) = q c02 6 (a3 - u4) sin 6 cos 6 - c4 sin2 6. So we have from [A.3] that F ( p ) - f(cp) 2 0 for any 9. We put 6 ( 0 ) = 9(0). Then by virtue of the comparison theorem (see page 27 of [4])we get, for 0 5 t < T ,
6 ( t ) i 9@).
(4.11)
W ( t )has necessarily infinitely many zeros. Let cy and p be successive zeros < /3 and 6 ( a ) = 0. Then we get S(p) = K , which contradicts 0 < 9 < 7r/2 and the inequality (4.11). Therefore u > 0 and w > 0 in 0 for any t > T is impossible. Then T < 00 and at least either u > 0 or w > 0 becomes non-positive for some t. The argument of proving that at least one of u(x,t ) < 0 or w(x,t ) < 0 holds for some (x,t)is almost similar to the corresponding part of the proof of Theorem 3.1. Thus we obtain that at least one of u or w changes its sign.0 of W . We assume that a
Zeros and Signs of Solutions for Some Reaction-Diffusion Systems
333
References 1. M. Bather, On the real solutions of systems of two homogeneous linear differential equations of the first order, Trans. Amer. Math. SOC.1 (1900), 414-420. 2. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol.1, Interscience, 1953. 3. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, 1998. 4. P. Hartman, Ordinary Differential Equations, Birkhauser, 1982. 5. E. Kamke, Uber Sturms Vergleichssatze fiir homogene lineare Differentialgleichungen zweiter Ordnung und Systeme von zwei Differentialgleichungen erster Ordnung, Math. Z. 47 (1942), 788-795. 6. K . Kreith, Oscillation Theory, Lect. Notes in Math. No.324, Springer, 1973. 7. H. Uesaka, Oscillation of solutions of nonlinear wave equations, Proc. Japan Acad., 72 (1996), 148-151.
Constructions of Approximate Solutions for Linear Differential Equations by Reproducing Kernels and Inverse Problems M. Asaduzzamanl, T. Matsuura2, and S. Saitoh3 Department of Mathematics Faculty of Engineering Gunma University Kiryu 376-8515 Japan asad0eng.gunma-u.ac.jp Department of Mechanical Engineering Faculty of Engineering Gunma University Kiryu 376-8515 Japan matsuurabe.gunma-u.ac.jp Department of Mathematics Faculty of Engineering Gunma University Kiryu 376-8515 Japan ssaitohc0math.sci.gunma-u.ac.j p
Summary. In this paper we introduce a unified method of solving inverse problems in some general linear differential equations numerically and as a prototype example we shall show a practical real inversion formula for the Gaussian convolution.
1 Introduction Let L : H K + H be a bounded linear operator between Hilbert spaces. We consider the best approximation problem
where g E H is given. If there exists an element f* E HK which attains the infimum (1) then the problem (1) is called solvable otherwise it is called unsolvable. If HK is a reproducing kernel Hilbert space admitting a reproducing
336
M. Asaduzzaman, T. Matsuura, and S. Saitoh
kernel K ( p , q ) on a set E then whether the problem (1) is solvable or not, the problem
is always solvable for all X > 0 and we can obtain a method of obtaining the member fi,g of the smallest norm in H K which attains the infimum (2). The problem (2) is called a Tikhonov regularization for the problem (1). If the problem (1) is solvable then fg* := 1imx-o fi,g in HK and fg* is the element of the minimum norm which attains the infimum (1)([9]). If the problem (1) is unsolvable then limx-,o fi,g does not exist in H K . Even though the problem (1) is unsolvable, for a value of A, we can think of fi,gas a generalized solution of Lf = g .
2 Background Theorems Theorem 1 ( [ 2 , 5 ] ) . Let H K be a Hilbert space admitting the reproducing kernel K ( p , q ) o n a set E . Let L : HK + 'H be a bounded linear operator between Hilbert spaces. Then, for g E 'H the problem
as solvable if and only if L*g E H k , where Hk is the reproducing kernel Hilbert space admitting the RK k ( p , q ) = ( L * L K ( . , q ) ,L * L K ( - , P ) ) o~n, E . Furthermore, i f the problem (3) is solvable then f,*(P) = W*g, L * W . ,P ) ) H k
is the element of HK with the smallest n o r m which attains the infimum (3).
Theorem 2 ([7]). Let H K , L , 'H and E be as in Theorem 1 and let V be the underlying vector space of H K . For X > 0 introduce a structure in V and call it HK* as follows:
(fl,f 2 ) H K A = ' ( f l , f 2 ) H K + ( L f 1 ,L f 2 ) x .
(4)
Then HK* is a Hilbert space with the reproducing kernel K x ( p , q ) o n E satisfying the equation K ( * 4, ) = ( X I + L * L ) K x ( . ,Q ) , (5) where L* is the adjoint of L : HK
+ 'H.
Theorem 3 ([7]). Let H K , L, 'H and E be as in Theorem 1 and let Kx be as in Theorem 2. Then, for any X > 0 and for any g E 'H, the approximation problem
Constructions of Approximate Solutions
337
is the member of HK with the smallest HK-norm which attains the infimum (6).
3 PDE and Inverse Problems We will be able to apply our theory to various inverse problems to look for the whole data from local data of the domain or from some boundary data. Here, we will refer to these problems with a prototype example in order to show this basic idea, clearly, from [6]. We recall a Sobolev imbedding theorem ([l], pp. 18-19). In order to use the results in the framework of Hilbert spaces, we assume p = q = 2 there. Let Wl(G) (l = 0'1'2, ...) be the Sobolev Hilbert space on G, where G c R" is a bounded domain with a one piecewise-smooth continuously differentiable boundary
r. We assume that I c > e - - .n2
Let m = 0 , 1 , 2 , ... such that
Let D C G W be any 1 times continuously differentiable manifold of dimension m. Then, for any u E W i ( G ) ,the derivative ( D " u ) ( z )E La(D)(rcE D), where la1 5 Ic, and we have the continuity of the imbedding operator IIDQuIIL2(D)
5 M11UllW2((G)y ( M > o;u E wi(G))*
(10)
Of course llullWi(G)
5
IIulIW2((Rn),
(11)
by using and we can construct the reproducing kernel for the space Wi(Rn) Fourier's integral for 21 > n. Then, for any linear differential operator L with variable coefficients on G satisfying IlLullL2(G)
< lIullW~(G)
(12)
and for any linear (boundary) operator B with variable coefficients on D satisfying
338
M. Asaduzzaman, T. Matsuura, and S. Saitoh ~ ~ B u ~ ~< ~~ z~ (u ~~ )~ W ~ ( G ) ~
we can discuss the best approximation: For any L2(D) and for any X > 0,
F1
(13)
E L2(G) , for any F2 E
If F1 = 0 and X is very close to zero then the problem may be interpreted that we wish to construct the solution u of the differential equation
Lu=O
on
G
on
D.
satisfying
Bu=
F2
Our general theory gives a practical construction method for this inverse problem that from the observation F2 on the part D, we construct u on the whole domain G satisfying the equation Lu = 0.
4 Gaussian Convolution In this section we will consider the integral operator Lt : H s bs
+ L z ( R ) defined
for given t > 0. Here H s is the first order Sobolev Hilbert space on the whole real line with norm defined by
J-CCl
admitting the reproducing kernel
We now consider the best approximation problem: For any given g E L2(R) and for X > 0.
Then for the RKHS H K consisting ~ of all the members of H s with the norm
Constructions of Approximate Solutions
339
the reproducing kernel Kx(z,y ) can be calculated directly by using Fourier’s integrals as follows:
Hence the unique member of H s with the minimum Hs-norm which attains the infimum (18) is given by
For f E H s and for g(<) = (Ltf)((), we have the favourable formula
uniformly on R ( [ 7 ] ) . Twenty years ago, the last-named author gave a surprising characterization of the image of (15) for L2(R) = L2(R,dx) functions in terms of an analytic function and established a very simple complex inversion formula. The paper created a new method and many applications to general integral transforms in the framework of Hilbert spaces and various analytic extension formulas ([5]).However, in particular, its real inversion formulas are very involved and one might think that its real inversion formulas will be essentially involved for catching “analyticity” in terms of the data on the real line as in the real inversion formulas of the Laplace transform. This is a typical and famous illposed problem. See [7] for more details. For example, recall the classical real inversion of the Gaussian convolution formula: For a bounded and continuous function f(z) and for t = 1, e-02 ( ~ ~ f ( = z )f(z) ) pointwise on
w.
R
([41, P. The real inversion formula (21) will give a practical formula for the Gaussian convolution. We will show experimental results by computers in Figures 1 and 2. There, we will see that in order to overcome the high “ill-posedness” in the real inversion and in order to catch “analyticity” of the image of (15) we must work hard; that is, we must take a very small X and we must calculate the integral (21) with difficulty in the numerical sense. Computers help us to calculate the integral for very small A. Meanwhile, for any X > 0 and any t > 0, we shall define a linear mapping
340
M. Asaduzzaman, T. Matsuura, and S. Saitoh
Fig. 1. For t = 1 and for g(z) = X [ - ~ , ~the I , inversion (21) for X = lO-”(the smaller one) and X = 10-23(the larger one).
Fig. 2. The images of Figure 1 by the Gaussian convolution: the bold curve corresponds to the larger curve of Figure 1 and the other one corresponds to the other one.
by MA,t(g) = fi,g.Now, we consider the composite operators L t M ~ , tand MA,&. Using Fourier’s integrals it can be shown that for f E Hs,
Constructions of Approximate Solutions and for g E L2(R),
lim MA,tLt = I
A+O
and lim LtMx,t = I .
A+O
The precise meaning of (26) and (27) is given as follows: For any f E H s lim (MA,tLtf)(Z)= f(z)
A 4 0
uniformly on R ([7], Section 3). The precise meaning of (26) and (28) is given as follows: For any g E R ( L t ) R(Lt)'
+
lim LtMA,tg = g
A+O
in L2(R) ([9]). See for example, [3] for the Tikhonov regularization. In order to see (27) numerically, we consider an example: Let f ( ~ = ) e-x2 then f E Hs. At t = 1 , we see from Figure 3 that lim ( M , , t L t f ) ( z )= f(z).
X+O
Now we give another experimental result to see the behaviour of lim LtMA,t
A-0
on L2(R) \ R(Lt).Here, we consider g(z) = x1-1,11then g E L2(R) \ R(Lt).
342
M. Asaduzzaman, T. Matsuura, and S. Saitoh
x = 10-~ x = 10-3
0.4
1.
/
0. 0. 0. 0.
Fig. 3. The figure below shows the graphs of (LtMA,,g)(z) at t = 1 for different values of X and g (z) .
Constructions of Approximate Solutions
343
References 1. Berenzanskii, Ju. M., Expansions in Eigenfunctions of Selfadjoint Operators, Translations of Mathematical Monograph, 79( 1968), AMS. 2. Byun, D-W, and Saitoh, S., Best approximation in reproducing kernel Halbert spaces, Proc. of the 2nd International Colloquium on Numerical Analysis, VSP-Holland (1994), 55-61. 3. Groetsch, C. W., Inverse Problems in the Mathematical Sciences, Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden (1993). 4. Hirschman, I.I., and Widder, D.V., T h e Convolution Transform, Princeton University Press, Princeton, New Jersey, (1955). 5. Saitoh, S., Integral Transforms, Reproducing Kernels and Their Applications, Pitman Res. Notes in Math. Series 369,Addison Wesley Longman Ltd (1997),
UK . 6. Saitoh, S., Matsuura, T., and Asaduzzaman, M., Operator Equations and Best Approximation Problems in Reproducing Kernel Halbert Spaces, J. of Analysis and Applications, 1(2003), 131-142. 7. Saitoh, S., Approximate Real Inversion Formulas of the Gaussian Convolution, Applicable Analysis, (to appear). 8. S. Saitoh, Constructions by Reproducing Kernels of Approximate Solutions for Linear Differential Equations with La Integrable Coeficients, International J. of Math. Sci. (to appear). 9. Saitoh, S., Applications of Reproducing Kernels to Best Approximations, Tikhonov Regularizations and Inverse Problems (in these Proceedings).
Ultrasound as a Diagnostic Tool to Determine Osteoporosis James L. Buchanan’, Robert P. Gilbert2, and Yongzhi Xu3 Department of Mathematics U. S. Naval Academy Annapolis, MD USA jlbuchanan(9hotmail.com Department of Mathematical Sciences University of Delaware Newark, DE 19716 USA gilbertQmath.udel.edu Department of Mathematics University of Tennessee Chattanooga, T N 37403-2504 USA Yongzhi-XuOutc.edu
Summary. Cancellous bone is known to be poroelastic in structure. Ultrasonic wave propagation in cancellous bone can be formulated by using Biot’s equations. In this paper we present some results in our ongoing research on the reflection and transmission of ultrasonic waves in cancellous bone. We investigate the relations among reflected waves, transmitted waves and Biot coefficients. We present an algorithm for the determination of the porosity of cancellous bone.
Introduction Bone tissues can be classified into two types. Bone with a low volume fraction of solid (less than 70%) is called cancellous bone. Bone with above 70% solid is called cortical bone. Cancellous bone is a two component material consisting of a calcified bone matrix with fatty marrow in the pores. Hence mathematical models of poroelastic media are applicable [4],[ 6 ] . Biot developed a general theory for the ultrasonic propagation in fluidsaturated porous media [l], 121, 131. Recently, McKelvie and Palmer 171, Williams [8], and Hosokawa and Otani[6] discuss the application of Biot’s model for a poroelastic medium to cancellous bones. Buchanan, Gilbert and Khashanah [4]discussed the determination of the parameters of cancellous
346
James L. Buchanan, Robert P. Gilbert, and Yongzhi Xu
bone using low frequency acoustic measurements. Following their model in this paper, we consider a piece of cancellous bone submerged in fluid. Ultrasonic wave is produced from the transmitter from one side and received by a hydrophone from other side. We apply Biot’s system of compressional wave equations for a dispersive dissipative fluid-saturated porous medium in the time domain to this model. We model the reflection and transmission of ultrasonic waves in cancellous bone and use this information to determine the various Biot coefficients.
1 The Biot Model Applied to Cancellous Bone The Biot model treats a poroelastic medium as an elastic frame with interstitial pore fluid. In our paper the following notation is used: Elastic frame displacement: u(z,y, z ) = (uz,uy,u,) Fluid displacement: U ( z ,y, 2) = (Uz,U,, Uz) Frame dilatation: e = ~7 . u fluid dilatation: E = v U Even though it is clearly not the case, we make the simplifying assumption that the elastic frame is isotropic. The six stress components are denoted by ozz,oyy,ozz, ozy,ozz,oyz. The fluid stress in the pore fluid is o(z,y,z,t) = -ppp, where pf is the pressure of the pore fluid, and 6 , is the fraction of fluid area per unit cross section. p describes the porosity of the medium. The motion of the frame and fluid within the bone are tracked by position vectors u and U respectively. The constitutive equations and dynamic equations are given below [l],[2], [3], [4]. The constitutive equations:
(T
:=
xv . U I + ~ ( V U+ V U ~+)Q V . V I + S I
+ RV*V
s = QV*U or
Dynamical equations:
The Fourier transform of these equations we write as
+
+ +
pV2u B[(x p)e
V[Qe
+
+
+w2(p1lu p12U) - iw b(u - U) = 0 RE] +w2(p12u p22U) ibw(u - U)
QE]
+
+
(4)
Ultrasound as a Diagnostic Tool to Determine Osteoporosis
347
where p11 and p22 are density parameters for the solid and fluid, p12 is a density coupling parameter, and b is a dissipation parameter. It is convenient to combine the viscous coupling terms into the mass terms following [9] P 1 1 := p11
+ -,ib W
-
p22 := p22
+ -,ib W
p12
:= p12
ib
- -. LJ
Define the new parameters
Note that p is the pore pressure [ 9 ] .In terms of these new parameters the harmonic equations of motion can be seen to take the form [ 9 ] ,
( +ip) vv .u + p nu + cuvp + p w 2 u = 0 np+.yp-av.
u = 0.
As in the case of elastic media there is a decomposition into scalar and vec= A 9 then the tor potentials. Namely,if we set the vector potential displacement u may be represented as
*
u = Wf
+ WS+ V A V A Q ’ ,
where @f,s are the scalar potentials of the fast and slow waves respectively. The resulting equations have four unknowns, namely the displacements u = [ U I ,212,2131 and the pressure p . This system is more convenient that the original Biot system involving both solid and fluid displacements. For the present case we will just use a one dimensional model where this advantage is not so obvious. However, in subsequent multi-dimensional modelling of bone it is a definitely more convenient system of equations. In the water, (x < 0 or x > L ) let PO be the acoustic pressure and uo the displacement. If the transducer is located at x = xs with waveform f(t), then the Fourier transformed acoustic pressure obeys
On the interface (z0 = 0 or xo = L ) , the displacement is continuous, and the normal stress and the pore stress in the bone is equal to the acoustic pressure in the fluid. Therefore, the displacements, pressure and stresses satisfy:
Symbol Parameter Density of the pore fluid pf Density of frame material p,. KF, ComDlex frame bulk modulus K f IFluid bulk modulus K , IF'rame material bulk modulus Porosity Viscosity of pore fluid Permeability
I
a IPore size Darameter I Table 1. Parameters in the Biot model
Po(%> = +O+)/P.
Here x; = 0- or L+, and x : = O+ or L-. The transient property of the wave implies that limlzl+m uo = 0. There are seven parameters to be determined in the one-dimensional Biot model. Naturally, in a one-dimensional model, the complex frame shear modulus p is not included. The other parameters X,R and Q occurring in the constitutive equations are calculated from the measured or estimated values of the parameters given in Table 1 using the formulas given in [4].; in this regard, see references [4] , [6]. Note that all seven parameters are functions of the porosity parameter p. In the next section we determine a relation connecting /3 to the reflected and transmitted waves.
2 Transient Reflection and Transmission of Waves Let fro, 00, 6, and Z be the Fourier transforms of P O , UO, e , and E , respectively. Then from ( 5 ) and ( 6 ) , the transformed equations in the water (x < 0 or z > L ) become 2 *
-w Po
2d2fio * = fs(x - zs), d x2
- co-
d $0 d x = -pow200.
Ultrasound as a Diagnostic Tool to Determine Osteoporosis
349
Correspondingly, In the bone (0 < z < L ) , the transformed equations from (4) become It is convenient to work with displacement potentials in the seabed since we may write the solid displacement as
d+f
d68
d24f dx2
24S + A.9-ddx2
u(x):= dx + -,dx and the pressure as P ( X ) := At-
(13)
In terms of these equations, as p = 0, the first of the Biot equations becomes
whereas the second of the Biot equations becomes
It is known that the fast and slow compression waves satisfy Helmholtz’s equations, i.e.
For this to happen the two potential equations must be linear combinations of the Helmholtz equations, i.e. if we define [9] I
a f , , := X
+ %Af,s,
2
df,, := yw A f , , - w o o
then we see that this can only occur for
Consequently,
’+
- Sw2Af,,- WQO , aoAf,s Af,, which is a quadratic equation for Af,,. Hence, A f , , and a f , , are algebraic W2P
-
expressions of their arguments. Since the Biot coefficients are rational functions of p, the Af,, and a f , , are algebraic-rational functions of p. algebraic expression of its arguments On the interface, (20 = 0 or xo = L ) , we have the transmission conditions @O cO(x0) = --(x,) ax
= P.;(Xo+)
+ (1 - P ) & ( X ; : ) ,
350
James L. Buchanan, Robert P. Gilbert, and Yongzhi Xu
Here x; = 0- or L+, and z,' = O+ or L-. In terms of the displacement potentials the displacement in the water may be written as
where the coefficients Bp,, are defied as
the displacement boundary condition then becomes
(C;eik.x:
- c;e-iksxg
+ ( 1 - P) kp ( c { e i k f x+ c i e - i k f x) + (1- P) k, (cieiksx+ c i e - i k s x ).
(20)
In order to reformulate the pressure-stress transition condition, we first note that the total stress can be written as Cxx
+
=
Cf4f + c s 4 s
where
This permits the right-hand side of the pressure-stress transition condition to be written as P O ( 4 = Cf4f(.O+) Cs4s(x,'), (23) or
+
The pore pressure transition condition Ppo(x,) = ~(x,')becomes
Ultrasound as a Diagnostic Tool to Determine Osteoporosis
351
where
In the water bath the acoustic pressure is
+ cge-ikox,
x < xs x, < x < o x > L,
where k,2 = G. We know solve for the coefficients c;’”’~using the transition CO conditions. After solving for the and we obtain the reflected and the transmitted waves in the form Reflected wave:
po(x,t>= - J 2lr - W
c;(z,w)e-iwx/co
e i w t dw, x < 0.
(26)
Transmitted wave:
c.(z, w ) e - i w x / c o ei w t dw, x > L.
(27)
Using the displacement potentials we may write the system of boundary conditions as in matrix form as
where the matrix E has the form -1 e 2 i k ~ z s -1 0 1 e - 2 i k , x , -1 0 0 5 3 0 ik eikoL 0 0 0 pow2 0 1 1 0 0 0 0 ,ik0L 0 - P P Q+R &+R 0 p 0 0 Q+R where
0 0
0 0
0 0
0 0
E35
E36
E37
E38
E45
E46
E47
E48
-Bf
-Bf
-Bs
-Bf
E65
E66
E67
E67
Df DfeikfL
D
Df f
,-ikfL
D
D S
.ik.L
Df Dse-ik,L
352
James L. Buchanan, Robert P. Gilbert, and Yongzhi Xu
if . c := [cy,c:,c3,c4, O o c lf , c 2 f , cs1 , c ;]* ,and b := [0, -ezkoxs l O , O , O , 0, 0lT. k The components of the matrix are given as
3 Determination of Porosity ,B from Transmitted Waves We consider recovering the porosity p by measuring the transmitted field arising from a point-source placed in a tank of water containing a specimen of bone. In this paper we consider using only the transmitted wave to determine the porosity p. For a point-source generated, incident wave f ( t )from source x = xs,the transmitted wave at x > L is
1
00
po(x,t ;p) =
c4(w,p)eiwx/coei"tdw.
(30)
--oo
If the transmitted wave is received at x = x, is p*(x,, t ) ,then c4(w,p)eiwx~lco= 27.r
/* -m
p*(xT,t ,p)euiwtdt.
(31)
Since the data are measured at x, > L , we use the interface condition at x = L to determine p. First, we consider the uniqueness and stability of determining P. From the previous sections we know that A, R, Q, ~ 1 1p ,1 2 , p 2 2 , and b are polynomials or rational functions of p, and hence, A f , , , B f , s ,D f , s ,etc. are algebraic-rational functions of p. From (19) we know that q5f,s are composed of functions of exponential and algebraic-rational type of p. Define
+
F2(P) := ( B f ( P ) B,(P)( L ) -
(33)
and F3(P) :=
p1 (Df(P)+ Ds(P))- c4e
iwL/co
If we allow P to be a complex variable in a region containing the set
(34)
Ultrasound as a Diagnostic Tool to Determine Osteoporosis
353
then Fl, F2, and F3 are analytic functions of P with at most a finite number of isolated singularities. If Po > 0 is a zero of Fi(P) (i = 1,2,3,) then it must be an isolated zero. Proposition 1(1) For a specimen of bone with porosity PO > 0, if p o ( x , t ;PO) is the corresponding transmitted wave defined by (12), (13), (10) and d f , s are defined by (19), then Fi(Po) = 0, i = 1,2,3.
(2) (Local uniqueness) There exists a constant ri Fi(P) = 0 for some i = 1 , 2 or 3, then P = Po. (3) (Local stability) There exists constants Si mi 2 1, (i = 1 , 2 or 3,) such that
> 0, if IP - ,801< ri and
> 0, Ni > 0 and integer
Proof (1) follows from equations (16)-(18) and (2) is based on the assumption that Po is an isolated zero. To prove (3),we use some of the properties of an analytic function in the neighorhood of a zero; see also [ 5 ] . Based on Proposition 1, if we can have a good initial guess of PO,the local stability will ensure that we can recover Po by numerically solving one of the equations Fi(P) = 0, for i = 1 , 2 or 3. Or we can determine
PO by minimizing one of the following cost functions
Because of the nonlinear dependence of Biot's parameters on P, it is not at all clear if there is a unique global solution for p. However, we can reduce the risk of converging to some local solution for ,B by using multiple frequencies w . We define the following cost functions for determination of P.
=s"
JW>
--M
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James L. Buchanan, Robert P. Gilbert, and Yongzhi Xu
Obviously for exact data 0 = J,7”(/?0) 5 J,7”(p).Moreover, since t h e zero of Fi(/?)is isolated, i t is very unlikely t h a t some ,d # ,do is t h e zero of Fi(,d;w ) for all w . Numerical experiments are under implementation and will be presented in another paper.
References 1. Biot, M. A,, “Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range,” J. Acoust SOC.Am. 28, 168-178 (1956). 2. Biot, M. A., “Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Higher-frequency range,” 3. Acoust SOC.Am. 28, 179-191 (1956). 3. Biot, M. A., “General theory of acoustic propagation in porous dissipative media,” J. Acoust SOC.Am. 34, 1254-1264 (1962). 4. Buchanan, J.L., R. P. Gilbert and K. Khashanah, “Determination of the parameters of cancellous bone using low frequency acoustic measurements,” (submitted) Computational Acoustics, (2002). 5. Buchanan, J.L., R. P. Gilbert and Yongzhi Xu, “Transient reflection and transmission of ultrasonic waves in cancellus bone”, Applicable Analysis, (2003). 6. Hosokawa, A. and T. Otani, “Ultrasonic wave propagation in bovine cancellous bone,” Acoust SOC.Am. 101, 558-562 (1997). 7. McKelvie, T.J. and S.B. Palmer, “The interaction of ultrasound with cancellous bone,” Phys. Med. Biol. 36, 1331-1340 (1991). 8. Williams, J.L. L‘Ultrasonicwave propagation in cancellous and cortical bone: prediction of some experimental results by Biot’s theory,” J . Acoustic SOC. Am. 91, 1106-1112 (1992). 9. Zimmerman, C., “Scattering of plane compressional waves by spherical inclusions in a poroelastic medium”, J. Acoustic SOC.Amer. 94, 527-536, (1993).
Remarks on Quantum KdV Robert Carroll Department of Mathematics University of Illinois Urbana, IL 61801 USA rcarrollQmath.uiuc.edu
Summary. Various quantum versions of equations of KdV and KP type are examined with emphasis on KdV.
1 Introduction Many q-differential operators arise in the study of q-special functions, Casimir operators, quantum groups, and representation theory. There are also natural origins via quantum integrable systems and the quantization of classical integrable systems. The latter is often expressed via a q-hierarchy picture akin to the standard Hirota-Lax-Sat0 formulation and this has many canonical aspects. On the other hand one can produce a great number of q-differential operators by more or less ad hoc manipulation of noncommutative differential calculi or by variations of classical Lie group methods applied to quantum groups. We examine first the hierarchy picture briefly and notice that although the standard methods generalize quite readily the resulting KP or KdV equations for example seem to have an infinite number of terms whereas many “ad hoc” derivations from differential calculi or e.g. MaurerCartan arguments have only a finite number of terms and there is no clear way to determine if in fact such equations have any intrinsic meaning. We show that the standard derivation of KdV via vector fields on the unit circle and the smooth dual space of Lax operators can be extended to a q-situation using a q-Virasoro algebra and we produce a corresponding qKdV equation with an infinite number of terms; this seems to be a fairly canonical derivation and we suggest that it could be equivalent to the hierarchy qKdV equation (not proved).
2 K P and KdV Hierarchies For classical KP and KdV one has (cf. [l,2, 4, 5 , 6, 11, 131
Robert Carroll
356
L = d + Cun+l(t)a-m;
w = 1 + Cw,a-n;
1
L = waw-l;
(2.1)
1
-
where r is the famous tau function with (A4) u = azlog(r) with u u2 in (2.1). TypicaI forms for K P and KdV are (A5)ut-u,,,+6uu,-3a-1uyy =0 and ut -u,,,+6uuz = 0. For KdV one has L t = L2 and the hierarchy picture is much simplified; we discuss this below in the context of qKdV.
3 Q-Hierarchies For qKP we go to [l,2, 6 , 8,9, 14, 16, 18, 19, 20, 22,231; the hierarchy picture is straightforward but the resulting equations are much more complicated. Thus one writes e.g. (A6) a,f(x) = [ f ( q x )- f(x)]/(q - 1). and D f ( z ) = f ( q x ) . Then in the hierarchy constructions z and t = tl are both used in the first variable position with
where r is an ordinary tau function for KP. One takes then an
c
-
(a, a/at,, . . . ,
apt,) oc)
L,
= 3,
+
an(t)qn;
0
(note
where
a0
# 0 appears here and u
[X-l]
-
w, = [(L;)+, L,] = [B:, L,]
ul). Further
(6 =
(34
xy &Ak)
= (l/nAn). Note also (A7) $, = W,e,(zA)et;
L,$,
= A$,
with
00
L,
=
wqaqw;l; w,= 1 + p j a , - j ; ajw, = - ( L p V q ; 1
(3-4)
Remarks on Quantum KdV We see that
3,acting on $Jq simply hits exp,(zX)
357
so
. Recall also the Schur polynomials
CPnwn = exp(C(t, A));
CP,(Z,t ) A n = exp,(zX)eE(tJ); $k(xc,
t)=pk(t +c(x)).
(3.7)
Next the classical Hirota bilinear identity is
f $*(t,~ > $ ( Xt '), ~ X= o
j
T(t
+ y + [ ~ - 1 1 > 7 -( t y - [ ~ - l l > e - 2yixi~ = o
(3.8) and this leads to classical Hirota equations (A8) &an7 7 = 2p,+1(8)7 T (Hirota notation - also 8 = (&, (1/2)&, . . .)). One can produce analogous formulas in q-theory but their usefulness is limited and not even clear (cf. [7, 81 for an extensive discussion). The problem lies in the fact that u = d2 log T does not generalize but instead
-
The classical Hirota equations are compatible with and lead to formulas involving log7 (and hence u ) but in the q-situation (3.9) makes this impossible. In fact I have never seen a qKP equation analogous to (A5)for example. The problems are more easily seen with qKdV and there one has (recall
Wzc>= f(P4)
L; = a; + (g - l)xua,
+ u;
a,& log[T*(x,t)D7q(z,t ) ] ; L, = a,+ so + Sld,-1 + * . . .
u=
(3.10)
Robert Carroll
358
Considerable calculation leads to so = ( q -
(u1 = (q - 1)zu)
l)za,a, log7,;
+ W2(a,“u)+ Wl(a,u) - [@WO) + ~ l ( a , w O ) ] ; w2 = D2so + u1 = D2so + Dso + so; w1 = ( q + l)(Da,so) + T 2 S 1 + [(Dso)+ + u; wo = a,‘., + ( q + l)(Da,sl) + ula,so + Ul(DS1)+ us0 + D2s2
atu = (a,”.)
(3.11) (3.12)
SO](DSO)
and the determination of the si requires infinite series calculations based on formulas like s1
+ Ds1 = u - a,so
00
- so2
=f
+ s1(z) = C(-l)”f(q”z)(3.13) 0
Similarly (A9) Dsg+s2 = -a,s1 -sosl -slD-lso, etc. This seems to suggest that an explicit form of qKdV (as in (A5)) arising in the hierarchy picture with coefficients specified in terms of u (or SO) will involve an infinite number of terms.
4 KdV and
S1
Following [3], let V e c ( S 1 )denote the Lie algebra of smooth vector fields on S 1 and then the Virasoro algebra is V i r = V e c ( S 1 )@ R = 2.D @ R with (note the minus sign convention involving f’g - f g ’ )
(
/
[(f(z)a,, a ) , ( g ( z ) & , b)l = (f’g - f9’)az,
-
S‘
f’g’ldz)
(4.1)
(D Witt algebra). Here Jsl f’g’’dx is called the Gelfand-Fuks cocycle, where a cocycle on a Lie algebra g is a bilinear skew symmetric form c(.,.) satisfying (A10) C c ( [ fg], , h ) = 0 over cyclic permutations of f,g ,h. This means that j = g @ R (central extension) with commutator [(f,a),(9,b)] = ([f,g],c ( f ,9)) satisfies the Jacobi identity of a Lie algebra. Now the Euler equation corresponding to geodesic flow is a 1-parameter family of KdV equations. To see ; smooth on S1 and how this arises consider ( A l l ) Vir* = { ( u ( z ) d z 2 , c ) u c E R}. Then (A12) < (v(z)a,,a), ( u ( z ) d z 2 , c )>= Jsl v(z)u(z)dz ac. The coadjoint action of (fa,,a) E V i r on (udz2,c) E Vir* is (ad: : g* -+ g*, a d : ~ ( v = ) ad,^)) (A13) adifax,a)(udz2, C ) = (2f’u”u’+cf”’)dz2, 0)
+
which arises from the identity (A14) < [(fa,,a),(ga,, b ) ] ,(udx2,c) >=< (ga,, b ) , adifax,a)(udx2, c) >. Now for S1 there are no boundary terms in integration (for single valued functions) so, integrating by parts,
Remarks on Quantum KdV
359
Now a function H on g = Vir determines a tautological inertia operator A : Vir -+ Vir* : (u&,c) + (udx2,c) and hence a quadratic Hamiltonian on Vir* via
Following [3] the corresponding Euler equation is & = - ~ d ~ - ,(m ~ m E 6 ) which here takes the form (A15) &(udx2,c) = - ~ d ~ - , ( ~ ~ ~ ~ , ~ ) ( u d 2 This becomes with (A16) (fa,,a) = A-'(udx2,c) = (u&,c) an equation (A17) dtu = -2u'u - uu' - cu"' = -3uu' - cu"' where c is independent of time.
5 q-Versions 5.1 q-Virasoro
We recall first from [9, 10, 211 the following information regarding q-Virasoro constructions. Thus work on S1 with ( q # 0, kl)
We adapt the formalism of [21] as follows. Let D, = -zn+ld with d : z m 4 qm [m]zrn--l so a 8 , where ~ T ~ ( z= ) f ( q z ) . Generally we will think of z = eie E S1 so ( 1 / 2 7 4 Jsl z"dz = (1/27r) J zn+'dO = 6(-l,o) which will be D, = -zn+laq7 written as (A18) J z n = 6(-1,0). Write also (A19) l , and it is known that q-brackets are needed now where (A20) [lm,ln], = qm-"emen - q"-mlnlm = [m- n]lm+n.For a central term in a putative Vir, one wants (cf. [9, 10, 211) a formula (A21) ~ [ m + l ] [ m ] [ m - l ] S ~ + (see ~ , o below for an optimal term). First we want to formulate the q-bracket in terms of vector fields as follows (the central term will be added later). This can be done as a direct calculation using the basic definition of d above (cf. also (5.9) below). Thus N
N
[ Z n a , zmd]q N
qn-mZnd(zmd) - qm-nzmd(znd) =
(54
= (q"[m] - qrn[n])zrnfn--l = [m- n]zm+n--l a = [n- m](-zm+n--la)
a
Ca,z" and w C b m z m ; and define a bracket in Vec(S1) Let now v via (A22) [va,wa], = - C a,bm[n - m ] ~ ~ + ~We - l adefined . a bracket of vector fields in [9, 101 so that from (A56) there resulted a correspondence (A23) v'w - vw' -[.az, wdz] -[.a, wa], = -{(TIJ)(~,w) - ( T W ) ( ~ , V ) ) T . This dangling T creates some complications and is removed below. N
N
N
N
Remark 5.1 In [21] one defines the q-analogue of the enveloping algebra
360
Robert Carroll
of the Witt algebra Z? as the associative algebra U,(Z?) having generators Virasoro algebra is defined as the associative algebraU,(Vir) having generators em ( m E Z) and relations ( q # root of unity)
em (m E 2) and relations (A20). The q-deformed
+
where < m >= q" q-" and 2, = q2"emE (thus 2 is an operator which we examine below and we refer to [9, 10, 211 for the central term). Then Vir, is a Z graded algebra with deg(e,) = m and d e g ( t ) = 0. One U,(Vir) also introduces in [21] a larger algebra U(V,) = associative algebra generated by J*', E, d, (rn E Z) with relations N
J J - l = J - l J = 1; J d , J-'
= qmdm; t J = J e ; Edm = qmdmt;
(5.4)
The subalgebra of U(V,) generated by ek = d, J and 6' = t J (m E Z) is the same as U,(Vir). It is stated in [21] that V, is the universal quantum central extension of Z?, and thus (5.4) is better adapted for optimal algebraic and geometric meaning; it is this aspect which we emphasize below. Now we mimic the framework of Section 4 and it is interesting to note that an ordinary integral Jsl will suffice. One does not need a Jackson type integral in order to deal with integration by parts. Thus we observe that (A24) f = C fnzn = f-1; sd,f = ( q - q-')-l(f0 - fo) = 0. Since d , ( f h ) = ( ~ f ) ( a , h ) ( a , f ) ( ~ - l h we ) have an integration by parts formula
ssl s
+
a
8 , ~ )(A25) J f d h = This can be written as (recall 8 , ~ Now ~ .think of U,(Vir) and elements (fa, u ) with (A261
N
- J h8f
for
8=
[(fa,4,( g 4 b)l = (-[fa,gdlqd, $(fa,ga>>
. The central term could be defined tentatively via e.g. (A27) J ( 7 d 3 f ) ( 7 g ) Z = $(fd,gd) (where one notes that (A28) $ ( f d , g d ) = q-l J g d 3 f E ) . We will want to put the central operator E into the integral, acting on f, and will see below that E ' 7 for example and T ~ F ( z= ) F ( q 2 z ) T 2 so it eventually automatically passes to the right in our qKdV type equations. Hence for the moment think of E = T~ put into (A27)via e.g. 7a37-2tf E 7d3f and ignored at the end except when exhibiting formulas like (5.3)on generators (see also remarks below). N
We recall now from [21] (second paper)
Remarks on Quantum KdV
= qmdmdn J
- qndnd, J = q-n Jdmdn
-
361
Jdndm
Further Ed, = qmd,E suggests E = r here. Indeed d , is being used as L',r-' and dropping the minus sign momentarily, from L', = .zm+'a,r we get then (A29) d, = zrnf1aq. Then, using (A30) a,r = q d , , one obtains 7 d m = r(zrn+la,) = qmzm+la,r and F ' d , = q-,dm7-l. In addition Jd,J-l = qmdm corresponds to J d , = qmd,J so we identify J = r. Writing em = d , J = dT, we can also easily see that the brackets [d,, dn] above are exactly the q-brackets (A31) [lm,Cn], = qmtmln - qntnL',. Now in [21] (second paper) a Jacobi type identity is used involving an ) z E $ C d n . This seems to be better operator o(x)= ( 1 / 2 ) ( 7 + ~ - ~ ) ( 2 for phrased in terms of an operator (A32) r ( d p ) =< p > d p which avoids the need to carry 7 around to other terms. Then we can check that the rule in (5.4), rewritten as (A33) [ d m ,dn] = [m- n]dm+n ymbm+n,ot will yield
+
[ [ d m , d n ]r(dp)] ,
+ [ [ d p , d m ]r, ( d n ) ] = -
+ [[dn,
d p ] ,r ( d m ) ]
sm,n,p= 0
(5.7)
This is based on two identities; one, stated in [5],is (A34) [m-n][m+n-p]< p > + [ n - p ] [ n + p - m ] < m > + [ p - m ] [ p + m - n ]=O. Thesecondis
[P + 1l"P
-1 "
- nl
+ [m+ 1l[ml[m- 11[n- PI+
+
(5.8)
+
+[n l][n][n - 1][p- m] = 0; ( m n + p = 0 ) The proofs are essentially straightforward (cf. [15] for details). Thus V, will be a genuine central extension of D,, with a reasonable Jacobi identity (5.7).
Remark 5.2 Now in (5.2) - (A23) we recall L', -zrn+la = -zrn+laqr and a dm formulation would drop the r. Thus work with 8, instead of a = aqr with (A35) dqzP+l= qp+1zP+1dq+[p+1]~P7-1 based on a,f = (7f)aq+(aqf)7-l. Then N
[d,,
d,] =
la^, zn+'aq] = qmdmd,T
-qnZn+'dqZrn+'dq7 = [TI -
- qndndmr = qm zm+1 aqzn+1t3q~-
(5.9) = [m- n]d,+,
In this context the 6m+n,o term does not arise. Note here r-la, = qi3,r-l and (A36) q m f l [ n 11 - qn+'[m 11 = [n- m].Let now v C vn+1zn+l and w = C w r n + l z m + l ; then
+
+
=
c
N
vn+1wm+1[n- m].n+m+laq
Going back to (5.9) this corresponds then to (A37) [d,, wa,]= [(rv)(a,w) (rw)(a,v)]a, and this replaces (A23).
-
rn
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Robert Carroll
Now we build in a cocycle term by using (5.4) in (5.9) so that a term arises of the form (A38) [zm+la,,zn+'aq] = [d,, dn] = [rn - n]dm+n T ~ ~ , + , , O T . For [d,, wa,] we get then an additional term
+
c
=
%+l~,+lYnb,+n,0T
Then for a = 1 / [ 2 ] [ 3and ]
(T
c
+
[n l ] [ n ] [-n11' 'Un+lW-n+l
+ 7 - l )=~ (qn ~ + q-n).zn
'PIPI
(5.11)
=< n > zn define (5.12)
=
One checks that
4([.%
c
~n+lW-n+lYn = 4(.8,,
wa,)
4 is antisymmetric and satisfies the q-cocycle condition
+ 4([waq,uaq1,r(va,)>+ 4([u%
7 - 4 1 , r(.Llaq>>
4
J
7
r ( w a q > )= 0
(5.13)
Consequently (cf. [15]for further details and [12]for a sketch) Theorem 1 The term q5(vaa,wa,) in (5.12) is a q-cocycle and following the constructions in (9, 101 one has a possibly canonical qKdV equation in the f o r m = 1/[21[31) Ut
+ C'a:(T + T-'>-'a,U + a q ( U T U ) + T-1Udq7-12L
(5.14)
Proof We modify slightly the constructions in [9, 101 and write Q
< [fa,,a > ,(9%
a,(u,4 >= -4 /if%,ga,1. + c q 4 ( f % , ga,> =
= -q/[(.f)(aqg)
- (.f>(a,f)lu + caq/@,"(.r
s
s
(5.15)
+ T-')-'a,"f
s
We note from [lo]that (A39) q fg = F 1 f F 1 g and aqf = 0 while from (A40) a,(gfu) = a,f7-l(fu)+(Tg)aq(fu)(via aq(ab) = (Ta)a,b+(a,)T-'b). Consequently (5.15) becomes (A41) J g [ ~ - l a (, T U T ~f) 7-l(ua,f) aqca,2(T ~ - ' ) - ' a , f ] . Putting f = u we obtain the Euler equation as in [9, l o ] ,namely (A42) qut = -qcaa;(T T-~)-~~,u-T-~~,(Tu -~-l(ua,u). T~u) Using also .-la, = qaqT-l we obtain (5.14) with c' = ac. Remark 5.3 In view of the expression (A43) (T+T-~)-' T C ( - l ) n ~the 2n equation (5.14) involves an infinite number of terms (much as are indicated for qKdV in the hierarchy picture in [9, 101. Since we now have a derivation with all of the classical algebraic and geometrical structure duplicated it seems that (5.14) could be a good candidate for a canonical form. Since the hierarchy qKdV seems to be considered canonical one could anticipate an equivalence. However the KdV equation arises in many different ways in mathematics and
+
+
+
+
N
Remarks on Quantum KdV
363
physics and it is perhaps not obvious that the hierarchy picture itself should be canonical; however the analytic approach of [20] and the algebraic (oper) I approach of [17] do point in this direction.
References 1. M. Adler, E. Horozov, and P. van Moerbeke, Phys. Lett. A, 242 (1998), 139-151 2. M. Adler and P. van Moerbeke, solv-int 9912014; CMP, 147 (1992), 25-56 3. V. Arnold and B. Khesin, Topological methods in hydrodynamics, Springer, 1998 4. R. Carroll, Quantum theory, deformation, and integrability, North-Holland, 2000 5. R. Carroll, Topics in soliton theory, North-Holland, 1991 6. R. Carroll, Calculus revisited, Kluwer, 2002 7. R. Carroll, Complex Variables, 82 (2003), 759-786 8. R. Carroll, Math.QA 0208170; 0211296 North-Holland, 1985 special functions, 63 (1978), 156-167 9. R. Carroll, Inter. Jour. Pure Appl. Math., 5 (2003), 177-211 10. R. Carroll, math.QA 0301361 11. R. Carroll and Y. Kodama, Jour. Phys. A, 28 (1995), 6373-6387 12. R. Carroll, Proc. Conf. Symmetry Nonlinear Math. Phys., Kiev, 2003, Part 1, 356-367. 13. R. Carroll, Nucl. Phys. B, 502 (1997), 561-593; Springer Lect. Notes Physics, 502, 1998, pp. 33-56 14. R. Carroll, Proc. NATO Conf., Yerevan, 2002, to appear 15. R. Carroll, math.QA 0303362 16. E. Fkenkel, IMRN, 2 (1996), 55-76 (q-alg 9511003) 17. E. F’renkel and D. Ben-Zvi, Vertex algebras and algebraic curves, Amer. Math. SOC., 2001 18. L. Haine and P. Iliev, Jour. Phys. A, 30 (1997), 7217-7227 19. P. Iliev, Lett. Math. Phys., 44 (1998), 187-200; Jour. Phys. A, 31 (1998), L241L244 20. B. Khesin, V. Lyubashenko, and C. Roger, Jour. Fnl. Anal., 143 (1997), 55-97 21. K. Liu, Jour. Algebra, 171 (1995), 606-630; CR Math. Rept. Acad. Sci. Canada, 13 (1991), 135-140; 14 (1992), 7-12 22. M. Tu, solv-int 9811010 23. M. Tu, J. Shaw, and C. Lee, solv-int 9811004
A Mathematical Model of Ductal Carcinoma in Situ and its Characteristic Patterns Yongzhi Xu Department of Mathematics University of Tennessee Chattanooga, T N 37403 USA Yongzhi-Xu(Outc.edu
Summary. This paper studies the characteristic stationary solution of a three dimensional mathematical model of ductal carcinoma in situ (DCIS). DCIS refers to a specific diagnosis of cancer that is isolated within the breast duct, and has not spread to other parts of the breast. We modify a spherical model to describe the tumour growth inside a cylinder, a model mimicking the growth of a ductal carcinoma. The model is in the form of a free boundary problem. The analysis of stationary solutions of the problem shows interesting results that are similar to the patterns found in DCIS.
1 Introduction Ductal carcinoma in situ (DCIS) refers t o a specific diagnosis of cancer that is isolated within the breast duct, and has not spread t o other parts of the breast. There are two categories of DCIS: non-comedo and comedo. Non-comedo type CDIS tends t o be less aggressive. The most common non-comedo types of DCIS are: (1) Solid DCIS: cancer cells completely fill the affected breast ducts. (2) Cribiform DCIS: cancer cells do not completely fill the affected breast ducts; there are gaps between the cells. (3) (a) Papillary DCIS: the cancer cells arrange themselves in a fern-like pattern within the affected breast ducts. (3) (b) Micropapillary DCIS: the cancer cells are smaller than papillary DCIS cells.(More information may be found in http://imaginis.com.) In [14],we studied a simplified one-dimensional and two-dimensional model modified from a model proposed by Byrne and Chaplain [7], (also see Friedman and Reitich [lo]), for the growth of a tumour consisting of live cells (nonnecrotic tumour) to describe the homogeneous growth inside a cylinder, a model mimicking the growth of a ductal carcinoma. The model is in the form of a free boundary problem. Assuming some symmetry of tumor growth, we analyze the growth tendency of DCIS. The analysis of stationary solutions
366
Yongzhi Xu
of the problem shows interesting results that are similar to the patterns found in DCIS. (There are many other models of tumour growth. For examples, see [l],[2], [3], [4], [5], [6], [8], [ll],[12], [13], and the references cited there.) In this paper we further study the characteristic stationary solution of the three-dimensional model and find that they have features that resemble the Cribiform, Papillary and micropapillary DCIS.
2 Mathematical Model of DCIS We assume the tumour to be within a cylinder and to occupy a region
B ( t ) = ( ( T , 6 , z)lO < z < s(r,6 , t ) ,0 5 r < T O ,0 5 6 5 27r} at each time t ; the growing boundary of the tumour is given by z = s(r,8, t ) , an unknown function of T , 6 and t. As tumour growth strongly depends upon the availability of nutrients its diffusion through the growing material is introduced in the description of model. Similar to [lo] we model the tumour growth using dimensionless nutrient concentration b(r,z , t ) which satisfies a reaction-diffusion equation of the form
Here the constants ag and F denote the dimensionless nutrient concentration in the nearby cells/vasculature (if there is any), and the rate of nutrient transfer per unit length, respectively. Hence ~ ( C T-B &), which may be positive or negative, is the transfer of nutrient from/to the neighborhood, and X o d is the is the ratio of the nutrient nutrient consumption rate. c = T~iffzlsion/Tgrowt~ diffusion time scale to the tumour growth time scale. Typically c << 1. (cf. [l]and [ll].) In the DCIS case, we assume that b(r,8, z , t ) satisfies the following boundary conditions.
a&
+
- a ( a~ &) dr
a&
= 71,when
T
= TO,
+
- P(ag - 6 ) = 72,when z = 0, 8.2
&(T,
d(r,6 , s(r,8, t ) ,t ) = el,
(2.4)
8, z , 0 ) = b0(r,6 , z ) , in B(0);
(2.5)
where B(0) = { ( r ,8, z)lO < z < so(r,6 ) ,0 5 r < T O ,0 5 6 5 27r) is the initial region of the tumour with so(r, 0) given. &o(r,8, z ) is the initial nutrient concentration. a,p, y1 and 7 2 are constants that reflect the properties of the duct.
A Mathematical Model of Ductal Carcinoma
367
Condition (2.2) represents an impedance condition for the nutrient concentration on the duct wall. The condition (2.3) reflects tumour growing tendency. If ,6 = 0, and 7 2 = 0, then the tumour is assumed symmetrical about z = 0. The external nutrient concentration 61is assumed to be a constant in (2.4). (2.5) is the initial state of DCIS. The smoothness of 5 implies that dd _ - 0, when r = 0,
dr
6(r,9 , z , t ) = &(r,9 + 27r, z , t ) . The mass conservation consideration implies the relation
:[L2yo 12r1'' 1
(2.7)
([lo])
S(T,t)
s ( r , 9, t ) r d r d ] =
p(b)rdzdrdo
(2.8)
where p(d) denotes the cell proliferation rate within the tumour. For inhibitorfree model (ref [7] and [lo]) we have p ( 6 ) = p(& - a)
(2.9)
where p and 6 are positive constants. Hence, (2.8) becomes
The goal of this paper is to study stationary solutions of (2.1)-(2.7),(2.10) for different choices of parameters that resemble the patterns of DCIS. We found that for different choices of parameters, the model has solutions that resemble the cribiform pattern, the papillary pattern and the micropapillary pattern, respectively. The dynamic properties of the one-dimensional model is discussed in [14]. The dynamic properties of the general model will be study in another paper.
3 Stationary solutions and their patterns In this section we present analysis of stationary solutions for some choices of parameters. Then we visualize the solutions and compare them with the DCIS patterns. Assume in (2.1) that = 0 and $$ = 0, we obtain equations of stationary solutions:
ab - + a(ag - 6) = 71,when r = ro, dr
368
Yongzhi Xu dl3
- - @(aB - 6 ) = y2, when
az
z = 0,
&(r,0, s(r,0)) = 81, 6% - = 0, when r = 0,
dr &(r,8, z ) = &(r,8 62T
$,,,
6To
+ 2n,z ) .
(& - 5)rdzdrdO = 0,
(3-7)
where z = s(r,8) denotes the stationary free boundary. Let
when r = ro,
(3.9)
- when z = 0,
(3.10)
x dU
- - pa = 7 2 -
az
x
(3.11) dU
(3.12)
when r = 0, dr u(r,0, z , t ) = u(r,0 2n,z , t ) . - = 0,
+
12T
(3.13)
LTo
(3.14)
( a - 5)rdzdrdO = 0. We study the special case that aX0Q3
71 = -, 7 2 = -
moaB
x
n
,c1=-
.
r +xo’
and assume that the free boundary is flat z = z s . Equations (3.8)-(3.14) become
(3.15) dU dr dU - - @a = 0, when z = 0, az
- - acr = 0, when r = ro,
(3.16) (3.17)
(3.18)
A Mathematical Model of Ductal Carcinoma
aU _ -- 0, when r = 0, ar o(r,B,~ , t=) o(r,0
i2= lro izs(o
369
(3.19)
+ 27r, z , t ) .
(3.20)
- 6)rdzdrde = 0,
(3.21)
where the free boundary z = z, is to be determined. Let
o(r ,z ) = R(r)O(e)Z ( z) . Then R ( r ) ,O(0) and Z(z) satisfy
(3.23)
2"
+ (-A
- c"2 = 0, 0 I z
I z,,
(3.24)
+
O(0) = O(e 27r) follows 7 = n for some natural number n. R ( r ) and Z(z) satisfy boundary conditions:
R' - aR = 0, at r
= ro,
(3.25)
R' = 0, at r = 0.
(3.26)
2' - PZ = 0, at z = 0,
(3.27)
2 = 0, at z = z,,
(3.28)
where E is a parameter to be determined. Fkom (3.24),
That Z(z) is not identical to zero requires that psin(Jnz,)
+&qcos(J=z,)
Hence tan( &K&) =-
Jq
For given 5, if z , is a solution of (3.29), then
and
P
= 0.
(3.29)
Yongzhi Xu
370
From (3.23),
R(r)= CJn(ET) + m l ( E T ) ,
where JO and YOare first kind and second kind Bessel functions of order n, respectively. (3.25) and (3.26) imply that D = 0 and E satisfies the eigenvalue equation: EJn+l(ETO)
+
(
Q
-
3
-
Jn(Er0)
= 0.
(3.31)
If [ and zs satisfy (3.29) and (3.31), then
dh,072; E, 2s)
(3.32)
072; E , zs)
(3.33)
= Jn(ET)
[
c o s ( J ~ z )+
P
sin(J=z)
d=K-pzs
1
sin(n0)
are the characteristic solutions of the stationary model. There may be a finite number or an infinite number of $, and z, that satisfy (3.29) and (3.31). The stationary solutions with flat free boundary have representation 00
4T7
6,4
[ M E , z s ) 4 h 6,z ; E , 2 s )
=
+ &(E,
.s)4$-,
072; E , zs)]
,
n=O E,zs
(3.34) where 0: and cr; are given by (3.32) and (3.33). Following we look at a single term of (3.34). For each n, there are four different combinations of E and r :=
dq-.
Case 1: E2
2 0 and r 2 0 .
Equation (3.31) has a sequence of solutions
Em such that
If X < 0, there is M 2 1 such that for m < M , (& < -A. Therefore, for m < M , rm= > 0. The equation (3.25)
d
q
tan(-r,z,)=
1 --rm
P
A Mathematical Model of Ductal Carcinoma
371
-
has solutions z, = a% (k = 1 , 2 , 3 , . .) for each m < M . The case n = 0 corresponds to that cancer cells are symmetrical about the axis T = 0, which has been studied in [14]. Here we show a pattern for n = 5. Example 1: To show the pattern of the solutions of free boundary problem, we choose the following parameters as an example. n = 5, C T = ~ 3, XO = 2, TO = 1, a = 2, P = 0.5, = 10.2735,X = -120, T = = 3.8.
<
Jq
Fig. 1. E2 2 0 and T 2 0, “Cribiform DCIS”
We show patterns of the corresponding nutrient concentration 6(r,z ) = z) in Figure 1. We make an assumption that if the nutrient concentration is lower than a certain level, there will be no tumour cell present. This kind of patterns mimic the Cribiform DCIS (spread out evenly with gaps). U(T,
+&
Case 2: S2 2 0 and
T
= i1T1.
Equation (3.31) has a sequence of solutions
For each Ern such that
t$ > -A.
tan(T,z,)
T,
=
tnsuch that
id-.
= tan(ilT,lz,)
Note that = i tanh(l.r,lz,).
The equation (3.29) tan(-r,z,) = has a unique solutions if and only if
-IT,/
1
- T ~
P < P < 0.
Example 2: We choose the following parameters as an example. n = 5 , o g = 3,Xo = 2,ro = 1 , a = 2 , P = -15,c = 10.27,X = 5 0 , = ~ = 12.5i. This kind of patterns mimic a kind of tumour that moves along the duct but does not grow in size. It would probably not be noticed or not be considered as a kind of DCIS.
id-2
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Yongzhi Xu
Fig. 2.
<
Case 3: = 2lEl and
T
t22 0 and T = i 1 ~ “Moving ( tumour”
> 0.
The equation (3.31) becomes n 2ltIJn+1(2l<\ro) (a- -Jn(iIElTo)
+
TO
(3.35)
= 0,
which has at most one solution. If 0. The equation (3.29)
Jq
< -ti
then
TO
=
1 tan(.roz,) = --TO
P
has a sequence of solutions z, = f&, k = 1,2,3, . .
a.
Example 3: We choose the following parameters as an example. 3(a) (Figure 3) n = 8 , a ~ = 3,Xo = 0.0003,ro = 1,a = 8.000045;P = -0.0005, = 0.0282, X = 0.0004, T = = 0.0262. 3(b) (Figure 4) n = 32, a~ = 3, XO = 1,TO = 1,(I! = 8.005; P = -0.05, ( = 0.57452, X = 0.2, T = = 0.3606.
Jn
<
Jm
Fig. 3. [ = il[l and r
>0
LLBaby tree”
These kinds of patterns mimic the Papillary DCIS and the micropapillary DCIS (Baby tree pattern).
Case 4:
t = iltl and 7 = Z I T ) .
The equation (3.31) becomes ~lSIJn+1(2lElTo)
+
n
((I!
- -Jn(2l
= 0,
(3.36)
A Mathematical Model of Ductal Carcinoma Fig. 4.
6 = iltl and T > 0 “Baby tree”
which has at most one solution. If
d q ’==
373
is a solutio, and X
> -<:, then
TO =
Z(To(.
The equation (3.29) 1
tan(Toz,) = ---TO has a unique solution z, if and only if
-1701
P < /3 < 0.
Example 4: We choose the following parameters as an example. n = 5 , a ~= 3,Xo = 4,ro = 1,cu = 2,P = -15,< = 0.5745i,X = 5 ; =~ = 2.1601i.
Fig. 5.
6 = ilel and T = i 1 ~ 1“Benign DCIS”
This solution mimics a kind of DCIS that is neither growing nor moving. This is a benign tumour and reasonably not included as a pattern of DCIS.
4 Conclusion In this paper we present some preliminary analysis and graphical examples for a free boundary problem model of DCIS. F’rom the analysis and numerical examples we show that this mode1 has characteristic solutions that mimic some typical patterns of DCIS, including cribiform DCIS, papillary DCIS, and micropapillary DCIS. The analysis also shows that there may be two other kinds of patterns that resemble the non-growing tumour. These interesting results show that the free boundary problem model of DCIS is worthy of further study. In particular, the study of patterns of 0 given by (3.34) and the dynamic properties of the free boundary problem may lead to better understanding of the model.
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Yongzhi Xu
References 1. Adam, J. A., A simplified mathematical model of tumour growth, Math. Biosciences, 81, 229-244, 1986. 2. Adam, J. A., A mathematical model of tumour growth, 11. Effects of geometry and spatial nonuniformity on stability, Math. Biosciences, 86, 183-211, 1987. 3. Adem, J. A. and N. Bellomo, A Survey of Models for tumour-Immune System Dynamics, Birkhauser, 1997. 4. Breward, C.J.W., H.M.Byrne and C.E. Lewis, The role of cell-cell interaactions in a two-phase model of solid tumour growth, J. Math. Biol. 45, 125-152, 2002. 5. Burton, A. C., Rate of growth of solid tumours as a problem of diffusion growth, J. Math. Biol. 30, 157-176, 1966. 6. Byrne, H.M., The effect of time delays on the dynamics of avascular tumour growth, Math. Biosciences, 144, 83-117, 1997. 7. Byrne, H.M. and M.A.J. Chaplain, Growth of nonnecrotic tumours in the presence and absence of inhibitors, Math. Biosciences, 130, 151-181, 1995. 8. Byrne, H.M. and M.A.J. Chaplain, Growth of necrotic tumours in the presence and absence of inhibitors, Math. Biosciences, 135, 187-216, 1996. 9. F'riedman, A., Partial differential equations of parabolic type, Prentics-Hall, Inc. 1964. 10. F'riedman, A. and F. Reitich, Analysis of a mathematical model for the growth of tumours, J. Math. Biol. 38, 262-284, 1999. 11. Greenspan, H. P., Models for the growth of a solid tumour by diffusion, Studies in Appl. Math., 52, 317-340, 1972. 12. Greenspan, H. P., On the growth and stability of cell cultures and solid tumours, J. Theor. Biol., 56, 229-242, 1976. 13. Shymko, R.M. and L. Glass, Cellular and geometric control of tissue growth and mitotic instability, J. Theor. Biol., 63, 355-374, 1976. 14. Xu, Y . , A free boundary problem model of ductal carcinoma in situ, Discrete and Continous Dynamical Systems - Series B, Vol. 4, No. 1 (2004), 337-348.
Classical Dynamics of Quantum Variations Margo Kondratieva* and Sergey Sadov Department of Mathematics and Statistics Memorial University of Newfoundland St.John’s, Newfoundland A1C 5S7 Canada [email protected]
Summary. An ODE system of order 5 called the two-system of the one-dimensional anharmonic oscillator is considered. The system is Hamiltonian with respect to a degenerate Poisson bracket and can be viewed as a l-parameter family of canonical systems with 2 d.0.f. on symplectic leaves, and also as a perturbation of an integrable system, degenerate in the sense of the KAM theory. Neither the KAM theory, nor the Birkhoff normal form are directly applicable, and the perturbation analysis is carried out by means of a non-Hamiltonian normal form. The system possesses a formal integral in any order of the perturbation theory and thus long-persistent adiabatic tori. The analytical description of such tori, a place of exact periodic orbits between them, and supporting numerical results are presented.
Introduction According to the Kolmogorov-Arnold-Moser (KAM) theory, a small Hamiltonian perturbation of a generic integrable Hamiltonian system preserves the majority of non-resonant invariant tori that exist in the unperturbed system; the tori just get slightly distorted [l,App. 81. As a consequence, in a generic near-integrable Hamiltonian systems with two degrees of freedom (d.0.f.) there is no ‘diffusion’, i.e. even those trajectories that do not themselves lie on invariant tori are locked between such tori. The action variables of the original system are adiabatic invariants for the perturbed system: they remain almost constant for a long time. In the absence of diffusion, their values stay close to the initial values permanently. The situation is different if the nondegeneracy conditions are not met. Then the original action variables even in a perturbed Hamiltonian system with 2 d.0.f. can be unbounded as functions of time [13]. In this paper we consider a dynamical system (9) in the 5-dimensional phase space, which is a Hamiltonian system with respect to a degenerate LiePoisson bracket (7). The system has two integrals: the Hamilton function and *Research supported by NSERC grant 261412-03
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the Casimir integral (see e.g. [lo]). The level surfaces of the Casimir integral are symplectic manifolds, called symplectic leaves. The system restricted on the symplectic leaves becomes a standard canonical system with 2 d.0.f. (except for the special one-point leaf, the origin). Our system contains a parameter E and we consider it as a perturbation of a simple integrable system with E = 0. The unperturbed system is strongly degenerate: all its orbits are periodic with the same period. It turns out that the perturbed system possesses, besides the two exact integrals, a family of very long-standing adiabatic invariants, which come as truncations of a formal integral (a divergent series whose formal derivative due to the system is 0). However, a true third integral does not exist, so an interesting question stays open: what can be said about diffusion in our system? In the absence of theoretical predictions, one can try to draw some conclusions from numerical experiments. But one must be very cautious. On the one hand, for small values of the perturbation parameter numerical results can not be trusted on the very long time intervals needed to reveal an evolution of the adiabatic tori. On the other hand, the behaviour of the system for not-so-small values of E may not reflect features of smaller perturbations. In this paper we do not attempt to investigate this question proper; instead, we set up a stage for such an investigation. As explained in Sect. 2, instead of considering perturbation of the global system as E --t 0, one can analyze the system with E = 1 in a small vicinity of the equilibrium point. Local analysis near the equilibrium for Hamiltonian system is usually carried out by means of the Birkhoff normal form, in both resonant and nonresonant cases. In particular, the Birkhoff normal form allows one to find adiabatic invariants as formal integrals [6, 4.31. This approach doesn’t work in our situation, since the only stationary point of the system is a singular symplectic leaf, thus neither of the canonical systems obtained after reduction on nondegenerate symplectic leaves has a stationary point. For this reason we use a non-Hamiltonian normalization procedure in the 5-dimensional space. In a general situation this procedure simplifies the system under consideration (e.g., eliminates fast variables), but it does not necessarily yield formal integrals. Our system has three special properties: its eigenfrequencies are in the strongest possible resonance, it has one less than maximum possible exact integrals, and it is volume-preserving. It was shown in [9] that a system with such properties has a formal integral independent of the known exact integrals. The structure of the paper is as follows. First we briefly motivate appearance of the system in a more general situation (Sect. 1).Then we discuss basic simple properties in the concrete case (Sect. 2). The main results are contained in Sect. 3 (analysis near the equilibrium). Sect. 4 gives a complementary perturbation analysis in the vicinity of a periodic solution of a finite amplitude. Numerical results in Sect. 5 illustrate and extend theoretical predictions.
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1 Two-System of a Hamiltonian System A Hamilton function H ( Q , P ) defined on R2”gives rise to a Hamiltonian system and consequently to its linearization X = J H ’ ( X ) , Y = J H ” ( X ) Y , X = ( Q ,P ) , Y = ( A Q ,A P ) .
(1)
Here H’ = ( a ~ H , d p H ) ,H ” ( X ) is the Hesse matrix of the second partial derivatives evaluated at X ( t ) , and J is the standard symplectic matrix of order 2n, with properties J 2 = - I , Jt = - J . A system called the two-system comes into play from two different directions. It was originally obtained as a ‘quantum system in variations’, see Appendix. Here we’ll bring out another, purely classical construction. Consider the new Hamiltonian F(x,y) in R4”,
1 2
F(x,y) = H ( z )+ -yt * H”(x)y.
(2)
Note that the second term is the time-dependent Hamilton’s function for the system in variations (the Y-subsystem in (l)),assuming x = X ( t ) is known. However, the Hamiltonian system produced by Hamilton’s function F(x,y) as a whole is not (l),but rather the system
System (1) is not Hamiltonian in general. In a sense, (3) is its Hamiltonian perturbation. Let us elaborate. Though the y-parts of both systems (1) and (3) look alike, a significant difference is that in (1)the x-part is an independent subsystem, while in (3) it receives a feedback from the y-part proportional to the magnitude of the variation vector y. Thus, if the variation y(t) is small, then the evolution equation for x(t) is close to that for X ( t ) . Typically, even in integrable systems, variations Y ( t )grow indefinitely with time and eventually trajectories of system (3) don’t trace those of system (1). The two-system is an extension of the system (3), in which (3) describes the dynamics on one invariant subspace. For simplicity, from now on we assume n = 1. The two-system in the general case is discussed in [8], [14], [12]. Consider a 2 x 2 symmetric matrix M and a matrix @ connected with M as
Then @ is an element of the symplectic Lie algebra sp(2, R). The phase space of the two-system is R2@sp(2,R) and the dynamics is defined by the equations
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(We use the standard notation: trA for trace of a matrix A, and [A,B] = AB - BA for the commutator of two matrices.) The system ( 3 ) is a closed subsystem of ( 5 ) with M = y y t being a matrix of rank 1. Theorem 1 (The Hamiltonian form of the two-system.) Let x = ( q , p ) and a , p , y be taken from ( 4 ) . System ( 5 ) can be written in the form 7i = {q,'Fl}n, where 7 = (q, P , a,P, $7 'Fl(q7 P , a , P, 7)= H(x)
and
{ a ,
1
+~t'(H~WM),
(6)
.}a i s the following degenerate Lie-Poisson bracket (see e.g. [lo])
{f,g)n=V,f-QV,g,
Q=J@w,
2a4p 0271. -4p-2y 0
0 w = [-2a
(7)
The function 6 = det M = a y - p2 is the Casimir function f o r the bracket { - , -}n and thus an integral of motion f o r the system ( 5 ) .
2 Two-syst em of the anharmonic oscillator. Elementary properties P) = P2/2 The anharmonic Hamiltonian H(Q, according to ( 6 ) , the Hamilton function
+ Q2/2 + &Q4/4
implies,
and the corresponding two-system, which is the subject of our study q =p,
d! = 2p,
+ &q3)- 3 q a , j = -(1+ 3&q2 )a + y,
p = -(q
i, = -2P(1+ 3 q 2 ) .
(9)
The system has two independent integrals: E , and S from Theorem 1. System (9) with E = 0 is integrable; it is the union of two independent subsystems. There are two families of trivial periodic solutions: (i) L1 : Q = ,B = y = 0. The governing equation for q is 4 = -4 - q 3with general solution expressed in terms of elliptic functions. (ii) L2 : q = p = 0. The orbits are parametrized by the values of the integrals E , S subject to constraint A2 := E2 - 6 1 0:
a(t)= E + A cos 4,
p(t) = - A sin 4, y ( t ) = E - A cos 4,
(10)
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where 4 = 2 ( t - t o ) . Invariant coordinate subspaces that carry the families LI and C2 are those of Lyapunov’s theorem [3, Part I1 9 4.21, hence the notation L. The intersection of the families L1 and L2 is the stationary point q = p = a = p = y = 0. System (9) has many periodic solutions besides those of the families L1 and L2, but they cannot be written in a closed analytical form. In the next section we study a neighbourhood of the origin, establish existence of adiabatic tori and an invariant set A filled with non-Lyapunov periodic solution. Then in Sect. 4 we investigate stability of solutions of the family L 2 in the linear approximation. System (9) is invariant under the scale transformation E
+
Eh-2,
( P , Q)
-+
Therefore, analysis of the case 0 assumptions E = 1, h + 0, and
( a ,P, r)
(24 q)h,
-+
( a ,P, y ) h 2 .
< E << 1 is equivalent to the analysis under
q = O ( h ) , p = O ( h ) , a = O(h2),
p = O(h2), y = O ( h 2 ) ,
(11)
carried out in Sect. 3. In Sect. 4 we don’t take a etc. small (but assume
IQLIPI
1).
3 Local Analysis Near the Origin The main result of this analysis is existence of the third formal integral. More precisely, there exists a formal power series C(z, . . . ,r) with principal part = C, of order h4, such that its truncation CN for any N 2 4 satisfies CN = O ( h N f 2 ) Consequently, . a small neighbourhood of the origin foliates into adiabatic tori. We will enumerate those tori by the values of E , 6, C. Some tori degenerate to periodic orbits that are truly (not asymptotically) invariant. The germ at the origin of the invariant set A formed by these periodic orbits is computable. The main step is the reduction of (9) to an integrable system of order 2 by the normal form method. Details are worked out in [14]; here we just briefly summarize the procedure. The normal form near 0 is constructed and a redundant set R, S, T , E , 6 of resonant variables is introduced. Here E and 6 are the integrals and we have the relation
( ( E - R / 2 ) 2- 6) R2 = S2 + T2. The first truncation of the system in the resonant variables is
1
3E
T = - (R3- 3Eh2 - 33R - 2& 4
+ 2kE2 + 43E).
(12)
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(The 'hat'-variables belong to the truncated system.) From the first two equations in (13), an integral of motion is found 3,. 6' = 8 + 2ER..- -R2
4
Eliminating
8 from (12) and
5 P ( R ; .. .) = - - R 4
16
+2
= const.
(14)
(14) we obtain T 2 = P ( k ; E , 6,
3c 3+ 3132 + 6)
~- ~ (2
ii2
c),where
+ ~ C E R- c 2 .
(15)
Non-negative roots of P(&) determine trajectories of the system (13). Lemma 1 (a) Given ( E ,6,e)suppose that polynomial (15) has roots R2 2 R1 2 0 and P ( k ) > 0 for R1 < k < R 2 . The system (13) has a periodic solution for which min A(t) = R1, max A(t) = R2. (b) Every solution of (13) is periodic and corresponds t o a certain pair of roots of P ( k ) as described. (c) Stationary points of (13) are in 1-to-1correspondence with triples (El6, C ) for which there exists a nonnegative double root of P(R;El6, 6'). The 2-dimensional subset of Ri,6,cin Lemma l(c) is called the discriminant set A. It consists of two (intersecting) parts. The first part is C = 0; in this case the polynomial P[&) has double root 0. The second part corresponds to values, for which P(R) has a positive double root; it admits a simple rational parametrization
In the coordinates ( E
= (sgn E ) E 2 ,6, C), the
set A is conical.
Fig. 1: Trace of the set A on the sphere 62 + C2+ E2 = 1: (a) E
> 0,
(b) E
<0
The set A separates three open regions in the parameter space, which are denoted as Ro,R2, R4.The polynomial P ( k ;E , 6, C) has exactly k distinct
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381
positive roots, if ( E ,6, C) E R k . Projective coordinates of the characteristic points on Fig. 1 are: A ( l , l , O ) , B(O,l,v), C(1, & O ) , D(l,O,l),E(1,-3,0), F ( 0 , v, l ) , where v = (-3 - &)/2 [14, 121. So far we have analyzed the first truncation of the system in resonant variables. It turns out that the described structure of solutions is preserved in higher truncations, at least locally. The following nontrivial statement is a consequence of a theorem from [9].
Theorem 2 Higher-order truncations of the normal form of system (9) are trajectory-equivalent near the origin t o system (13). That is, a neighbourhood of the origin in the phase space is a union of invariant tori enumerated b y the pairs of positive roots of the polynomial P and periodic orbits enumerated b y the points of the set A. Finally, small periodic orbits indexed by points of the set A do exist in system (9), though exact 2-dimensional invariant tori do not exist. The union of the periodic orbits is the analytic invariant set A [3, 111.3.41.
Theorem 3 System (9) possesses a family of periodic solutions near the origin, which are in 1-to-1 correspondence with some neighborhood of zero of the set A. It contains Lyapunov solutions of small amplitude: the family C1 corresponds to the point D o n Fig. 1, and the family C2 corresponds t o the line C = 0 . The periodic solutions of (9) are small perturbations of periodic solutions of the truncated normal f o r m that correspond to stationary points of the system (13).
4 Stability Analysis for the Family L2 in the Linear Approximation The system in variations for the two-system (9) at the periodic solution (10) splits into two independent subsystems. The system for the variations ye, y p , yr is linear and its solutions are 7r-periodic, hence bounded. The dynamics of the variation yg obeys the Mathieu equation
d2Y, dr2
-+ ( a - 2bcos2r)yq = 0,
u=1
+ ~ E Eb,=
3&
(17)
The solution (10) is stable in the linear approximation iff any solution of (17) is bounded, which is equivalent to the well-known trace condition for the corresponding monodromy matrix M: ltrMI 5 2. The stability zones for Eq.(17) in the plane of parameters ( a , b) are well known, see e.g. [2, Fig. 20.11. The corresponding zones in the ( E ,6) plane are shown on Fig. 3. The region S > E2 marked as ‘forbidden’ does not represent any data in the phase space of the two-system.
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5 Numerical Results One can visualize dynamics in the two-system via the Poincark section. Consider the 3-dimensional level set with k e d values of the integrals E and 6 and let V be its 2-dimensional subset where ,8 = 0. The set V can be parametrized by p and q. The Poincark map 17 : V + V takes a point where the trajectory of the two-system intersects the hyperplane ,8 = 0 in the direction d,8/dt > 0 to the next such point on the same trajectory. Numerical results are shown in Fig. 2 and Fig. 4.Here 6 = 0. The analysis in Sect. 3 predicts that for small values of E , there exist two stable periodic solutions with initial conditions qo = po = 0 and qo = 0 , p o
M
k f i , and
4
an unstable periodic solution with qo = 0, po M f $E. They all are clearly visible in Fig. 2a. As E grows, we observe (Fig. 2b) loss of stability of the point (0, 0), which corresponds to the solution of family L2,in agreement with Sect. 4. The second stable solution still survives in Fig. 2b. The distinct loop present in all the three figures represents the solution of the family Ll and is given by the equation p 2 q2 q 4 / 2 = 2E. Fig. 4 shows a chaotic trajectory whose initial point lies beyond the region shown on Fig. 2a. It is remarkable however that there are rather big areas filled with apparently quasiperiodic solutions (closed ovals on the Poincare section). More graphics and details can be found in [la].
+ +
-. t-I*l2 -0.8 -0.4
..I
.:--.=-a
0.0
0.4
... .
.)
0.8
Fig.2: Poincar6 sections for E
=
1, 6 = 0, (a) E = 0.1, (b) E = 1.0
Classical Dynamics of Quantum Variations
383
-12.01
p Fig. 3: Stability diagram for family L2
Fig. 4: Section Fig. 2(a) zoomed oul
Appendix: The Two-System in Quantum Mechanics
Let @ ( q , t ; t i ) be a solution of the Schrodinger equation (SE) iha& = H (q, -itia,) !& (with, say, Weyl's ordering in H ) , and let X ( t ) = (Q(t), P(t)) be a solution of the corresponding classical Hamiltonian system, see (1). Consider the mean values
a t ) = (*I 4 I@), P ( t ) = (*I
P(t) = (*I
- iha,
( q - Q(t))(--ihdq- P ( t ) ) 1%
I@),
4 t >= (@I (4 - 4(t)I2I@), y ( t ) = (*I
(--iwl- fW2I*).
Using notation (4) in Sect. 1, we say that the solution @(q,t ;ti) is a localized state, if for some TE > 0 and all t E [O,TE]the following properties hold (cf. [5, p. 2351): limq(t) = Q ( t ) , limp(t) = P ( t ) and @(t)= O(h) as ti --t 0. Proposition (see [5, (5.22)]) If @(q,t ;ti) is a localized state, then f o r t E [O,TE]the discrepancy between the r.h.s. and the l.h.s. of the two-system (5) is o(ti3l2).
The bound TE known as the Ehrenfest time may grow with ti 0: TE logh-' [11, 71. The two-system was obtained in [8] as a post-classical truncation of an infinite ODE system, formally equivalent to the SE and involving all higher ~ which it approximates moments of !P. The question about time T E during the actual values of the quantum means is open. From Proposition 5.4 in [14] it follows that T E << ~ h-'. --f
N
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References 1. Arnold V. I. Mathematical Methods of Classical Mechanics, Springer (1989). 2. Abramowitz, M. & I. Stegun, eds., Handbook of Mathematical Functions. Nat. Bureau of Standards, Washington DC, 1964. 3. Bruno A. D. Local Methods in Nonlin. Diff. Equations. Springer (1989). 4. Bagrov V. G., Belov V. V., Kondrat’eva M. F. Quasiclassical trajectorycoherent approximation . . . Theor. and Math. Phys. 98 (1994), 48-55. 5. Bagrov V. G., Belov V. V., Trifonov A. Yu. Semiclassical trajectory-coherent approximation in quantum mechanics. Ann. Phys. 246 (1996), 231-290. 6. Broer H., Hoveijn I., Lunter G., Vegter G. Bifurcations in Hamiltonian Systems. Lecture Notes in Math. 1806. Springer (2003) 7. Bambusi D., Graffi S., Paul T. Long time semiclassical approximation of quantumjlows: aproof of the Ehrenfest time. Asymptot. Anal. 21 (1999), 149-160. 8. Belov V. V., Kondrat’eva M. F. Hamiltonian systems of equations for quantum means. Math. Notes 56 (1994), 1228-1236. 9. Bruno A. D., Sadov S. Yu. O n a formal integral of a divergence-free system. Math. Notes 57 (1995), 803-813. 10. Goriely A. Integrability and Nonintegrability of Dynamical Systems. World Scientific (2001). 11. Hagedorn G. A., Joye A. Exponentially accurate semiclassical dynamics.. . Ann. Henri Poincar6 1 (2000), 837-883. 12. Kondratieva M. F., Sadov S. Yu. Two-system of a Hamiltonian system. EPrint: arXiv.org/rnath.DS/0410237. 13. Nekhoroshev N. N.An exponential estimate of the time stability of nearly integrable Hamiltonian systems. Rus. Math. Surveys 32 (1977), 1-65. 14. Sadov S. Yu. O n a dynamical system arising from a finite-dimensional approximation of the Schrodinger equation. Math. Notes 56 (1994), 960-970.
On the Zeros of a Transcendental Function Mark V. DeFazio and Martin E. Muldoon Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada rnuldoonQyorku.ca
+
Summary. We examine the zeros of the function g ( z ) = z pe-' where p E C . In particular, we discuss the movement of the zeros as ,B moves over straight lines through the origin and find that all the zeros of g ( z ) are in the left half-plane if and only if
+ I argPI < ~ / 2 .
1 Introduction: A Transcendental Function In this paper we study the location in the complex plane of the zeros of the function g(z) = z pe-". (1)
+
According to Liouville's theorem [3], the zeros are not elementary transcendental functions of p. When p = 0, there is a single zero at the origin, while for p = 00, g ( z ) has no zeros in the finite part of the plane. For every other complex p, g ( z ) has a countably infinite number of zeros. So it is of interest to examine how the zeros change with changes in the parameter p. Here we study their behaviour as the parameter p varies over straight lines through the origin in the complex plane. It is convenient to distinguish four cases according as to whether the line of variation of ,B is the real axis, or the imaginary axis or lies in the odd or even quadrants. The behaviour of the zeros when ,B is real is qualitatively different from the other three cases. We use our results to show I arg,BI < 7r/2. that the zeros of g ( z ) have negative real part if and only if The interest in the kind of question discussed here arises from the stability theory of delay differential equations where the location of the complex zeros of a transcendental function becomes important; see, e.g., [l],[2], [4]. In particular, stability requires that the zeros be located in the left half plane, Re(z) < 0. This paper goes beyond the scope of the talk given by the first-named author at ISAAC-4. That talk dealt mainly with the case where p is real. It
+
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Mark V. DeFazio and Martin E. Muldoon
was a question from Ilpo Laine at the end of the talk that led us to consider the zeros of g ( z ) for ,8 complex as well.
2 The Zeros of z
+ pe-”
b=O-
1
1 I
.
\
I
b>O
X
b
b>O
Fig. 1. Part of the graph of 2 = -ycot(y- 1) (not drawn to scale), with horizontal asymptotes y = n7r 1 ; in case Q = 1, the zeros of g ( z ) lie on this graph. The alternate transverse branches describe the paths of zeros as b increases from -ca to 0, and from 0 to infinity. The convex branch is special; one of the zeros moves along it from 1 ioo to 1 - 7r ica as b increases from -ca to 00.
+
+
+
Let us put polar coordinates on the parameter space by writing p = beia with -00 < b < $00 and 0 5 a < T . Thus for a fixed a, allowing b to vary from --oo to +-oo causes @ to trace out the line at angle a > 0 to the positive real axis, starting from the lower half @-plane, and passing through the origin when b = 0. When a = 0, @ is real and it traverses the real axis in the usual sense. The imaginary axis is traced out when a = 7r/2 and @ traces out a line in the odd and even quadrants when 0 < a < 7r/2 and 7r/2 < a < 7r, respectively. We first note that all the zeros of g ( z ) are simple except for a zero of order two at z = -1 in the special case p = l/e. We also note that g ( z ) has real zeros only if @ is real. Now let z = z iy and note that
+
On the Zeros of a Transcendental Function g(z) = z
+ be-"
+
cos(y - a) i (y - be-" sin(y - a)).
387
(2)
= 0, n/2. In case cy = 0, we have g(Z) = g ( z ) and the zeros of g ( z ) occur in complex conjugate pairs. In case a = n/2, we have ,B = ib for b E R and g ( z ) = z be-" sin y i (y be-" cosy). Thus if z is a zero of g ( z ) with ,6 = -ib then F is a zero of g ( z ) with p = ib. In both cases, the locus of the zeros of g ( z ) is symmetric with respect to the real axis.
Let us consider two special cases: cy -
+
+
+
P k a: I 9
6
7
-9
3
-6
6
9
.axis
-3
-6
-9
Fig. 2. The graphs of u = y csc(y - 1) and u = beYcot(Y-l) for b = 1,2.
To investigate the zeros of g ( z ) further, we equate real and imaginary parts in g(z) = 0 to get z = -be-" cos(y - a), (3) and y = be-" sin(y - a).
(4)
To solve this system of equations we need to consider two separate cases, a = 0 (i.e., p real) and a # 0 (i.e., p non-real).
3 Non-Real ,8 Let us first consider p non-real. For 0 by (4) to get
< cy < T ,y # 0 and we can divide (3)
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Mark V. DeFazio and Martin E. Muldoon
z = -ycot(y - a ) .
(5)
If we now substitute (5) in (3) we get y csc(y - a ) = b e Y c O t ( y - a ) .
(6)
r
Equation ( 5 ) represents a curve with multiple branches in C . All the zeros of g ( z ) must lie on which is depicted in Figure 1 in the special case a = 1. For 0 < a < T , r has horizontal asymptotes L, = { z E C I Im(z) = a nT}, n an integer. Each strip of the complex plane between two consecutive asymptotes L,-l and L, contains a single branch I?, of For n # 0, these branches are transverse, crossing the imaginary axis at (a+[n- 1 / 2 ] ~ ) iThe . transverse branches have y an increasing function of z for n = 1 , 2 , . . . and a decreasing function of z for n = -1, -2,. . . . To is concave to the right, crossing the . that for a = ~ / 2 I'o , imaginary axis at the origin and at ( a - ~ / 2 ) i Note does not actually cross the imaginary axis, but is tangent to it at the origin and lies entirely in the right half plane { z E C I Re(z) 2 0). The branch To has its vertex at 77 = - cos2(( - a ) (i where ( is the unique solution of 2y = sin2(y - a ) . It is of interest to note that as a increases between 0 and T , the point q traverses the circle of radius 1/2 and center z = -1/2 from z = -1 - Oi to z = -1 Oi, in the positive direction, passing through the origin when a = ~ / 2 . We now turn to equation (6) and consider the intersections of the two curves u = ycsc(y - a ) and u = b e Y c o t ( y - a ) and how they depend on b. (Figure 2 illustrates the cases a = 1 and b = 1,2. The U-shaped and inverted U-shaped branches represent the curve u = y csc(y - 1).The general shape of the graphs of u = b e Y c o t ( Y - ' ) follows from the discussion above of equation ( 5 ) ; see Figure 1.) We see that, for a fixed a , 0 < a < T , is the locus of the zeros of g ( z ) as b varies over R. Figure 2 suggests that, in case b < 0, (6) has a unique zero in ( a nT, a ( n 1 ) ~for ) n = 1 , 3 , . . . and n = -2, -4,.. ., while in case b > 0, ( 6 ) has a unique zero in (a nT, a ( n 1 ) ~for ) n = 0,2,. . . and n = -3, -5,. . .. To verify this, we have to show, for example that, with b > 0, the equation (6) y csc(y - 0)= b e Y C o t ( Y - a )
r
+
r.
+
+
r
+
+
+
+
+ +
+
has a unique root on the interval (a,a T ) . Now the rhs + 0 and the lhs + +oo as y -+ (a T ) - . Also both the rhs and the lhs + +oo as y -+ a + . +oo much faster (like b e l l ( Y - " ) as opposed to just l / ( y However the rhs a ) ) .Thus it is clear that equation ( 6 ) has a root on the interval (a,a T ) . It remains to show that this root is unique. To this end, consider
+
--f
+
+
We have f(a+)= --oo and f ( ( a T)-) = +m. The uniqueness of the root will be clear if we can show that f'(z) > 0 at all points where f(x) = 0. A direct computation shows that at such a point
On the Zeros of a Transcendental Function
389
+
f ' ( 5 ) = csc(z - a ) [ 2 (1 - zcot(5 - a ) ) 2 > ] 0. The direction of the movement of the zeros on each branch can be further analyzed by looking at d z / d b for z a zero. Then z = -pe-" = -beia-" and
Thus
--
dz Z db b(1 + z ) b where we have written z = 5 gives dy = db
+
-1
+
Y
+ x ) +~ y2 and } = -1 x2 + + y2
%L{ldb
1+5-iy l+z 11 212 iy. Equating real and imaginary parts thus
b (1
5 l+x ( l + ~ ) ~ + y ~b ( 1 + x ) 2 + y 2
b
-
-1 (5
-
b
+ +)2 +
y2
-
t
( l + ~ ) ~ + 'y ~
For a zero on a transverse branch the formula for d y / d b tells us that in the upper half plane, the imaginary part of the zero decreases when b < 0 and increases when b > 0 and the situation is reversed in the lower half plane. Let us now consider d x / d b . All of the transverse branches are exterior to the circle with center -1/2 and radius 1/2 and thus the real part of the zeros is strictly increasing(decreasing) for b > O(< 0 ) . Now the concave branch F , crosses the circle with center -1/2 and radius 1 / 2 at the origin when b = 0 and at the vertex 7 when b = bo = ke-cos2(c-a) where we take the plus sign for 0 < a < $ and the minus sign for $ < cx < n-. A careful analysis of d z / d b shows that the real part of the zero is strictly decreasing for b < bo and strictly increasing for b > bo. The conclusion is that for b < 0, we have a single zero, z,, on each branch for n = 2 , 4 , . . . . that starts from (nr cx)i 00 when b = -00 and goes to ( ( n- 1)n a)i - 00 when b = 0 crossing the imaginary axis at the point ( a ( n - 1/2)7r) i when b = - ( a ( n - 1 / 2 ) r ) . Similarly, still with b < 0, we have a single zero, z,, on each branch for n = -1, - 3 , . . . . that starts from ( ( n- 1)n a)i 00 when b = -00 and goes to (nn a)i - 0;) when b = 0 crossing the imaginary axis at the point - ( a - ( n 1/2)7r) i when b = - (Q:- ( n 1 / 2 ) ~ ) . , of the zeros on the transverse branches Thus, in the case 0 < a < ~ / 2 all are in the left half plane Re(z) < 0 for a - 3n/2 < b < 0. Also we have Re(z,) is close to -00 when b is close to 0-, n # 0. We can do a similar analysis for 7r/2 < a < n to show that all of the zeros on the transverse branches are in the left half plane Re(z) < 0 for a n / 2 > b > 0. Also we have Re(%,) is close to -m when b is close to 0-. Let us now turn to the motion of the zero on Fo. We see from (6) that there is a single zero 20 on and that it starts on the upper part at ai 00 when b = -00 and ends up at (a-n-)i+wwhen b = +m. The zero zo is in the
r,
+
+
+
+
+
+
+
+
r,
+
+
+
+
390
Mark V. DeFazio and Martin E. Muldoon
left half plane Re(z) < 0 for b between 0 and ( n / 2 - a ) ,0 < a < w, a: # w / 2 . When a = n / 2 , the branch is tangent to the imaginary axis and zo is always in the right half plane. It is clear then that, as b increases from -00 to 00, to is the last zero to leave the right half plane and the first to re-enter. Thus all of the zeros of g ( z ) are in the left half plane for b between 0 and ( w / 2 - a ) ,0 < a < w , a # w / 2 .
ro
4 Real
p
Let us now consider the case where p is real. For a = 0, equations ( 3 ) and ( 4 ) become x = -be-" cosy (7) and y = be-" sin y.
Now equation (8) has the solution y = 0, in which case (7) becomes 2 = -be-" and describes the real zeros of g ( z ) , and the solution y = be-" sin y which together with (7) gives the system
x = -ycoty
(9)
and y csc y = beYcotY
which describes the non-real zeros of g ( z ) . The situation for real zeros is easy. The real zeros of g ( z ) are the 2-coordinate of the points of intersection of the graph of y = z with y = -be-". For b 5 0, g ( z ) has only one simple, positive real zero xt. As b increases from -00 to 0-, xcb+ decreases from 00 to O+ and x.b+ approaches 0 and like -b. As b increases past 0 , xb+ continues to decrease and a second negative real zero, xb , starts out from -00 and these two zeros approach each other, meeting when b = 1 / e and merge into the zero = xT/e = -1 of order two. As b increases past l / e , this double real zero splits into 2 simple complex zeros. These zeros depart at right angles to the real axis . Now we turn to the complex zeros. The analysis is similar to that in Section 3. The curve 2 = -ycot y has an infinite number of transverse branches but unlike the cases considered above, the concave branch occupies a strip that is 2w units wide. (See Figure 4.) Let us call the transverse branch in the strip kw < Im(z) < (k l)n, and the concave branch in the strip -w < Im(z) < w, I'o. If we compare the graphs of u = y cscy and u = beycoty we see that there is one zero on each r k for k odd when b < 0 and as b varies from -00 to 0 this zero moves from ( 2 k 2)wi 00 to ( 2 k 1)wi - 00, crossing the imaginary axis from right to left at (4k 3 ) n / 2 i when b = -(4k 3)w/2. Furthermore, the last pair of non-real zeros leave the right half plane Re(z) > 0 when b = -3w/2 but the real zero is in the right half plane Re(z) > 0 for all
+
r,,
+
+
+
+
+
On the Zeros of a Transcendental Function
Fig. 3. The graphs of y = 2 and y = -be-”
391
for b = -1,l/e, 1.
b < 0. For b > 0 there is one zero on each r k for k even, k # 0 and this zero moves from 2kni - 00 to (2k 1 ) n i + 00 crossing the imaginary axis from left to right at (4k l ) n / 2 2 when b = (4k l ) n / 2 . We also see that there are no zeros on TOfor b < 1 / e and two zeros on To for b > l / e . These two zeros split off from the double zero at z = -1 and move off to kni 00 as b goes to +00, crossing the imaginary axis at f i n / 2 when b = n / 2 . Thus all of the zeros of g ( z ) are in the left half plane for 0 < b < n / 2 . Note that for b close to but not equal to zero, g ( z ) has one zero near the origin and a countable number of zeros in a neighbourhood of -00.
+
+
+
+
5 A General Stability Result It is clear form the preceding discussion that all of the zeros of g ( z ) = z+pe-” are in the left half plane Im(z) < 0 for b between 0 and x / 2 - a. We formally state this result as follows.
Theorem 1 All the zeros of the function g ( z ) = z +,Be-’, ,O = betaY,0 5 a < have negative real part if and only if I,OI + I arg /?I < n / 2 .
T,
+
The region 1/31 1 argpl < n/2 is illustrated in Figure 5 . It is bounded by the piecewise polar curve given by r = n/2 - 101, 101 5 n/2 consisting of segments of the spirals T = n / 2 f 0.
392
Mark V. DeFazio and Martin E. Muldoon
Fig. 4. Part of the graph of 2 = -ycot y (not drawn to scale); in case a = 0, most zeros of g ( z ) lie on this graph. The alternate transverse branches describe the paths of zeros as b increases from -m to 0, and from 0 to infinity. The convex branch is special; two initially real zeros move along it as b increases from 1/e to 00.
Remark The doubling of the width of the convex branch in the case a = 0 (Figure 4) as opposed to the case a # 0 (Figure 1)may seem counter-intuitive.
It can be understood if one considers that as a
-+ Of, the two branches T o a.nd TI in Figure 1 coalesce into a single convex branch To in Figure 4. The separate simple zeros on the branches To and TI in Figure 1 are replaced by a pair of zeros which are initially real ( b < l/e) then collide to form a double zero (b = l/e) and the two complex conjugate zeros ( b > l/e). The limit a -, O+ brings no qualitative change in the other branches and the zeros lying on them.
6 A Result of Hale and Verduyn Lunel In [2], the function
h(t) = t
+ a + be-t
is considered. We consider a to be real and fixed and let b vary over R. In [2, Theorem A.5, p. 4161, there are given necessary and sufficient conditions for all the zeros of h ( z ) to have negative real parts. The conditions are - l < a and - a < b < < s i n < - a c o s < ,
(12)
On the Zeros of a Transcendental Function
393
I
0.4 -
0.2-
0
1.6
-0.2-
-0.4
-
Fig. 5 . The region 101+ 1 argpl < 7r/2 for which
g(t) = t
+ ,Be-'
is stable.
where C is the unique root of -5 cot C = a. We will obtain this result using the results of Section 4 and obtain the simpler form -asecC for the upper bound for b in (12). Let us note that h(t) = 0 if and only if g b e a ( t a) = 0 where we have written g b ( Z ) = g ( z ) = z be-' to emphasize the dependence on b E R . Thus h(t) is stable if and only if all the zeros of g b e a ( z ) lie in the left half plane Re(z) < a. A first reduction occurs when we note that if a 5 -1, there will always be a zero in Re(z) 2 a by following the trajectory of the real zero that starts from +cm as b increases over R. Thus a must be greater than -1. Now if a > -1, it is clear from our discussion of the movement of the zeros of g ( z ) that the positive real zero of gbea (2) that starts from +oo crosses the line Re(z) = a from right to left when b = -a, as b increases. Furthermore, at that point all of the zeros of g b e a ( z ) lie in the left half plane Re(z) < a and stay there until the pair of zeros on cross the line Re(z) = a for some positive value of b. The only question left is how big b is allowed to get. Consider the zero on the upper half of To. It will cross the line Re(z) = a when Im(z) = where C is the unique solution of a = -y cot y and 0 < C < 7r. Note that
+
+
0
< C < 7r/2for - 1
< a < O and
7r/2
< 7r for
a >O.
To find the value of b when it crosses the line Re(z) = a we note that 6 = (be")e-" = (be")eC
C =
C csc C = -a
sec C
(13)
394
Mark V. DeFazio and Martin E. Muldoon
by (10). It is now simple trigonometry to show that --a sec C = C sin ( - a cos C and (13) gives us --a sec ( > 0 as expected by our discussion above. Acknowledgment. We are indebted to a referee for pointing out a number of errors and omissions in an earlier version of this article.
References 1. 0. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O.Walther, Delay Equations: Functional-. Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, vol. 110, Springer-Verlag, 1995. 2. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, 1993. 3. M. Matsuda, Liouville’s theorem on a transcendental equation log y = y/z, J. Math. Kyoto Univ. 16 (1976), no. 3, 545-554. MR 55 #2864 4. J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, vol. 119, Springer-Verlag, 1996.
Evaluation of Sylvester Type Determinants Using Block-Triangularization Olga Holtz Institut fiir Mathematik, MA 4-5 Technische Universitat Berlin Strai3e des 17. Juni 136 D-10623 Berlin Germany holtz(9math.W-Berlin.DE
Summary. It is shown that the values of Sylvester type determinants considered by Askey in [l]can be ascertained inductively using simple block-triangularization schemes.
1 Introduction Richard Askey in [l]shows two ways, one matrix-theoretic and another based on orthogonal polynomials, to evaluate determinants
x
1
N x
2
0
N - l ~3
,
l o
2 XN 0 l x
which were first considered by Sylvester [2]. In addition, he obtains several generalizations of Sylvester’s determinants and explores their connection t o orthogonal polynomials. The purpose of this note is t o show how the determinants from [l]can be evaluated in yet another way, based on partial information about left or right eigenvectors of the corresponding matrices coupled with a simple similarity trick. In all cases except one, only one (the most obvious) eigenvector is used. The exceptional case is the Sylvester determinant itself, where two eigenvectors are readily available and hence used to derive the result.
396
Olga Holtz
2 Sylvester's Determinant and Two Close Variants Let us start with the Sylvester determinant. We want to prove that N
rI.(
DN+1(4 =
+N -2 d
(1)
j=O
(which is formulas (2.3), (2.4) from (11).Since the values of D1 and Dz agree with (l),it is enough to show that
DN+l(Z) = .(
-N)(x
+ N)DN-l(Z)
(2)
for N _> 1. The Sylvester determinant D N + is ~ the characteristic polynomial of the matrix - D N + ~ i.e., , D N + I ( Z )= (x A), where
nXEa(DN+l) +
0 o . . * o0 2 o * * * o0 ON-1 0 3*.*00 0 0 N-20***00
-0
1
N O
0 0
0 0
.. .
.. .. .. .. . . . .
0 0
0.'. 1 0
O * * * ON
-
1 11 1 1 1-1 1-1 ... (-1)N 0 0 1 o... 0 ?&+I:= o 0 0 I . . . 0
.. .. .. .. . . . . . . . -0 0 0 o . . .
-
. 1
therefore reduces D N + to ~ a block lower-triangular form [diag(T-N)
0 MN-1
where 0 3+(-N+1) 0 -N+1 N-2 0 4 0 0 N-3 0 5 0 0 N-4 0 0 N-5 0 0 0 N 0 0 0
0 -N+l... 0 ... 0 0 ... 0 ... 0 6 0 7 ... -6 0 ..*
Evaluation of Sylvester Type Determinants Using Block-Triangularization
397
Now it remains to show that M N - is ~ similar to Div-1. But this is indeed so, since S ~ I , M N - ~ S N = -DN-I ~ where
' 1 0 -1 0 0 0 1 0-1 o . . . 00 1 O-l... 00 0 1 0 . a . 0 0 0 0 1...
... .. . .. .
.. .. .. . . .
This proves (2) and therefore (1).
A related determinant 0
a
2
...
N(a-1) 2-1 2a ... O (N-l)(a-l)x-2...
. 0 0
0 0
-
0
0
0 0 0
0 0 0
0 0 0
.. . . . 2(a-1) z - ( N - 1 ) 0 a-1
...
0 U 0 *.. N(" - 1) -1 2a . . . 0 ( N - l ) ( a - 1) -2 . . .
0 0 0
0 0 0
.. 0 . . . 2(" - 1) - ( N 0 ... 0 a-1
Nu 2-N
0 0 0
..
0 0
0
0
for the matrix that satisfies B ~ + i ( x=) To prove that N
BN+l(Z) = n [ x
1) N u
-N
nIXEa(BN+l) (x + A).
+ ( N - 2j)a - N + j ]
(3)
j=O
(formula (2.8) from [l]),we will show that BN+I(Z) =
+
(x N a - N ) B N ( x - a).
(4)
The vector (1,1,1,.. . ,1)is a left eigenvector of B N + corresponding ~ to eigenvalue N u - N . The similarity transformation
BN+1 H TV+lBN+lT;t,
398
Olga Holtz
where
-
-1 1 1 1 1 0 100 0 00 10.*-o TN+I:=0 0 0 1 ... o a
*
.
a
*
.
.. .. .. .. .. .. .... ..
-0 0 0 0
*..
1-
reduces B N + to ~ a block lower triangular form
where
- N ( a - 1) 2a - N ( a - 1) ( N - l ) ( a - 1) -2 0 ( N - 2)(a - 1)
'-1
0 0
0 0
- N ( a - 1) . . * - N ( a - 1) ... 3a 0 . .. -3 4a . .. . ( N - 3)(a - 1) -4 0 ( N - 4 ) ( a - 1) . . .
-
-
where
SN
1-1 0 o . e . 0 ' 0 1-1 o . - . o 0 0 1-1-..0 := 0 0 0 1 . * . o
.. .. .. . . .. .. . . . . . .
0
0
0
0
"'
1
This proves (3).
0
As is shown in [l],the formula (3) for B N + implies ~ the formula (1) for the Sylvester determinant D N + ~since ,
Also, (3) gives a formula for another related determinant,
Evaluation of Sylvester Type Determinants Using Block-Triangularization
399
AN+I(2):= 1 0 o... 0 2-2 2 O... O 0 -( N - 1 ) ~ - 4 3 . . . 0
0 0 0
2
-N
0 0
0 0
0 0 0
.. .
.. . . .. . . .
0 0
0...-2~-2(N-l) N 0 ... 0 -1 X-2N
(5)
via the relation AN+l(z) =
2 N + ~ BN+1
(;)I
*
a=
3 Determinants for Krawtchouk and Dual Hahn Polynomials The determinant for the Krawtchouk polynomial K ~ + l ( z ; N p ,) is given by K N + l ( X ; P ,N
) :=
With the usual notation ( u ) k for the shifted factorial (see [l,formula (3.10)]), we need to prove [l,(3.25)]: KN+l(z;P,N
)
(6)
(-2)N+1-
For that, it is enough to show that K N + l ( W ,N
) = (-qGV(v, N
-
1).
As usual, we need to find an ‘obvious’eigenvector of the matrix K N + 1 :=
(7)
400
Olga Holtz 0 P"
- 1)
PN +2( 1 - 2p)
...
0
0
...
0
0
...
0
0
0
corresponding to eigenvalue 0. In this case, it happens to be the right eigenvector (1,-1,1, -1,. . . , ( - l ) N ) TWe . then use the transformation
1 -; ;:::::: 0 1o . . . o 001...0
1 -1
. ..
.. .. * .. . . ... .. *
000
(-1)N
* * *
1
to transform K N + ~into a block upper-triangular form
where
-
PN+PN + ( I - 2P) -PN
+ 2(1 - P )
.'.
0
+2(1 - p2p) N p(N - 2) . . .
0
3(1 - p ) ... ... 0
0 0
p(N-1)
PN -PN
Now, M N is similar to the matrix ICN
via the transformation
0 0
0
+ I , viz.
Evaluation of Sylvester Type Determinants Using Block-Triangularization
401
1 000---0 1 100 - - - 0 0 110---0 00 1 1 ••• 0
(9)
0 0 0 0• • •
This proves (7) and hence (6). The determinant for dual Hahn polynomials,
N(-f _
3(AT + 6 - 2)
+ AT(7 + 7) (7 -6 + 6)'
where A(z):=-z(:r + 7 + 5 + 1), appears in [1] as the first example beyond Krawtchouk polynomials in terms of simplicity of recurrence coefficients. This determinant seems much harder to evaluate, but this in fact requires just an additional shift of parameters. The formula for RN+1 is [1, (4.5)]: RN+1(X(x);~f,S,N)
= (~x)N(x +7 + 6 + l)N+1.
(10)
It can be proved from the relation
;*f,6,N) = X(x)RN(\(x) + (>v + 6 + q;'Y + l,6 + l,N-l). (11) The latter can be checked exactly as above, i.e., by applying the transformation TJV+I of the form (8) to the matrix 0
_2(
0
402
Olga Holtz
obtaining a block upper-triangular matrix
where MN(-y,6) :=
N ( 7 + 1) +N(7 + 3) ( N - l)(y -(y - 6 - 2)
+ 2)
0
0
+ + 1)+ (y + 6 + 2)1 by
and then transforming M N ( ~ S) ,into RN(Y l , d
M N ( Y6),
++
S N M N ( Yd)SF1, ,
where SN has the form (9). This proves (11) and (10).
0
4 Determinants for Hahn, Racah, and q-Racah Polynomials The last examples appearing in [l]are three types of orthogonal polynomials, whose recurrence coefficients are fractional rather than polynomial. Recall that the recurrence coefficients appear as entries on the main diagonal and the first sub- and super- diagonals of the tridiagonal determinant that gives the value of the corresponding polynomial. The alternating sums of these coefficients are still zero, just like in the two examples we just considered in section 3. In other words, the determinants we now consider all have the form
Evaluation of Sylvester Type Determinants Using Block-Triangularization
403
where
b, = -a, -c, for all n. The coefficients a, and c, for Hahn polynomials are
(13)
(n+Q+p+l)(n+Q+l)(N-n) (an a p 1)(2n Q p 2) n(n a P N l ) ( n P) c,= (2n a P)(2n Q + p + 1)'
a, =
+ + + + + + + + + + + + + +
with X(x) = -2, while the coefficients for Racah polynomials are
+ + 1)(n + a + p + l ) ( n + y + 1)(N- n ) (2n + a + p + l)(2n + + p + 2) -n(n + a + p + N + 1)(n + + p - y)(n + p) = (an + a + P ) ( 2 n + a + p + 1)
a, =
(n a
Q
c,
Q
+ +
with p y 1 = -N and X(x) = -x(x q-Racah polynomials are a, =
7
+ y + S), and the coefficients for
(1 - abq^+l)(l - q,+1)(1 - q."-N)(1 - cq,+l) (1 - ubq2,+1) (1 - ubq2,+2) cqPN (1 - q n ) ( l - bq")(l - abc-lq")(l - abq,+N+1)
c, = -
b
(1 - abq2,) (1 - abq2,+1)
(14) 1
(15)
with bdq = q - N and X(x) = -(-qPZ)(l - q"+lcd) (see [l,Sec. 41). Recall that Hahn and Racah polynomials are just limiting cases of qRacah polynomials [l,Sec. 41. Precisely, if we denote Hahn polynomials by H N + ~ ( X ( ~ ) ; Q , ~Racah , N ) ,polynomials by R A ~ + l ( X ( z ) ; a , p , y , N and ) , qRacah polynomials by QRAN+l(X(x); q, a , b, c, N ) , then we see that
So, it is enough to evaluate q-Racah polynomials. The formula is given in [l] in the form Q R A N + ~ ( X ( ~q,a, ) ; b,c, N ) = (-l)N+l(q-Z;
q)N+1(qz+lCd; q)N+1,
where ( ; )k is the q-analogue of the shifted factorial [l,formula (4.13)]: k- 1
( a ; q ) k:=
H(1- aqj).
j=O
Its equivalent form that is more suitable for an inductive proof is
404
Olga Holtz 11
QRAN+l(X(x);q,a, b, c, N) = J](A(a;) - A(n)).
(16)
n=0
This formula will be proved once we show that QRAN+1(X(x)-q,a,b,c,N) N
= X(x}q- QRAN(X(x
- 1); q, aq, 6, cq, N - 1),
(17)
where X(x) corresponds to the parameters (q, aq, b, cq, N — 1) so that
#«•> cq
- q X(x) + 1 +
bq N
(18)
bq1
To prove (17), let us start with an observation about our ansatz matrices GN+I satisfying (12) and (13). Suppose that such a GN+I is transformed using the matrix TN+I given in (8). Then, as we already saw, TA7 ^(/jv-nT/v-n is block upper triangular: TO * Next, the transformation form -i O.Q + Cl Cl
0 0 0
0,1
given by (9) reduces MN to the tridiagonal (19) 0
tti + C2 02 C2 C12 + C3
0 0
0 0
0 • • 0 • •• a3 • •
0 0 0
0 • • CJV-2 0 • • 0
0 0 0
0 0 0 ajv-i
«N-2 + Cjv-l
CN-1
I
IN-I +cN _
Proving (17) therefore amounts to showing that the matrix (19), with the entries an and Cn coming from the determinant QRAN+l(X(x); q, a, b, c, N), is similar to the matrix -(GN + ^(x ~ I)^AT) ~ A(x)/jv, where £/jv is an NxNmatrix of the form (12)-(13), with the entries an and cn coming from the determinant QRAN(X(x — 1); q, aq, b, cq, AT — 1). (Note that, due to (18), the difference A(x) — -X(x — 1) does not depend on a;.) The needed similarity is realized by the diagonal matrix AN := diag ((1 - abq2), i(l - abq4),..., _L-(1 - abq2N)j . so that
Evaluation of Sylvester Type Determinants Using Block-Triangularization 1
405
+ X(S
= +iN - I)I~) - qZ)iN. (20) q Verification of (20) is straightforward for off-diagonal entries. For diagonal entries, it reduces to verification of the identity an
+
C N + ~-
1-
1 - ubq2n+4 1 - abq2, +q2cn 1 - abq2n+2 ’ 1 - abq2,+2
where a, are given by (14) and c, by (15). This last identity can be checked USing MATLAB Symbolic Math Toolbox. This finishes the proof of (17) and (16). 0
The determinant for Racah polynomials is therefore
and the determinant for Hahn polynomials is
Acknowledgements This work was supported by the Alexander von Humboldt Foundation and by the DFG research center “Mathematics for key technologies” in Berlin.
References 1. R. Askey, Evaluation of Sylvester tvpe determinants using orthogonal polynomials, Proceedings of the 4th ISAAC Congress, 2003, xxx-xxx. 2. J. J. Sylvester, Nouvelles Annales de Mathkmatiques, XI11 (1854), 305, Reprinted in Collected Mathematical Papers, vol. 11, 28.
Square Summability with Geometric Weight for Classical Orthogonal Expansions Dmitrii Karp Institute of Applied Mathematics Far Eastern Branch Russian Academy of Sciences 7 Radio Street Vladivostok, 690041 Russia dmkrp0yandex.ru Summary. Let fk be the k-th Fourier coefficient of a function f in terms of the orthonormal Hermite, Laguerre or Jacobi polynomials. We give necessary and sufficient conditions on f for the inequality Ck1fk('ok < 00 to hold with 8 > 1. As a by-product new orthogonality relations for the Hermite and Laguerre polynomials are found. The basic machinery for the proofs is provided by the theory of reproducing kernel Hilbert spaces.
1 Introduction The goal of this paper is to find necessary and sufficient conditions to be imposed on a function f for its Fourier coefficients in terms of classical orthogonal polynomials to satisfy the inequality m
k=O
with 0 > 1. So, we have three problems corresponding to the following three definitions of f k : 00
fk =
/f(x)Hk(x)e-'ldz, -03
and
(2)
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Dmitrii Karp
-1
Here W k , IL; and P ~ " is the k-th orthonormal polynomial of Hermite, Laguerre and Jacobi, respectively [19]. For the sake of convenience we use orthonormal instead of standardly normalized versions of the classical polynomials. In each case f is defined on the interval of orthogonality of the corresponding system of polynomials. We will use (Pk as a generic notation for either of the three types of polynomials. Classes of functions with rapidly decreasing Fourier coefficients in classical orthogonal polynomials have been extensively studied. We only mention a few contributions without any attempt to make a survey. The series of papers [9][ll]by E. Hille is devoted to the Hermite expansions. Among other things, Hille studied expansions with f k vanishing as fast as exp(-~(2k 1)'12). The functions possessing such coefficients are holomorphic in the strip IQzI < T . Hille provided an exact description of the linear vector space with compact convergence topology formed by these functions where the set {Wk : k E WUO} is a basis. Its members are characterized by a suitable growth condition. In addition, he studied the convergence on and analytic continuation through the boundary of the strip. The convergence domain of the Laguerre series with fk exp(-.rk 1/2) is the interior of the parabola % ( - z ) ~ /=~ r / 2 . Rusev in [17] described the linear vector space with compact convergence topology formed by functions holomorphic in { z : % ( - z ) ~ /<~ 7/2} where the set {ILL : k E N U 0) is a basis, and Boyadjiev in [4]studied the behavior of the Laguerre series on the boundary of the convergence domain. If the coefficients (2) or (3) decrease as fast as exp(-.rks) with q > 1 / 2 the function f is entire. The spaces comprising such functions for Hermite expansions have been characterized by Janssen and van Eijndhoven in [12]. If the Fourier-Hermite coefficients fk decline faster than any geometric progression, the suitable characterization is provided by Berezanskij and Kondratiev in [2] (see Corollary 1.1below). Byun was the first to study the Hermite expansions with condition (1) - see remark after Theorem 1. For the Laguerre expansions with lim supk IfkI'lk < 1 Zayed related the singularities of the Bore1 transform f k z k in [20]. of f with those of F ( z ) = For the Legendre expansions with limsupk Ifk('lk = < 1, Nehari relates the singularities of f on the boundary of the convergence domain { z : Iz 11 Iz - 11 < [ 5-l) to those of F ( z ) = f k z k in [16]. Gilbert in [7] and Gilbert and Howard in [8] generalized the results of Nehari to the Gegenbauer expansions and to expansions in eigenfunctions of a Sturm-Liouville operator. Functions satisfying (1) apparently form a proper subclass of functions with limsupk lfkli 5 19-4. On the other hand the condition (1) itself cannot be expressed in terms of asymptotics of fk. Consequently, our criteria for the validity of (1) are of different character from those contained in the above
+
N
Ck
+
+
<
Ck
+
Square Summability for Classical Expansions
409
references. Although they also describe the growth of f for the Hermite and Laguerre expansions and its boundary behaviour for the Jacobi expansions, our growth conditions are given in terms of existence of certain weighted area integrals o f f and cannot be expressed by an estimate of the modulus, while the restriction on the boundary behaviour is given on the whole boundary and not in terms of analysis of individual singularities.
2 Preliminaries Throughout the paper the following standard notation will be used: N,R, R+ and CC will denote the positive integers, the real numbers, the positive real numbers and the finite complex plane, respectively. Since the coefficients (2)-(4) do not change if we modify f on a set of Lebesgue measure zero, all our statements about the properties o f f should be understood to hold almost everywhere. If, for instance, we say that f is the restriction of a holomorphic function to (some part of) the real axis, it means that f is allowed to differ from such restriction on a set of zero measure. All proofs in the paper hinge on the theory of reproducing kernel Hilbert spaces (RKHS), so for convenience we briefly outline the basic facts of the theory we will make use of. For a Hilbert space H comprising complex-valued functions on a set E , the reproducing kernel K ( p ,q ) : E x E -+ C is a function that belongs to H as a function of p for every fixed q E E and possesses the reproducing property
for every f E H and for any q E E. If a Hilbert space admits the reproducing kernel then this kernel is unique and positive definite on E x E: n
i,j=l
for an arbitrary finite complex sequence {ci} and any points pi E E. The theorem of Moore and Aronszajn [l]states that the converse is also true: every positive definite kernel K on E x E uniquely determines a Hilbert space H admitting K as its reproducing kernel. This fact justifies the notation H K for the Hilbert space H induced by the kernel K . The following propositions can be found in [l,181. Proposition 1 If H K is a Hilbert space of functions E arbitrary non-vanishing function on E , then
K&, q ) = s ( p ) s ( q ) K ( pQ, )
-+
C and s is an
(6)
is the reproducing kernel of the Hilbert space HK, comprising all functions on E expressible in the form f s ( p ) = s(p)f ( p ) with f EHK and equipped with inner product
410
Dmitrii Karp
Proposition 2 Let El C E and K1 be the restriction of a positive definite kernel K t o El x E l . Then the R K H S H K ~comprises all restrictions t o El of functions f r o m H K and has the n o r m given by I I f l l l ~ K=~ m
in{llfll~~;= f lf~i ~ , f E HK).
(8)
Proposition 3 If RKHS HK is separable and {$k : k E N} is a complete orthonormal system in H K , then its reproducing kernel is expressed by
where the series (9) converges absolutely for all p , q E E and uniformly o n every subset of E , where K(q,q) as bounded.
The relation K1 << K2 will mean that K2 - K1 is positive definite. This relation introduces a partial ordering into the set of positive definite kernels = H K will ~ be understood on E x E. Inclusion H K ~c H K ~and equality H K ~ in the set-theoretic sense, which implies, however, that the same relations hold in the topological sense as stated in the following two propositions.
Proposition 4 Inclusion H K ~c H K ~takes place iff K1 << MK2 f o r a constant M > 0. I n this case M 1 / 2 ) ) f l )_>l f o r all f E H K ~ (Here . llflll and 11 f 112 are the norms in H K ~and H K ~respectively). ,
Proposition 5 Equality H K ~= H K ~takes place iff mK2 << K1 << MK2 f o r some positive constants m, M . In this case m1/211fl115 llfllz I M 1 ~ z ~ ~ f ~ ~ Kernels satisfying Proposition 5 are said to be equivalent which is denoted by K1 M K2. It is shown in [l]that the RKHS H K induced by the kernel K ( p , q ) = K1(p,q)K2(p,q ) consists of all restrictions to the diagonal of E' = E x E (i.e., the set of points of the form { p , p } ) of the elements of the tensor product HI = H K @ ~I H K ~The . space HK is characterized by
Proposition 6 Let K ( p , q ) = Kl(p,q)Kz(p,q)and let { $ k : k E N} be a complete orthonormal set in H K ~ Then . the R K H S H K comprises the functions of the f o r m W
f(p) = c f L ( p ) $ k ( p ) , k=l
00
fl E HKi,
lfk111? < 00.
(10)
k=l
The n o r m in HK is given b y
where the minimum is taken over all representations o f f in the f o r m (10) and is attained o n one such representation.
Square Summability for Classical Expansions
41 1
3 Results for the Hermite and Laguerre Expansions Functions satisfying (1) form a Hilbert space with inner product 00
k=O
This space will be denoted by Re for the Hermite expansions and by C i for the Laguerre expansions. The sets { 8 - k / 2 H k } k E N u o and {8-k/2L",k,Wuo constitute orthonormal bases of the spaces 'He and L i , respectively. For each space we can form the reproducing kernel according to (9):
The explicit formulae for these kernels are known to be [3]:
(13)
(Mehler's formula) and
(Hardy-Hille's formula). Here I, is the modified Bessel function. The kernels (13) and (14) are entire functions of both z and 21. The space 'He comprises functions on R, while the space C i comprises functions on . ' R Hence, we consider the restrictions of the kernels (13) and (14) to R and R+,respectively. Applying Proposition 2 with E = C and E' = R or E' = R+ we conclude that the spaces 'He and LE are formed by all restrictions to R and R+,respectively, of entire functions from the spaces generated by the kernels (13) and (14). We can drop minimum in (8) due t o uniqueness of analytic extension and, consequently, the norm induced by inner product (11) equals the norm in HHK@or in H , q . Next, we observe that both kernels (13), (14) are of the form s ( z ) s ( ~ ) K ( z T i ) with non-vanishing functions S H ( Z ) = e-z2/(e2-1) and s ~ ( z )= e - z / ( e - l ) . Hence we are in the position to apply Proposition 1. In compliance with (7) the norms in HHK* and H ~ , are Y known once we have found the norms in the spaces induced by the kernels
and
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Dmitrii Karp
Ql//2+1
-1/
LK,(zii)
=
~
0-1
(z..)-q/
(E) 2-
Both of them depend on the product z;ii and are thus rotation invariant. Rotation invariance of the kernel implies radial symmetry of the measure with respect to which the integral representing the norm is taken. For the kernel (15) the measure and the space are well-known. It is the Fischer-Fock (or the Bargmann-Fock) space .To of entire functions with finite norms
where the integration is with respect to Lebesgue's area measure. By Proposition 1 the final result for the Hermite expansions now becomes straightforward.
Theorem 1 Inequality (1) with fk defined b y (2) holds true for all restrictions to R of the entire functions with
and only for them. For inner product (11) this leads to the expression
(19) Since the polynomials Wk are orthogonal with respect to this inner product, we obtain the following orthogonality relation for the standardly normalized Hermite polynomials Hk: Hk(z)H,(z)exp
0+1
0-1
do = 6 k , , ~ ( 2 0 ) ~ k ! .
(20) These results for the Hermite expansions have been essentially proved by DuWong Byun in [5], although the emphasis in his work is different and it seems that the orthogonality relation (20) has not been noticed. The following corollary is an immediate consequence of (18).
Corollary 1.1 Inequality (1) with fk defined by (2) holds true for all 0 > 1 iff f is the restriction to R of an entire function F satisfying IF(z)I 5 CeElzIz for all E > 0 and a constant C = C(E)independent of z .
Square Summability for Classical Expansions
413
A direct proof is given in [2]. For the Laguerre expansions the situation is a bit more complicated. The kernel (16) is a particular case of a much more general hypergeometric kernel. The spaces generated by hypergeometric kernels are studied in depth in [15]. For the kernel (16) we get the space of entire functions with finite norms
where K , is the modified Bessel function of the second kind (or the Macdonald function). Application of Proposition 1 brings us to our final result for the Laguerre expansions. Theorem 2 Inequality (I) with fk defined by (3) holds true for all restrictions to R+ of the entire functions with
and only for them. The orthogonality relation for the standardly normalized Laguerre polynomials L i that follows from this result is given by
Corollary 2.1 Inequality (1) with fk defined by (3) holds true for all 6 > 1 iff f is the restriction to R+ of an entire function F satisfying IF(z)I _< Ce“lzI for all E > 0 and a constant C = C(E)independent of z . This corollary can be easily derived from (22) with the help of the asymptotic relation [3]
4 Results for the Jacobi Expansions The space of complex-valued functions on ( - 1 , l ) whose Fourier-Jacobi coefficients (4)satisfy (1) will be denoted by J:@. Pursuing the same line of argument as in the previous section, we form the reproducing kernel of this space found by Bailey’s formula [3]
414
Dmitrii Karp
where
c 00
F4(a, b; c, c’; t , s)
=
m,n=O
( c )n (c’ )m rn !n!
is Appell’s hypergeometric function and ~ ( a : , p= ) 2a+P+1T(a: l)/T(a: p 2). Define the ellipse E,g by
+ +
Ee
= {Z :
IZ
- 11 + )Z+ 11 <
+ l)r(p +
+ F1I2}.
(26)
Our first observation here is that both the series on the right hand side and on the left hand side of (24) converge absolutely and uniformly on compact subsets of Ee x Ee (see [13]). The implication of the uniform convergence is holomorphy of the kernel (24) in Ee x Ee with respect to both variables. Application of Proposition 2 with E = Ee, E’ = ( - 1 , l ) leads to the assertion that the space is formed by restrictions to the interval ( - 1 , l ) of functions holomorphic in Ee whereby the norm in &?’ equals the norm in H J K ~ ,due , to uniqueness of analytic continuation. Our main result for the Jacobi expansions will be derived from its particular case a: = p = X - 1/2 whereby the orthonormal Jacobi polynomials reduce to the orthonormal Gegenbauer polynomials C i . The reproducing kernel (24) reduces in this case to
Jta,’
- +q(e2
+ i ) x + 1 2Fl ( Tx, T+; X1 + A- *+
eZx(e2- 1) - 2ez;ii
2
1 4e2(1- Z 2 ) ( i - ~ 2 2’ (e2-zeZEiq2
)
)
(27) where 2F1 is the Gauss hypergeometric function and .(A) = & r ( X 1/2)/r(X 1). The series on both sides of (27) again converge absolutely and uniformly on compact subsets of Ee x Ee. Formula (27) can be obtained from (24) by using a reduction formula for F4 and applying a quadratic transformation to the resulting hypergeometric function. Details are in [13]. Let aEg denote the boundary of the ellipse Ee. We introduce the weighted Szego space AL2(aEe;p ) with continuous positive weight p ( z ) defined on ~ E Q as the set of functions holomorphic in Eg , possessing non-tangential boundary values almost everywhere on dEe and having finite norms
+
+
Square Summability for Classical Expansions
415
We will write AL2(aEe) for AL2(aE8;1). For X = 0, the orthonormal Gegenbauer polynomials reduce to the orthonormal Chebyshev polynomials of the first kind T k :
c i ( z ) = Tk(Z)
= (2/T);Tk(z) = (2/7r)3
cos(karccOSz), I% E
N, (29)
C:(Z) = T ~ ( z= ) ( l / ~ ) + T o ( z=) ( l / n ) + ,
where T k is the Ic-th Chebyshev polynomial of the first kind in the standard normalization.
+
Lemma 1 The polynomials (Ok 8-k))-1/2Tk/2 f o r m a n orthonormal basis of the space ALz(aE0;Jz2- 11-i).
Proof The boundary aEe is an analytic arc which implies the completeness of the set of all polynomials in AL2(dEe;1z2- 11-a) (see, for instance, [S]).To prove orthogonality we will need some properties of the Zhukowskii function z = (w+w-l)/2. This function maps the annulus 1 < IwI < fione-to-one and conformally onto E8 cut along the interval (-1,l) whereby the circle J w J= fi corresponds to aE8. The inverse function is given by w = z d n , where the principal value of the square root is to be chosen. We see by differentiation that the infinitesimal arc lengths are connected by the relation ldzl = 1w2 l [ l d w [ / ( 2 [ ~ 1Note ~ ) . also that z 2 - 1 = (w2 - 1)2/(4w2). Applying the identity Tk(z(W)) = wk w - ~we get by the substitution z = (w w-’)/2:
+
+
e(m-k)/2 +
27r
+
J-iu(k+m)dq
= + 0-k,
0
k#m, k = # 0, k=m=O.
Combined with (29) this proves the 1emma.O Denote G 0A -- Jx-1/2?A-1/2 0 . We are ready to formulate our main result for the Gegenbauer expansions.
Theorem 3 Let X 2 0. The space 0,” is formed by all restrictions of the elements of AL2(aEe) to the interval (-1,l). The norms in and AL2(dE0) are equivalent.
Qt
Proof The proof will be divided in three steps. Step 1. For the space HcK; induced by the kernel (27) with X to prove that HGK: = AL2 (aE8).
=0
we want
(30)
416
Dmitrii Karp
The weight 1z2 - 11-lI2 is positive and continuous on aEe so the norms in ALz(aE0; ( z 2- 11-4) and AL2(aEe) are equivalent and these spaces coincide elementwise. According to Lemma 1 and formula (9) the space AL2(8Ee;1z2It-$) admits the reproducing kernel given by
This kernel is equivalent to the kernel GK: due to (29) and inequalities
satisfied for any choice of n E N,ci E C and zi E Ee. Hence by Proposition 5 our claim is proved. Step 2. Consider the following auxiliary kernel: 1) 2Fl 02 - 20zE 1 7r-yo2 -
@(z,E) =
+
X+1 X+2
; +
1 402(1 - z 2 ) ( 1 - E 2 ) 5; (02 - 20zE 1)2
It is positive definite as will be shown below. Substitution X identity @(z,-iZ) = GK;(z,E). We want to prove that for all A, p > -f
H q = Hk;.
+
=
(31) 0 yields'the (32)
(33)
According to Proposition 5 we need to show that Kf x KE.Following the definition of the positive definite kernel (5), choose n E N,a finite complex , . Positive definiteness of the kernel sequence ci and points zi E Ee, i = 1 [402(1- z2)(1 - E 2 ) ] ' / ( 0 2 - 20zE+ 1)2k+1due to its reproducing property in the Hilbert space of functions representable in the form f(z) = (1 - z2)'g(z), where g belongs to the Bergman-Selberg space generated by the kernel (36) with X = 2k 1, and interchange of the order of summations justified by absolute convergence, lead to the estimates
+
Square Summability for Classical Expansions
417
where
r(x+ 1/2)T((X + 1)/2 + k ) T ( ( X + 2)/2 + k) > o. r((x+ 1)/2)T((X + 2)/2)F(X + 1/2 + k)k! This shows the positive definiteness of the kernel I?;. Using the asymptotic -
relation [3]
we obtain lim k-mc
4 uk
=1
+ 0 < sup kENo
The estimate from below is obtained in the same fashion with sup { u i / u ; } kENo
replaced by inf { a i / a ; } . This proves equality (33). Combined with (32) and kENo
(30) this gives:
H k ; = ALZ(aE8)
(35)
for all X > -1/2. Step 3. According to (27) and (31), the kernel GK; is related to the kernel by GKf(2,E)= B ; ( z , E ) I ? f ( z , E ) ,
I?:
where
(36) For X > 0 the function B ; ( z , E ) is the reproducing kernel of the BergmanSelberg space HB; [18]. This space comprises functions holomorphic in the disk ( z (< [(02 1 ) / 2 ~ 9 ] ~and / ~ having finite norms
+
where f k is the k-th Taylor coefficient of f. The functions
constitute a complete orthonormal system in H q . Note further that the closed ellipse is contained in the disk IzI < [(02 1)/20]1/2 due to the inequality
+
418 J(O2
Dmitrii Karp
+ l)/(20) > (04 + 0-+)/2, the right-hand side of which is equal to the
large semi-axis of the ellipse Ee. As stated in Proposition 6 the space HGKi is obtained by restricting the elements of the tensor product HB; 8 HR; to the diagonal of Ee x Ee and comprises the functions of the form
By (35) we can put ALZ(dE0) instead of HA; here. For any g E AL:!(dEe) consider the estimate
IISTkIIAL2(aEe) 2 = /Ig(')Yk(')121d'I
5 zF$'e IYk(')1211SII~L2(aEe)' (37)
aEe which shows that every product gyk belongs to ALz(aEe>and hence so does a finite sum of such products. Denote
M
M
M
M
Square Summability for Classical Expansions
419
ck
Since both (/gk(IiLa(aEB) and c k [ a ! ( k ) ] 2converge, the above estimates prove that the sequence Sn is Cauchy. It follows that HcK; c AL2(aEg). Inverse inclusion AL2(aEo)cHGK:is obvious, since I ( z ) = 1 belongs to HB; and so for any g ~ A L 2 ( a E o )the , product Ig = gEHGKo. 0 Now it is not difficult to establish our main result for Jacobi expansions.
-4.
Theorem 4 Let a , P 2 Inequality (1) with f k defined b y (4) holds true for all restrictions to the interval ( - 1 , l ) of the elements of AL:!(aEo) and only for them. Proof. Choose y > max(a,P}, then
JKz,P(z,z)<< MJK:,,(z,;iz) for some constant M > 0. Indeed, for an arbitrary n E ci and points zi E Ee, i = G,estimate using (24):
N,complex numbers
n
where
Interchange of the order of summations is justified by absolute convergence of the series (24). Application of formula (34) yields as k,1 -+ 00 aa'P A 0
-
((k+l ) a + P - 2 W - v - P )
=0 ((1
+ Z / k ) " - - f ( 1 + k/Z)P--f) = O(1).
k,l
Therefore sup {az,'//a:::} is positive and finite. Similarly by choosing -1 < k,lENo
q < min{a, P ) we can prove that
mJK;,, << JK&.
420
Dmitrii Karp
It remains to note that JK;,p = GKf, where X = ,L+ 1/2, and GKfis defined by (27). Now Theorem 3 gives the desired result. 0 When (Y and/or ,L belongs to (-1, -1/2), Step 3 of the proof of the Theorem 3 breaks down and the problem remains open. Corollary 4.1 Condition (1) f o r the Fourier-Jacobi coefficients (4) of a function f is satisfied for all 0 > 1 iff f is the restriction of a n entire function to the interval (-1,l).
The last theorem and Szego’s theory [19] suggest that the following much more general conjecture might be true. Conjecture Inequality (1) holds true f o r the Fourier coefficients in polynomials orthonormal o n (-1,l) with respect t o a weight w that satisfies Szego’s condition f 1 l n w ( z ) d z / d m > --oo i f and only i f f belongs t o AL2(889).
Acknowledgments The author thanks Professor Martin Muldoon and York University in Toronto for hospitality and support during the Fourth ISAAC Congress in August 2003 and Professor Saburou Saitoh of Gunma University in Kiryu, Japan, whose book [18] on reproducing kernels was the main inspiration for this research. This research has been supported by the Russian Science Support Foundation and the Far Eastern Branch of the Russian Academy of Science Grant 05111-r-01-046.
References 1. Aronszajn N. Theory of reproducing kernels, Trans. Amer. Math. SOC.68, 1950, pp.337-404. 2. Berezanskij Yu.M., Kondratiev Yu.G. Spectral methods in infinite-dimensional analysis. VoZ. 1, 2. Mathematical Physics and Applied Mathematics. 12. Dordrecht: Kluwer Academic Publishers. xvii, 1995. 3. Bateman Manuscript Project (edited by A. ErdClyi), Higher transcendental functions, Vol. 1-3, McGraw-Hill Book Company, Inc., New York, 1953. 4. Boyadjiev L. I. On series representation in Laguerre polynomials. C. R. Acad. Bulg. Sci., 45, no.4, 1992, pp.13-15. 5. Byun D-W. Inversions of Hermite Semigroup. Proc. Amer. Math SOC.118, no. 2, 1993, pp.437-445. 6. Gaier, D. Vorlesungen uber Approximation am Komplexen, Basel, Boston, Stuttgart: Birkhauser Verlag., 1980. 7. Gilbert R. P. Bergman’s integral operator method in generalized axially symmetric potential theory, J. Math. Phys. 5 , 1964, pp.983-997. 8. Gilbert R. P., Howard H. C. A generalization of a theorem of Nehari, Bull. Amer. Math. SOC.72, 1966, pp.37-39.
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9. Hille E. Contributions t o the theory of Hermitian series, Duke Math. J. 5,1939, pp.875-936. 10. Hille E. Contributions t o the theory of Hermitian series. II. T h e representation problem., Trans. Amer. Math. SOC.47, 1940, pp.80-94. 11. Hille E. Contributions t o the theory of Hermitians series 111: M e a n values., Inter. Jour. of Math. and Math. Sci. 3,no. 3, 1980, pp.407-421. 12. Janssen A. J. E. M., van Eijndhoven S. J. L. Spaces of type W, growth of Hennite coefficients, Wigner distribution and Bargmann transform, J. Math. Anal. Appl., 152,1990, pp.368-390. 13. Karp D.B. Square summability with geometric weight f o r Jacobi expansions, Far Eastern Journal of Mathematics, no.1, 2000, pp.16-27. 14. Karp D.B. Holomorphic spaces related to orthogonal polynomials and analytic continuation of functions, in: Analytic Extension Formulas and their Applications, Kluwer Academic Publishers, 2001, pp.169-188. 15. Karp D.B. Hypergeometric reproducing kernels and analytic continuation f r o m a half-line., Journal of Integral Transforms and Special Functions, 14, 2003, pp.485-498. 16. Nehari Z. O n the singularities of Legendre Expansions, J. Rational Mech. Anal. 5, no. 6, 1956, pp.987-992. 17. Rusev P. Analytic functions and classical orthogonal polynomials, Sofia, 1984. 18. S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman Research Notes in Mathematics Series 369, Addison Wesley Longman, Harlow, 1997. 19. G. Szego, Orthogonal polynomials, AMS Colloquium Publications 23,8th printing, 1991. 20. Zayed A. I. O n Laguerre series expansions of entire functions. Indian Jour. of Pure and Appl. Math. 12,no. 11, 1981, pp.1319-1325.
The First Positive Zeros of Cylinder Functions and of Their Derivatives Lee Lorch* Department of Mathematics and Statistics York University 4700 Keele Street Toronto, Ontario M3J 1P3 Canada 1orchOmathstat.yorku.ca Summary. I f & is the first positive zero of the cylinder function Cu(x) or of Cu'(x), it is shown here that either (i) &, > u for all u > 0 or that (ii) there exists p > 0 such that & > v,0 < u < p, while tP= p and tU < v,p < u < oa.This contrasts with
the well-known result that the positive zeros of higher rank all exceed IvI.Thus, the asymptotics (V -+ m) for cul and cL1 each divide into precisely two cases. Oscillation about u cannot occur. In (i) cul < u (respectively cL1 < u) for all large u,where R. Spigler [5] has found asymptotic representations. For (ii) where cul > u (respectively, cL1 > u),F.W.J.Olver [3] has determined the complete asymptotic expansions. His work covers also cuk and CLk, k = 2,3, . . . , since the quantities involved all exceed U.
1 Introduction and Statement of Results When the first positive zero cvl of the cylinder function Cv(x) (or cL1 of its derivative) remains less than v for all large v, R. Spigler [5] has determined the first few terms of the respective asymptotic expansions as v -+ 00. When the zero exceeds v, F.W.J.Olver [3] has found each complete asymptotic expansion, u + 00. There would appear to remain third cases in which cvl or cL1 oscillate infinitely often about v. However, these cases cannot occur, as Theorems 1 and 2 (alternatively, Theorems 1A and 2A) below, show. The subsequent respective zeros all exceed v where only Olver's results are relevant. The first positive zero of a cylinder function Cv(x) or of its derivative CL(x)has a distinctly less uniform behaviour than the zeros of higher rank. As customary, *This research was supported partially by the Natural Sciences and Engineering Research Council of Canada.
424
Lee Lorch
Cv(x,9)= C,(x)
= Jv(x)cos9 - Y,(x)sin9,
where the constant 9 satisfies 0 5 9 < 7r, and Jv(x),Yv(x)are the Bessel functions of first and second kind respectively. The k-th positive zero of Cv(x) is designated by C,k, of CL(x) by cLk, k = 1 , 2 , .... It is well-known, and follows readily from Bessel’s differential equation, x2y//
+ xy’ + (2- v2)y = 0 ,
(1)
satisfied by y = C,(x), that c,k > IvI and cLk > IuI, for k = 2 , 3 . . ., for any cylinder function, i.e., for any 9. For k = 1, the situation is more complicated. Clearly, it is always possible to find a non-trivial solution of ( l ) ,i.e., a value of 9, such that c,1 < v , for each fixed v > 0, and another non-trivial solution such that cL1 < v for each fixed v > 0. On the other hand, for C,(x)= J,(z), i.e., 9 = 0, or C v ( x )= Y,(z), i.e., 9 = 7r/2, cvl > v and cL1 > v [6, p. 485 (l)]. Indeed, M.E. Muldoon and R. Spigler [4]have shown more broadly that c,1 > v when 0 5 9 5 57r/6 and that cL1 > v,7r/6 5 9 5 7r. They observe that this point is implicit in Olver’s work [31. Regardless of 9, it has been established [I] that
dcvk
lim --1,k=1,2 dv
y-+m
,....
Here the following results will be established:
Theorem 1 If c,l is the first positive zero of C,(z), then either (i) c,1 > v, all v > 0 , or (ii) there exists p > 0 such that cpl = p, while c,1 > v,O < v < p, and C”1 < u , p < v < 00. Theorem 2 If cL1 is the first positive zero of C,’(z), then either (i) cL1 > v, all v > 0, or (ii) there exists X > 0 such that cil = A, while cL1 > u,O < u < A, and CL1 < v , x < v < 00. These results make clear that the asymptotic expressions due to F.W.J. Olver [3] and R. Spigler [5] cover between them the only possible cases for C v k and C L k , k = 1 , 2 , . . .. This is an issue only for k = l , as noted in the discussion of (1).For k=2, 3, . . . only the Olver results are relevant.
Zeros of Cylinder Functions
425
2 Proof of Theorem 1 We have already noted that case (i) of the conclusion can occur. Now we address the case in which there exists a > 0 such that cal 5 a. Here a lemma is helpful: Lemma 1 If cal 5 a, then v - c,1 increases in a 5 v < 00 so that c , ~< v for a < v < 00. Proof of Lemma. Abbreviating c,1 to c, we appeal to Watson’s formula [6, p. 508(3)1,
dc
= 2c dv
00
Ko(2~sinht)e-~,~dt
(3)
and to [2, Lemma] to obtain
We have used also the fact that KO(.)is a decreasing function of the argument, along with the evaluation of the last integral [6, p. 3881.
Thus, v - c,1 increases wherever cyl 5 u. This proves the lemma. Thus, once c,1 becomes less than or equal to v, it stays less for all larger v. Let us suppose, per impossibile, that c,1 5 v for all 0
< v < 00. Then
0 = C,(cyl) = J,(cul) cos B - Y,(c,l) sin 0. Letting u 1 0, it follows that
O=Co(0) = Jo(0)cosB-Yo(O)sinB, which is clearly impossible for 9 since Jo(0) = 1. In other words, there exists /3
# 0 , ~It.
is equally impossible for B = O,T,
> 0 such that cp1 > p.
Now, v - c,1 is a continuous function of v,O < v < 00, which is negative for v = /3 > 0 and positive or zero for v = a > 0. Further, a > p, since v - c,1 > 0, v > a. Accordingly, there exists p , p < p 5 a, such that v - c,1 = 0 when v = p and changes sign there. This completes the proof of Theorem 1. Remark. Combining Spigler’s asymptotics for c,1 [2, p.761 with the lemma, it may be said that v - c,l T 00,p < v < 00.
426
Lee Lorch
3 Proof of Theorem 2 This proof follows the pattern of that of Theorem 1 and hence will be abbreviated. As there, it suffices to consider case (ii), i.e., where there exists > 0 such that cbl 5 p. This leads to the use of Watson's formula [6, p. 5101 for dc'ldu, where c' = cLl: dc' O0 /'cosh2t - u2 -= 2 c ' l KO(2 c ' ~ i n h t ) e - ~ ~ ~ d t . c" - y2 du The analogue of Lemma 1 is
(4)
Lemma 2 If cbl < p, then u - cLl increases in /3 < v < 00, so that cL1 < u,p < u < 00. Proof of Lemma. We write,
dc'
(cosh2t - 1 ) i 2 ]K0(2c'sinht)e-~~~dt
O0
< 2c'
1-
K0(2c'sinht)e-~~~dt,
since c' < p. This last quantity is less than one, as argued in Lemma 1. This lemma then follows. The proof of Theorem 2 concludes in the same fashion as did that of Theorem 1.
4 Another Approach A practical construction of a cylinder function Cu(x) for which cul < v for v > p > 0 is suggested by Theorem l(ii). Simply fix p > 0 and find 8 such that cpl = p. That is, find 8 for which Cp(p)= 0, so that
For instance, with p = 1, it turns out that 8 = 2.628602096, which conforms with the Muldoon-Spigler result requiring 6 to exceed 5 ~ 1 6= 2.617993878. Accordingly, we have
Cu(~ =) -.8712806850 JU(x)- .4907850527Yv(.). A similar discussion applies to CL(x). Keeping in mind that cul cannot be less than u for all v > 0, as shown in the course of the proof of Theorem 1, we may reformulate that theorem as
Zeros of Cylinder Functions
427
follows.
Theorem 1A If cpl = p for a fixed p > 0, then c,1 < v for all p < v < m. Proof. From C,(p) = 0, it follows that
thereby fixing the value of 8 needed to define Cv(x). As Muldoon and Spigler have shown [4],8 is restricted to 5n/6 < 8 < 7r so that sin 8 > 0, cos 6’ < 0. Further [4],Jv(v)/Yv(v)is a decreasing function of v. Hence, with v > p , J,,(v)
---
sin8 coso
< 0, Yv(v) whence C,(v)= J,(v) cos6’ - Y,(v)sin8 < 0, v > p. For sufficiently small v > 0, C,(v)> 0, since sin 8 > 0. Hence, C,(x)vanishes for a positive x less than v , i.e., cvl < v , for v > p , as claimed. Correspondingly, Theorem 2 can be reformulated.
Theorem 2A If cL1 = p, for a fixed p > 0 , then cL1 < v for all p < v < 00. Proof. As above, we find that
fixing the value of 6’ at stake here. This value is restricted to 0 5 B Muldoon and Spigler have established [4].Here, therefore, sin 8 > 0, cos 8 > 0.
< ~ / 6 as ,
Moreover [4],J,’(v)/Y,‘(v) is an increasing function of v. Thus, for v > p ,
and so Cv’(v) = J,l(v) cos 8 - Y,’(v) sin 8 > 0, v > p. But Cv’(x) < 0 for sufficiently small IC. Thus, cL1 < v for v > p , as claimed.
5 A Contrast There is at least one issue on which c,1 and cL1 exhibit different behaviours. As formula (3) for dc/dv makes clear, c,1 is an increasing function of v. The
428
Lee Lorch
more complicated formula (4) for dc’ldv reveals that cL1 can actually decrease for some v. Taking cL1(< v) close to v > 0 produces a negative value for dc’/dv.
As a concrete numerical example, consider the cylinder function C,,((c) which is a constant multiple of
Clearly, cil = 1. Maple 6 furnishes the other values in the following table. According to it, cL1 decreases from v = 1 until about v = 1.03, and then increases. In each case, cL1 < v in accordance with Theorems 2 and 2A. Further, v - cL1 increases with v in accordance with Lemma 2.
1.02 0.9702512562 1.03 0.9692189887 1.04 1.05 1.06 1.07 1.08 1.09
0.9699641692 0.9718457547 0.9745347714 0.9778356836 0.9816210259 0.9858022216
P
References [l] A. Elbert, L. Lorch and P. Szego, Bessel functions in a quantum-billiard configuration problem, Analysis and Appl. 1 (2003), pp. 421-428. [2] L. Lorch and P. Szego, Monotonicity of the differences of zeros of Bessel functions as a function of order, Proc. Amer. Math. SOC.15 (1964), pp. 91-94. [3] F. W. J. Olver, A further method f o r the evaluation of zeros of Bessel functions and some new asymptotic expansions f o r zeros of functions of large order, Proc. Cambridge Phil. SOC.47 (1951), pp. 692-712. [4]M. E. Muldoon and R. Spigler, Some remarks o n zeros of cylinder functions, SIAM J. Math. Anal., 15 (1984), pp. 1231-1233. [5] R. Spigler, Alcuni risultati sugli zeri delle funzioni cilindriche e delle lor0 derivate, Rend. Sem. Mat. Univers. Politec. Torino, 38 (1980), pp. 67-85. [6] G. N. Watson, A Treatise o n the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London 1944.
Bergman Kernel for Complex Harmonic Functions on Some Balls Keiko Fujita Faculty of Culture and Education Saga University Saga 840-8502 Japan keikoPcc.saga-u.ac.jp
Summary. We studied holomorphic functions and analytic functionals on an “NPball” in the complex Euclidean space Cn+l,n 2 2, in [2], [5], [7] and [8]. Then we expressed the Bergman kernel for a Hardy space on the N,-ball by a double series by using of homogeneous harmonic extended Legendre polynomials in [2]. We showed that our results on Cn+’,n 2 2, hold on C 2 and we calculated the coefficients of the double series expansion of their Bergman Kernel for the 2-dimensional N,-balls in [3]. In this paper, we survey our results obtained in [3] and [4].
Introduction We denote by O(X)the space of holomorphic functions on a bounded domain X in Cn+’ equipped with the topology of uniform convergence on compact sets. P u t
HOW =
{f
E
s,
Wf);
If(W)l2dVX(W)
where dVx(w) is the normalized Lebesgue measure on Let A be the complex Laplacian on Cn+’;
A = -a2 + - + .a2 .. - az; az,z
+-,
<
I
7
X.
a2
az:+1
and we introduce the following subspace of the Hardy space H O ( X ) ;
H O a ( X ) = {f E H O ( X ) ; A , f ( z ) = 0). When a function f satisfies A , f ( z ) = 0, we call f a complex harmonic function. In the following, we call the Bergman kernel on H O a ( X ) the “harmonic” Bergman kernel.
430
Keiko h j i t a
For the 1-dimensional Euclidean ball D ( r ) = { z E C; IzI < r } , the Bergman kernel B:(z,w) for H O ( D ( r ) )and the harmonic Bergman kernel B:,A(z, w) for HOA(D(r)) are given by
For n 2 1, in general, it is difficult to express the Bergman kernel in a concrete form like (1) even if their existence is well-known. Special domains such as the complex Euclidean ball l?;+l(r) and the Lie ball Bn+l(r)(for the definition, see the following section), the Bergman kernels Bzsl(z,?iI) for HO(B"+l(r)) and B,"+l(z,Z) for HO(B"+l(r)) are known to be ,,-2n+4
B;,:'(z,W
= (.2
- z . W)n+2
7
n 2 0,
r4n+4
B,"+l(Z,U) =
(7-4 - 27-22.
w
+ z 2w2 )n+l'
2 O,
+ +
where z . w = z l w l + z2w2 * - . zn+lwn+l,z2 = z z. For a proof of (2) for n 2 2, see [6], for example.
1 &-Ball 1.1 Lie Ball
1.1.1 Lie Norm For z = (z1,z2, - .- ,zn+l) E Cn+', n = 0,1,2, - - -, the Lie norm L ( z ) on Cn*' is given by L ( z ) = J.1l 12
where
+ mF-m
1 1 ~ 1= 1 ~ z - Z. If z E C, then L ( z ) = 121, and if z = (q, 22) E C2,then L ( z ) = max{ Iz1 fizgl}.
(3)
Thus L ( z ) is equivalent to the supremum norm 11 /loo in C2. Therefore, we can say that the Lie norm L ( z ) , n _> 2, is another generalization of the 2dimensional supremum norm to a higher dimensional space. 1.1.2 Lie Ball We denote by
BnS1(r)the Lie ball: Bn+l(r) = { z E
cn+l; L(z)< T } .
Bergman Kernel for Complex Harmonic Functions on Some Balls
Therefore, by changing the variable
2 1 = z1+ iz2
B2(r) ei D2(r) = ( 2 = (21,22) E
and
22
431
= z1- iz2, we have
c2;lZ,( < T , 1221 < r } .
Thus we can see the Bergman kernel B z ( z , w ) for HO(B2(r)) by using the result (1) in l-dimensional case: The Bergman kernel B:(2, W ) for HO(&(r)) is given by
m,
Since A, = 4 a2 we haveHOA(&(r)) = {a+CbkZf+CclcZ;}. Thus the Bergman kernel B;,,(Z, W ) for HOa(Dz(r)) is given by
r4
Therefore, the Bergman kerneI B,"(z,w) and the harmonic Bergman kernel B:,,(z, w) are given by
B,2(z,w) = and
r8 (r4 - 2r2z
*
;i7T
+ 22G2)2 '
- 4r2z * + 2 2 5 2 ) (7-4- 2r22 * Ti? + z2G2)2
r8 - z252(4r4 B ? , A ( z , w, =
1.2 N,-Ball 1.2.1 N,-Norm
For p E R, consider the function
If p 2 1, then N p ( z )is a norm on Cn+l (see [l]or [9]). Note that N ( Z ) = d(llzl12 + lz211>/2, N2(4 = 1141*
(4)
432
Keiko Fujita
we call the N1-norm the dual Lie norm. In C 2 ,by (3), we have
NP(4 =
(
12.1
+ iz2y +
121 - 2 2 2 ( P
2
)
l',.
Thus the N,-norm is equivalent to the L,-norm I( ( I p in C 2 and we can say that the Np-norm ( n 2 2) is another generalization of the 2-dimensional L,norm to a higher dimensional space. In C , as N p ( z )= 121 for all p E R, there is nothing to say. Thus, in the sequel, we assume that n 2 1. Further, we consider N,(z) as L ( z ) because L ( z ) = lim N p ( z ) . p-00
1.2.2 The Tchebycheff Polynomial The Tchebycheff polynomial Tk(x) of degree k is defined by
We define the homogeneous extended Tchebycheff polynomial of degree k in Cn+l by
Then we extend the parameter k to a E R, and consider the function
Note that Fa:,,(z,w) is not a well-defined one-valued function but Ta:,,(z,F) is a well-defined one-valued function. Z )have )~~), Since fa:,,(z,Z) = ( L ( Z ) ~( ~~ Z ~ ~ / L ( we
+
N&)
= (Fp/2,,(z,F))+ , z E
cn+l.
Therefore, for p 2 1 the Np-norm is slightly related with the Tchebycheff polynomial.
Bergman Kernel for Complex Harmonic Functions on Some Balls
433
1.2.3 N,-Ball
Now, we define the N,-ball
B,"+'(T) by
B,n+l(r)= { % E c n + ' ; N p ( z )< r } , p _> 1. By (4),&+'(r) is the dual Lie ball and B;+'(T) is the complex Euclidean ball. We consider B&+'(r) as the Lie ball B"+'(r) because
n
~;+1(.)
= {.
c~+I;
P21
In case of the 1-dimensional Euclidean space, every N,-ball of radius T being equal to the disk D(r),the Bergman kernel and the harmonic Bergman kernel on the Np-ball in C are given in (1).
2 Bergman Kernel 2.1 Double Series Expansion
First we recall the definition of spherical harmonic functions. 2.1.1 Spherical Harmonic Functions
Let Pk,n(t)be the orthogonal polynomial of degree k whose highest coefficient is positive and determined by
where N ( k , n) is the dimension of the space of homogeneous harmonic polynomials of degree k in Cn+' : N ( 0 ,n) = 1, N ( k , n ) = (2k
+ n - l ) ( k + n - 2)!/(k!(n - I)!),
k = 1,2,.
a
.
For a reference, see [lo], for example. We call Pk,n(t)the Legendre polynomial of degree k and of dimension n + 1. We define the homogeneous harmonic extended Legendre polynomial &,n(z, w) of degree k and of dimension n 1
+
hv
Note that & , n ( ~ , w )= &,(w,z) and A,&,,(z,w) &(z, w)= T~,J(z, w) and we denote it by p k ( z , w ) .
= 0. When n = 1,
Keiko Fujita
434
2.1.2 Double Series Expansion of B,"+'(r, w)
It is well-known that the Bergman kernel B,"+l(z,w) is expanded into the double series: FORMULA 1
where 2r(Z aE,k-21
=
+ y ) r ( k + n - 1 + 1 ) ~ ( -k 21, n ) ( n+ l)!Z!T(k+ - 1)
2.2 Bergman Kernel for HO(B;+'(T))
For the Bergman kernel on HS(B;+'(r)), we proved the following theorem for n 2 2 in [2] and for n = 1 in [3]: Theorem 1 The Bergman kernel B;:'(z, w) f o r HC)(B,"+l(r)) is given by
k=O l=O
where
and dVp,T(<)denotes the normalized Lebesgue measure o n BF+'(r) and Sn is the real unit sphere in Rn+l.
Further, in the 2-dimensional case, we calculated the coefficients ,!?;fT and Theorem 1 is restated as follows:
in [3],
Theorem 2 The Bergman kernel B&(Z,w) f o r HC)(B;(r)) is given by
w21N ( k - 2 l , l ) r ( y r ( y + 1 ) ( 2 2 ) 1 ( w 2 ) 1 p k - 2 1 ( 2 , ~ ) = k=O 1=0 r(;+W(2 k - 2p 1 + 2 ) r ( Z 3 ) 2 F r 2 k
cc O0
B;,r(z,w)
where XI=
2
. j j j + i.J2.2w2
x2=
- ( 2 . w>2,
1 Note that we have pk,n(z,E)= - (Xf 2
x, = 2 2 . w, X'XZ
x1+
2 .
- i 4 2 2-2 w - (z . w -) 2 .
(5)
+ Xi) and
= z2w2, (1 - X1)(1- X2) = 1 - 22.
w +Z2E2.
Bergman Kernel for Complex Harmonic Functions on Some Balls
435
3 Harmonic Bergman Kernel 3.1 Harmonic Bergman Kernel for H O ~ ( f i ; f l ( ~ ) )
By the orthogonality of rem:
{Fk,n}k
and Theorem 1, we have the following theo-
Theorem 3 Let n = 1 , 2 , . . .. The harmonic Bergman kernel B,":: HOa(B;+l(r)) is given by ,
(2,w )
for
I
k=O
Similarly, by Theorem 2, we have the following theorem:
Theorem 4 The harmonic Bergman kernel BE , ,A ( z ,w ) f o r HOn(BE(r))is given as follows:
where
X1
and X 2 are given by (5) and F p ( X ) is the function defined by
If we find the concrete form of (6), then we will obtain the harmonic Bergman kernel for B : , T , d (w~), in a concrete form. 3.2 Examples for the Harmonic Bergman Kernel Example 1. (The Lie ball (see Section 1.1.2))
k=O
436
Keiko Fujita
Example 2. (The Euclidean ball) O0
k=O
(k + 2)!N(k, 1) 2kk!2r2k
Tk(z,W)
F2($)+F2(5)-1
-
=2
-
-2
W where Qz,~(-&.$$-, s F is) the polynomialin z.Z A T A T E C and C of degree 3 determined by
F aE z2
-2
Q z , ~ ( s ,= ~ )1 - 9t + 18ts - 3t2 - 12ts2 + 6t2s - t3. Example 3. (The dual Lie ball) 00
k=O
$s)
g,
where &1,.(5 is the polynomial in of degree 4 determined by
t) = 1
Q ~ , T ( s ,
+ 2s - 24t + 60st + 4t2 +18st2 - 80s2t - 4t3
5 .$ E C and p$ =2
+ 48st3 - 24s2t2+ 40s3t
-2
-
E C
t4.
References 1. M. Baran, Conjugate norms in C" and related geometrical problems, Disserta-
tiones Mathematicae, 377 (1998), 1-67. 2. K . Fujita, Bergman transformation for analytic functionals on some balls, Microlocal Analysis and Complex Fourier Analysis, World Scientific publisher, 2002, 81-98.
Bergman Kernel for Complex Harmonic Functions on Some Balls
437
3. K. Fujita, Bergman kernel for the two-dimensional balls, Complex Variables: Theory and Applications, 49 (2004), 215-225. 4. K. Fujita, Harmonic Bergman kernel for some balls, Universitatis Iagellonicae Acta Mathernatica, 41 (2003), 225-234. 5. K. Fujita and M. Morimoto, On the double series expansion of holomorphic functions, J. Math. Anal. Appl., 272 (2002), 335-348. 6. L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in Classical Domain, Moskow 1959, (in Russian); Translations of Math. Monographs vol. 6, Amer. Math. SOC.,Providence, Rhode Island, 1979. 7. M. Morimoto and K. Fujita, Analytic functions and analytic functionals on some balls, Proceedings of the Third ISAAC Congress, Kluwer Academic Publishers, 2003, 150-159. 8. M. Morimoto and K. Fujita, Holomorphic functions on the Lie ball and related topics, Finite or Infinite Dimensional Complex Analysis and Applications, Kluwer Academic Publisher 2004, 35-44 9. M. Morimoto and K. Fujita, Between Lie norm and dual Lie norm, Tokyo J. Math., 24 (2001), 499-507. 10. C. Muller, Spherical Harmonics, Lecture Note in Math., 17 (1966), Springer.
Applications of Reproducing Kernels to Best Approximations, Tikhonov Regularizations and Inverse Problems S. Saitoh Department of Mathematics Faculty of Engineering Gunma University Kiryu 376-8515 Japan. ssaitoh0math.sci.gunma-u.ac.jp
Summary. We shall show fundamental applications of the theory of reproducing kernels to the Tikhonov regularization that is powerful in best approximation problems in numerical analysis.
1 Introduction In the 2001 ISAAC Berlin Congress, the author [4]gave a plenary lecture in which he showed that the theory of reproducing kernels is fundamental, beautiful and applicable widely in analysis. After that, the author found fundamental applications of the theory t o the Tikhonov regularization that is powerful in best approximation problems in numerical analysis. In this survey article, we shall present their essence, simply. First, we recall a fundamental theorem for the best approximation by the functions in a reproducing kernel Hilbert space (RKHS) based on [1,3]. Let E be an arbitrary set, and let HK be a RKHS admitting the reproducing kernel K ( p , q ) on E . For any Hilbert space 3-t we first consider a bounded linear operator L from HK into 3-t. Then, we shall consider the best approximate problem for a member d of ‘FI. Then, we have Proposition 1.1 such that
For a member d of Z, there exists a function
inf llLf - dll7-l= llG- dII7-l f €HK i f and only if, f o r the RKHS H k defined by
j
in HK (1.2)
440
S. Saitoh
k ( p , q ) = (L*LK(.,q),L*LK(.,p))HK,
(1.3)
L*d E H k .
(1.4) Furthermore, i f the existence of the best approximation f satisfying (1.2) is ensured, then there exists a unique extremal function f* with the minimum norm in H K , and the function f * is expressible in the f o r m fcT(P) = (L*d,L*LK(.,P))H, on E .
(1.5)
In Proposition 1.1, note that
( L * d ) ( P )= ( L * d , K ( . , P ) ) H K= (d,L K ( . , p ) ) w ;
(1.6)
that is, L*d is expressible in terms of the known d, L , K ( p ,q ) and 'FI. In Proposition 1.1,even when L*d does not belong to H k , the function fd**(P)= (d, LL*LK(.,p))w
(1.7)
is still well defined and the function is the extremal function in the best approximate problem inf ( ( L * L f- L*d((H,, f€ H K
as we see from Proposition 1.1, directly. Let P be the projection map of 7-l to R ( L ) (closure). Then, there exists f in HK satisfying (1.2) if and only if Pd E R ( L ) . This condition is equivalent to d = Pd ( I - P ) d E R ( L ) R(L)'.
+
+
Further, this condition is equivalent to
Lf - d E R(L)'
= N(L*)
for some f E H K ; that is, for some f E H K ,
L*Lf = L*d. f : in (1.5) is the Moore-Penrose generalized inverse of the equation
Lf = d . In particular, if the Moore-Penrose generalized inverse fi exists, then it coincides with f:* in (1.7). Proposition 1.1 is rigid and is not practical in practical applications, because, practical data contain noises or errors and the criteria (1.4) is not suitable. Meanwhile, the representation (1.7) is convenient in these senses. However, the function f:*(p) is, in general, not suitable for the problem (1.1). Indeed, we shall give an estimate of IILfA* - d117-I.We shall show a good relationship between the Tikhonov regularization and the theory of reproducing kernels. For the Tikhonov regularization, see, for example, [2].
Applications of Reproducing Kernels
441
2 Tikhonov Regularization We shall introduce the Tikhonov regularization in the framework of the theory of reproducing kernels based on ([l];[3], pp. 50-53). However, from the viewpoint of Tikhonov regularization we shall give a further result constructing the associated reproducing kernels and a new viewpoint for the previous results. Let L be a bounded linear operator from a reproducing kernel Hilbert space H K admitting a reproducing kernel K ( p , q ) on a set E into a Hilbert space 7-L Then, by introducing the inner product, for any fixed positive A > 0
(f,S ) H K ( L ; X ) = A(f, S ) H K + (Lf,~S).HH,
(2.9)
we shall construct the Hilbert space H K ( L ; A )comprising functions of H K . This space, of course, admits a reproducing kernel and we shall denote it by K L ( ~q;,A). Then, we first have the elementary properties:
Lemma 2.1 The reproducing kernel KL(P,~;X) is determined as the unique solution K ( p , q; A) of the equation:
Note here, in general, that the norm of the RKHS HAK admitting the reproducing kernel AK(p,Q ) (A > 0) is given by (2.12)
and the members of functions of H X K are the same of those of H K . We shall consider that the reproducing kernel K ( p , q ) is known and we wish to construct the reproducing kernel KL(P,q; A). For this construction we can obtain a very effective method by using the Neumann series. We define the bounded linear operator 2 from H K into H K defined by ( W ( P ) = (Lf, LKp).H = (L*Lf)(P).
Then, from (2.10) we obtain directly Theorem 2.2 If I(L(I< A, then K ~ ( p , qA); is expressible in t e r n s o f K ( p , q ) by the Neumann series:
442
S. Saitoh
where ( I
+ $)-'
is a bounded linear operator from HK into H K satisfying l
<1
I I X I I1+5;
- .
1-IIiII
Of course, if the operator is compact, then we can apply spectral theory to the equation (2.10) without the restriction IlLll < A. In particular, if -1 is not an eigenvalue of the operator then ( I is a bounded linear operator and
i,
Furthermore, when -1 is an eigenvalue of result. See, for example, [2].
+ i)-l
2, we can obtain a further related
We shall consider the best approximation problem, for any given fo E H K and d E 'FI: inf { A l l f o - fll&
f€HK
+ Ild - Lf Ilah
(2.14)
in connection with the Tikhonov regularization for the equation L f = f , Then, we can obtain, from Proposition 1.1:
Theorem 2.3 In our situation, f o r any given fo E HK and d E 'FI, the generalized solution f * of the equations fo = f
in HK
d=Lf
in 'FI
and
in the sense
exists uniquely and it is represented by
Applications of Reproducing Kernels
443
In Theorem 2.3, in particular, we shall consider the best approximate function, for fo = 0 fX,d(P) =
(d,LKL(*,P ; W H ,
(2.17)
which is the extremal function in the Tikhonov regularization (2.15) for fo = 0. In general, in the Tikhonov regularization, the operator L is compact and the extremal functions are represented by using the singular values and singular functions of the selfadjoint operator L*L. So, the representations are, in a sense, abstract. And the behaviour of the extremal functions as X tends to zero is an important problem, because the limit function may be expected as a solution of the equation Lf = f as in the Moore-Penrose generalized inverse. From many examples in our situation ([5,6,7]), however we see that
and lim (d,LKL(P,4; X>>,
A 4 0
(2.19)
do not exist, in general.
3 Main Results We now give our main results in this paper: Theorem 3.1 For the two best approximate functions f i , d ( P ) in (2.17) and f:*(p) in (1.7) we have the estimate
Corollary 3.2 If LL* is unitary, then we have for the two best approximate functions f i , d ( p ) in (2.17) and f:*(p) in (1.7) we have the estimate (3.21) which shows that as X tends to zero, f i , d ( p ) tends t o f;*(p) with the order X and the convergence is uniform on any subset of E satisfying K ( p , p ) < 00.
For the best approximate function f i * ( p ) when there exists, we have fd**(P) = (L*d,L*LK(.,P ) ) H K = (L*LL*d)(p).
(3.22)
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S. Saitoh
For the image of f:*(p), we thus obtain the estimate
IILf:*
- dllx
i IILL*LL* - Ill Ildll7-l.
(3.23)
The quantity IILL*LL* -Ill may be understood as that of unitariness of the operator LL*.
Theorem 3.3 If L is a compact operator, then for the Moore-Penrose generalized inverse f:, lim f X , d ( P ) = fd*(P), (3.24) X-+O
uniformly on any subset of E satisfying K ( p , p ) < m.
Proof: Since L is compact, we have, from (2.10)
Then,
As we see by using the singular value decomposition of L , for the MoorePenrose generalized inverse f:, as X -+ 0, 1 L * d + f:, XI L*L
+
(see Section 5.1 in
in
HK
[a]). Hence, from the identity
we have the desired result.
Corollary 3.4 If g E N(L)', then (3.25)
uniformly on any subset of E satisfying K ( p , p ) < 00. Meanwhile,
Applications of Reproducing Kernels
445
Corollary 3.5 Ifd E 'FI belongs t o R ( H K ) , then lim LfI,d(p) = d
Ah0
in
7-L.
(3.26)
For several concrete applications of our general theorems, see the forthcoming papers [5,6,7].
References 1. D.-W. Byun and S. Saitoh, Best approximation in reproducing kernel Hilbert spaces, Proc. of the 2nd International Colloquium on Numerical Analysis, VSP-Holland (1994), 55-61. 2. C . W. Groetsch, Inverse Problems an the Mathematical Sciences, Vieweg & Sohn Verlagsgesellschaft GmbH, Braunschweig/Wiesbaden (1993). 3. S. Saitoh, Integral Transforms, Reproducing Kernels and their Applications, Pitman Res. Notes in Math. Series 369, Addison Wesley Longman Ltd (1997),
UK. 4. S. Saitoh, Theory of reproducing kernels, Analysis and Applications - I S A A C ZOO1 (H.G.W. Begehr, R.P. Gilbert and M.N. Wong (eds.)), Kluwer (2003). 5 . S. Saitoh, Approximate Real Inversion Formulas of the Gaussian Convolution, Applicable Analysis (to appear). 6. S. Saitoh, T. Matsuura and M. Asaduzzaman, Operator Equations and Best Approximation Problems in Reproducing Kernel Hilbert Spaces, J. of Analysis and Applications (to appear). 7. S. Saitoh, Constructions by Reproducing Kernels of Approximate Solutions for Linear Diflerential Equations with Lp Integrable Coefficients, International J. of Math. Sci. (to appear).
Equality Conditions for General Norm Inequalities in Reproducing Kernel Hilbert Spaces Akira Yamada Department of Mathematics Tokyo Gakugei University Nukuikita, Koganei, Tokyo Japan yamadaQu-gakugei.ac .j p
Summary. Let H K ( E ) be the reproducing kernel Hilbert space on E admitting a reproducing kernel K . Let $ ( z ) be an entire function with nonnegative coefficients. Then we have the following general norm inequality: for all f E H K ( E )
llW)ll&)5 4(llflla
(IK
where ( 1 . ( ( + ( K ) and ( 1 . denote the norms of the RKHSs H + ( K ) ( Eand ) HK(E), respectively. We shall give a fairly general condition for equality of this inequality.
1 Introduction A positive matrix on a set E in the sense of E. H. Moore is a function f (x,y) on E x E such that for any finite number of points {xi} in E and for any complex numbers { ci}, ~ c i c j f ( s i , x2j )0. i,j
We shall denote a positive matrix f by f >> 0. The reproducing kernel of a reproducing kernel Hilbert space (RKHS) is a positive matrix. Conversely, any positive matrix K ( x ,y) on E uniquely determines a RKHS H K ( E )admitting the reproducing kernel K(z,y) o n E . We shall denote by 11 . I I K the norm of H K ( E ) .The reader is referred to [l,151 for the general theory of reproducing kernel Hilbert spaces. If A: H I -+ Hz is a linear mapping from a Hilbert space H1 into a linear space H2 with closed kernel kerA c HI, then the range A(H1) is linearly isomorphic with the Hilbert space H1 8 ker A . Using this isomorphism we may introduce an inner product of A(H1) such that A(H1) becomes a Hilbert space isometric with H I 8 ker A. The Hilbert space A(H1)
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Akira Yamada
with this inner product is called the operator range of the mapping A. If K l ( z ,y) >> 0 and K2(z,y) >> 0, then we see that the sum K l ( z ,y) Kz(rt,y) and the product K1 (z,y)Kz(z,y) are positive matrices on E. The correspond) are closely related to the orthogonal ing RKHSs H K ~ + (KE~) and H K ~ (KE ~ sum H K ~@ H K ~and the Hilbert tensor product H K ~ 8 H K ~respectively. , In fact, the RKHS H K ~ + K ~ (isEgiven ) by the operator range of the mapping H K ~ C B H3Kf ~~ ~ g ~ f + g ~ H ~ ~ + ~ ~ ( E ) . H e n c e
+
Ilf + g l l k + K 2 5 llfll&, + 11g11a,-
(1)
Also, the RKHS H K ~ K ~ (isEgiven ) by the operator range of the mapping H K ~8 H K ~3 fi 8 gi H figi E H K ~ (KE )~.Hence
xi
IIfgllKrK2
xi
5 Ilf 8 SIlKl@K2 = IlfllKl IlgIlKz.
(2)
As a special case we note that IlfllcK
= llfllK/lh
xr=o
(3)
for any positive constant c. Let $ ( z ) = c,z’ be an entire function with nonnegative coefficients {c,}. Then $ ( K ) >> 0, and from ( l ) , (2) and (3) we have the following general n o r m inequality [2, 161: for all f E H K ( E )
IlW)Il$(K, 5 Icl(llfll?d.
(4)
In fact, S. Saitoh discussed in [16] a more general case than ours, where the coefficients {c,} are not constants but complex-valued functions. It is easy to see [2] that iff = Kq(= K(.,q ) ) for some q E E , then equality holds in (4).Although we have an example [12] that the converse does not hold in general, by examining various special but important RKHSs, there are many papers [2, 3, 4, 5 , 8, 9, 10, 11, 13, 141 proving that equality occurs in (4)if and only if f = Kq for some q E E . To solve the equality problem which determines the functions attaining the equality in (4),these papers required case by case arguments. In his book [15, p. 181 Saitoh raised the above equality problem as the first one expecting more general and satisfactory theory and he commented that it will be important. To answer this problem, we shall define a class of RKHSs and present a general theory concerning conditions for equality in the general norm inequalities. Our results will yield a unified treatment for the equality problems treated in the papers cited above. This is an expository article of our work in progress and the proofs will appear elsewhere. Finally, the author would like to thank Professor S. Saitoh very much for his encouragement and valuable comments on this paper.
2 Main Results Let rn be a fixed integer greater than or equal to two. Let Hj ( j = 1 , 2 , . . . ,rn) be a complex RKHS on E with a reproducing kernel KL’) at z E E. Let H
Equality Conditions for General Norm Inequalities
449
nj"=,
be the Hilbert tensor product @ F I H j which is a RKHS on Em = E and denote by HO its subspace { f E H I f l ~=r0 ) where ET is the diagonal {(x,x,. . . ,x) 1 x E E } of the set Em. For f and g E H define an equivalence relation by f g if and only if f 1~~ = g(E,"-.An element E H is called extremal if @g14j E H;. We remark that @F14j is extremal if and only if the identity (f,@ j & ) = ( 9 ,@ j & ) holds for any f , g E H with f g . The definition of the extremality above is closely related to the equality condition for norm inequalities between RKHSs. As is well known, if HI denotes the uniquely determined RKHS with the reproducing kernel J = 1 I&),then HI consists of functions that are restrictions to the diagonal E r 2 E of the functions in the tensor product H defined on Em. For 4j E Hj ( j = 1 , 2 . . . ,m ) we have the following norm inequality "N"
@Fl&
N
N
n?
II41.*.4rn(lHt I l l 4 1 @ . . * @ 4 r n (=l ~(141((H1*..lldmll~,.
Then equality occurs in the above inequality if and only if
is extremal.
Definition 1 Assume that 4 = @z14j is a nonzero element of the tensor product H . H is called regular (cf. [12/) i f and only iJ for each extremal 4, there is a point q E E and a constant cj E C such that q5j = cjKf' ( j = 1 , 2 , . . . ,m ) . Also, H is called weakly regular if and only iJ f o r each extremal 4, there is a point q E E such that, for each j ( j = 1 , 2 , . . . , m ) , one of the following holds: (i) q is a common zero of H j , that is, f ( 4 ) = 0 for all f E H j , (ii) 4j = C ~ K : ) for some constant c j E C. For convenience, we shall call Case (i) above the exceptional case. The main objective of the present paper is to obtain a useful general condition on RKHS which implies the regularity or weak regularity of a tensor product of a finite number of RKHSs. In what follows, we always assume that R is a @-algebraconsisting of complex-valued functions on E equipped with the sum and the product defined pointwise and that R satisfies the following properties (the existence of the identity is not assumed): (a) There is no common zero of R in E , (b) R separates points of E , that is, for each p , q E E ( p # q ) , there is f E R such that f ( P ) # f ( q ) . We shall say that R strongly separates points of E if R satisfies the above properties (a) and (b).
Lemma 1 Assume that 4 = @c14j E @ Z I H j is nonzero extremal. If each Hj has a dense R-invariant subspace Hi ( j = 1 , 2 , . . . ,m ) , then there is a unique @-algebra homomorphism A,: R -+ C satisfying, f o r all f E R and U E H ; ( j = 1 , 2 ,.", m), (fu, 4j) = A,(f>(u, 4j).
(5)
Akira Yamada
450
Remark 1 For proofs of Lemma 1 and Theorem 1 the assumptions (a) and (b) o n R are not needed. Also, we remark that i f R has the identity element 1, then f r o m (5) we have A + ( l ) = 1. Let R1 . R2 denote the complex linear subspace of R generated by products of an element of R1 and an element of R2, where R1 and Rz are linear subspaces of R.
Definition 2 Let H be a RKHS o n E . If (i) R n H is dense in H , and (ii) R - ( R n H) = R n H , then H is said to be R-dense. If H is R-dense for some function algebra R o n E , H is said to be ring-dense. Remark 2 If 1 E R, then the condition (ii) of the above definition is equivalent t o the one that R n H is an ideal of R. For convenience, when H is an R-dense RKHS on E , we write simply that
(H,E ) is R-dense. Definition 3 Let ( H ,E ) be R-dense. Then C-algebra homomorphism x: R -+ C i s called a bounded homomorphism of R i f x satisfies the following property: There is a constant C > 0 such that
The set of nonzero bounded homomorphisms of R is said t o be an H-hull of
E and is denoted by EH.
If Hj (j= 1 , 2 , . . . , m) is R-dense, then each R n Hj is a dense R-invariant subspace of H j and we can apply Lemma 1 to the tensor product @ H j . Theorem 1 For j = 1 , 2 , . . . ,m, let ( H j ,E ) be R-dense. If 4 = @T=lcjjE @FIHj is nonzero extremal, then there is a unique nonzero bounded ho~ ( 5 ) . Furthermore, for each j ( j = momorphism A+ E ~ ; , E H satisfying 1 , 2 , . . . ,m), either one of the following holds. (i) A+lRnH, =o, or (ii) there is a constant Cj
# 0 such that for each
f E R n H j , we have
K 4 j ) = CjA+(f>. Remark 3 Case (i) corresponds t o the “exceptional case” stated in Definition 1. An example of this case is given by Theorem 4 (iii).
xq(f )
= f ( q ) be the point evaluation of f E R at q E E . Then C is a @-algebrahomomorphism of R. Since R has no common zero on E (hypothesis (a)), xq # 0. Also, since H is a RKHS on E , xq is a bounded homomorphism of R. Hence, xq E EH.By hypothesis (b) of R the mapping E 3 q I-+ xq E EH is injective. From now on by identifying a point q with the homomorphism xq via the above mapping, we assume that E c E N .
Let
xq : R
-+
Equality Conditions for General Norm Inequalities
451
Definition 4 Let ( H ,E ) be R-dense and let ( H 1 , E l ) be R1-dense. Then ( H I ,E l ) i s called an extension of ( H ,E ) i f there is a mapping d j : R U H -+ R1 U H1 satisfying the following properties. (i) @ I H : H -+ H1 is onto isometry, (ii) @\R:R ---t R1 is an @-isomorphism, (iii) E c El, and (iv) @(f)lE = f for all f E R u H .
W e express the fact that (HI,E l ) is an extension of ( H ,E ) by writing ( H ,E ) 4 ( H I ,E l ) , or more precisely, d j : ( H ,E ) + (HI, El).
Definition 5 A ring-dense R K H S H is called maximal if ( H , E ) 4 ( H 1 , E 1 ) implies E = El. Every ring-dense RKHS has an essentially unique extension which is also maximal. To show this we begin with the following definition. If (H,E ) is R-dense, then for each f E H there is a sequence { f,} c R n H such that f, -+ f (v --t co) in H . For x E & H , define x(f) = lim, ~ ( f , ) This . definition is well-defined and we conclude that each homomorphism x E @H is extended uniquely to a function defined on the set R U H. For f E R U H , let be a function on EH defined by f(x)= ~ ( f )x, E E H . Denote by @ the mapping R U H 3 f + - + f , a n d p u t f i = d j ( H )a n d A = d j ( R ) .
f
Proposition 1
(A,& H )
is A-dense and is a maximal extension of (H,E ) .
From Proposition 1 we note that H is maximal if and only if E = E H , that is, any nonzero bounded homomorphism of R is the point evaluation at some point in E. Example 1 Let 12(E) be the Hilbert space {f E C E I f ( x ) = 0 f o r all but countably many x E E and CIEE I f (.)I2 < co} with the inner product (f79) =
c
f ( X ) r n
f o r f , g E t2(-q.
xEE
Then C2(E)is a R K H S on E whose reproducing kernel at p E E is S,(x) = S , (Kronecker’s delta). Let R be the C-algebra (without the identity) {f E C E I f (x)= 0 f o r all but finitely many x E E } . Then R strongly separates points of E and is dense in l z ( E ) .Since R . R = R, we see that (12(E), E ) is R-dense. If x: R --,C is a nonzero C-algebra homomorphism, then it is easy to see that x is the point evaluation at some point in E . Thus 12(E) is R-dense and maximal.
The following is our main theorem which partially answers Problem 3.1 of [15, p. 181.
Theorem 2 Assume that each H j ( j = 1,2,. . . ,m) is an R-dense RKHS o n E . If EH~ = E , then their tensor product @ G I H j i s weakly regular.
nj”=,
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Akira Yamada
Remark 4 Whenever ( H ,E ) is R-dense, we shall introduce the weak topology induced from the family R of functions as a topology for the set E, that is, we give E the weakest topology such that every function in R is continuous on E. Then we see that, i f (HI,El) is a maximal extension of ( H ,E), the weak topology for El is unique up to a homeomorphism.
3 Equality Conditions for the General Norm Inequality Let H be the RKHS HK(E)and let $ ( z ) = Cr=o c,z’ be an entire function with nonnegative coefficients {c,}. Denote by 11 . the norm of the RKHS H$ = H$(K)(E).Then we have the following general norm inequality [2, 161: For all f E H , Il$(f>ll$ 5 ICl(l1.f 112>* (6) By construction, $ ( K q )is a reproducing kernel of H$ at q E E (e.g. see [2]). Therefore, if f = Kq (q E E ) , then we have always equality in the inequality (6). To study the equality problem, we write $ as a composition $1 0 $2 where
c 00
$l(Z)
= co
+ 2,
$2(4 =
c,z’.
u= 1
From the chain of inequalities
lMf)II$I ~ l ( l l ~ 2 ( f ) l l $ Il ) $1(+2(llf 112>>
=
4(llf112),
we see that, to study equality condition for the general norm inequality induced from $ ( z ) , we need only to consider the following special cases: (i) $ ( z ) is a linear function, (ii) $(O) = 0. We recall some facts from the general theory. Let .)j and 11 . Ilj ( j = 1 , 2 ) be the inner product and the norm of a RKHS H j = H K ~( j = 1,2), respectively. Let 11 12 be the norm of H K ~ + KThen, ~ . we have the Pythagorean inequality (a,
IIf1+ fiII?z I IIf1II;
+ IIfzll;,
fl
E
HI, fz
E H2.
(7)
By using simple variation, we see [6, p. 321 that equality occurs in the inequality (7) if and only if ( f i ,h)l
= (f2,
h)2
for all h E HI n H2.
(8)
Combining Theorem 2 and the condition (8) we obtain the following
Theorem 3 In the notation stated above, assume the following: (i) H is a RKHS which is ring-dense and maximal, and (ii) {(i,j)1 cicj # 0 , 1 I i < j } # 8 and gcd{j - i I cicj # 0 , 1 I i
< j } = 1.
If co > 0 and 1 E H$2, then equality occurs in the general norm inequality (6) if and only i f f = Kq for some q E E . Otherwise, equality occurs i f and only if f = O or f = K , f o r s o m e q E E .
Equality Conditions for General Norm Inequalities
453
4 Special Cases and Applications As an application of Theorem 1, we first consider the case where the set E is a subset of the complex n-dimensional space @" and R is the restriction to E of the polynomial ring @ [ z l , .. . ,zn]. For simplicity, we use multi-index notation: z = ( z l , z z , . . . , z n ) , a = (cq,a2,.. . , a n ) ,z a = z s i , C [ z ]= @ [ z l , . . , zn]. A power series with center at the origin is denoted by C , a&.
n2,
Definition 6 Let H be a RKHS o n a subset E of @". then H is called polynomially dense.
If H is @ [ z ] I ~ - d e n s e ,
Let x : @ [ z ]-+ C be a @-algebra homomorphism with x ( 1 ) = 1. Putting ~ ( z=) ( ~ ( z l ~) ,( z z ). ., . ,x ( z n ) ) E @", for each polynomial f (2) = C , a,za we have x u ) = a a x ( 4 " = f(x(z>>.
c a!
Hence we see that any nonzero C-algebra homomorphism of C [ z ] l is~ a point evaluation. Consequently, when H is polynomially dense, we have the following necessary and sufficient condition for H to be maximal: Let q be a point in @". If there is a constant C > 0 such that 1 f (4)) 5 CJIf 1) for all f E @[z]n H, then q E E . We next give examples of RKHSs which are polynomially dense and show a sufficient condition for these RKHSs to be maximal.
Example 2 ( [ 5 ] ) For z,C E @" we put z< = ( z l c l , . . . ,zn&) E @". Fix a power series with positive coefficients ~ ( z=) C , caza!, (ca > 0, a E Z?), and assume that the domain of convergence D of the function q ( z E ) is nonempty. A function f holomorphic in the domain D has a power series expansion f (2) = C , a a z a o n D . Define the norm o f f by
Ilf112
=
c
IQa12
-7
a
ca
and let 3-1, denote the space of holomorphic functions in D with 11 f )I Then E,, is a Hilbert space with the inner product
<
< 00.
The function kc(z) = q ( z c ) is the reproducing kernel at E D f o r the space and hence 'H, is a RKHS on D. B y definition of the norm, 3-1, is clearly polynomially dense.
H ',
Proposition 2 If q(z2) = 00 f o r each z E 8D,then Ev is maximal.
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Akira Yamada
We next consider the case where the domain of definition E has complex dimension one. Now let E be a regular subregion of a closed Riemann surface S . Let us denote by RE the complex algebra of meromorphic functions on S which are holomorphic on
z.
Proposition 3 Let x: RE t C be a C-algebra homomorphism with x ( 1 ) = 1. Then there is a unique point q E F such that x(f) = f ( q ) f o r all f E R E . Definition 7 If a RKHS H o n E is RE-dense, H is said t o be meromorphically dense.
From Proposition 3 we see that a meromorphically dense RKHS H is maximal if and only if the following holds: If for some point q E there is a constant C > 0 such that lf(q)1 5 CIIf 11 for all f E RE n H , then q E E. As an application we study the regularity of tensor products of RKHSs of analytic functions or analytic differentials on a subregion of a closed Riemann surface. Now we assume that the set E is a regular subregion of a closed Riemann surface S. Consider the following reproducing kernel Hilbert spaces: (i) hl: The Hardy H 2 space on E with the norm 1) f / I 2 = JaE l f I 2 p where p is a positive continuous metric on aE. (ii) h2: The space of analytic functions f on E with finite Dirichlet norm 11 f [I2 = i JJE p d f A satisfying f ( a ) = 0 for a fixed point a E E , where p is a positive continuous function on E . (iii) h3: The Bergman space on E with the norm 11 f ]I2 = i pf A f where p is a positive continuous function on E .
JIE
For our purpose the following result [17, Theorem 81 is useful. Proposition 4 Let E be a regular subregion of a close Riemann surface S . For every holomorphic function f o n E and every constant E > 0 there is a function g E RE such that (1 f - g(lo0 < E o n E .
Using Proposition 4 and the existence of holomorphic functions on F with logarithmic singularity near dE we have
Theorem 4 The following hold: (i) R K H S hj ( j= 1 , 2 , 3 ) is meromorphically dense and maximal. (ii) For any integer v 1 2 and j # 2, h:” is regular, whereas hf” is weakly regular. (iii) Let E be the unit disk A = { IzI < 1) with a = 0 and p = 1. Then 4 R 2 is extremal in h2 8 h2 if and only i f + ( z ) = cz or 4 = ck, (q E A \ ( 0 ) ) f o r some c E C,where k,(z) = log(1 - Qz) is the reproducing kernel of h2 at q E A . Thus, hz 18 h2 is not regular.
-&
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References 1. N. Aronszajn, Theory ofreproducing kernels, Trans. Amer. Math. SOC.68 (1950), 337-404. 2. J. Burbea, A Dirichlet norm inequality and some inequalities for reproducing kernel spaces, Proc. Amer. Math. SOC.83 (1981), no. 2, 279-285. 3. -, Norm inequalities of exponential type for holomorphic functions, Kodai Math. J. 5 (1982), no. 2, 339-354. 4. -, Inequalities for reproducing kernel spaces, Illinois J. Math. 27 (1983), no. 1, 130-137. 5. -, Inequalities for holornorphic functions of several complex variables, Trans. Amer. Math. SOC.276 (1983), no. 1, 247-266. 6. L. de Branges and J. Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York, 1966. 7. H. M. Farkas and I. Kra, Riemann surfaces, Second edition, Springer, New York, 1992. 8. S. Saitoh, The Bergman norm and the Szego norm, Trans. Amer. Math. SOC.249 (1979), no. 2, 261-279. 9. -, The Dirichlet norm and the norm of Szego type, Trans. Amer. Math. SOC. 254 (1979), 355-364. Some inequalities for analytic functions with a finite Dirichlet integral on 10. -, the unit disc, Math. Ann. 246 (1979), no. 1, 69-77. 11. -, Some inequalities for entire functions, Proc. Amer. Math. SOC.80 (1980), no. 2, 254-258. 12. -, Reproducing kernels of the direct product of two Hilbert spaces, Riazi J. Karachi Math. Assoc. 4 (1982), 1-20. 13. -, A fundamental inequality in the convolution of Lz functions on the half line, Proc. Amer. Math. SOC.91 (1984), no. 2, 285-286. 14. -, Hilbert spaces admitting reproducing kernels on the real line and related fundamental inequalities, Riazi J. Karachi Math. Assoc. 6 (1984), 25-31. 15. -, Theory of reproducing kernels and its applications, Longman Sci. Tech., Harlow, 1988. 16. -, Natural norm inequalities in nonlinear transforms, in General inequalities, 7 (Oberwolfach, 1995), 39-52, Internat. Ser. Numer. Math., 123, Birkhaiiser, Basel, 1997. 17. S. Scheinberg, Uniform approximation by meromorphic functions having prescribed poles, Math. Ann. 243 (1979), no. 1, 83-93.
Comparing Multiresolution SVD with Other Methods for Image Compression Ryuichi Ashino', Akira Morimoto2, Michihiro Nagase3, and RBmi Vaillancourt4 Mathematical Sciences Osaka Kyoiku University Kashiwara, Osaka 582-8582, Japan ashinoQcc.osaka-kyoiku.ac.jp,
Information Science Osaka Kyoiku University Kashiwara, Osaka 582-8582, Japan morimotoQcc.osaka-kyoiku.ac.jp Department of Mathematics Graduate School of Science Osaka University Toyonaka, Osaka 560-0043, Japan deceased Department of Mathematics and Statistics University of Ottawa 585 King Edward Avenue Ottawa, Ontario K1N 6N5, Canada remiomathstat .uottawa.ca
Summary. Digital image compression with multiresolution singular value decomposition is compared with discrete cosine transform, discrete 9/7 biorthogonal wavelet transform, Karhunen-Loisve transform, and a hybrid wavelet-svd transform. The compression uses SPIHT and run-length with Huffmann coding. The performances of these methods differ little from each other. Generally, the 9/7 biorthogonal wavelet transform is superior for most images that were tested for given compression rates. But for certain block transforms and certain images other methods are slightly superior.
1 Introduction Image compression is important in digital image transmission and storage. Comparative studies of compression methods are found in [5] and [l].In [3], image compression with multiresolution singular value decomposition [6] is compared with discrete cosine transform, discrete 9/7 biorthogonal wavelet transform, Karhunen-LoBve transform, and a hybrid wavelet-svd transform.
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Compression uses Set Partitioning in Hierarchical Trees (SPIHT) [7] and runlength with Huffmann coding. These methods are briefly reviewed and their performance is tested through numerical experiments on several well-known images. It is found that these methods differ little from each other at moderate compression ratio. Generally, the 9/7 biorthogonal wavelet transform is superior for most images that were tested for given compression rates. But for certain block transforms and certain images other methods are slightly superior. Section 2 summarizes multiresolution analysis (MRA) and block algorithms. Section 3 describes the coding methods. In Section 4, we propose a hybrid method using the 9/7 biorthogonal wavelets with singular value decomposition (SVD). Table 1 lists the results of numerical experiments with these methods on five images and Table 2 in Section 5 list the results on a fingerprint image, for which visual inspection is done in Section 6.
2 Multiresolution Processing and Block Algorithms The analysis stage of a two-dimensional separable discrete wavelet transform produces the matrix X = UAVT, where the upper and lower half-parts of the orthogonal matrices U and V correspond to lowpass and highpass filters, respectively. The discrete wavelet transform divides the image into four parts as in the following procedure:
( P l ) The scaling function cp(z)cp(y) produces the top left part. (P2) The vertical wavelet function $(z)cp(y) produces the top right part. (P3) The horizontal wavelet function cp(z)$(y) produces the bottom left part. (P4) The diagonal wavelet function $(z)$(y) produces the bottom right part. The top left part is called an upprozirnution because it is smooth and has large values. The other three parts are called details because they emphasize horizontal, vertical, and diagonal edges, respectively. These three parts have small absolute values except for edges. A multi-level decomposition is obtained by applying this decomposition to successive approximations. Similar decompositions are achieved by the discrete cosine transform (DCT) and the SVD by means of the following block algorithm:
(BA1) A given image matrix X E Itmx"is divided into b x b submatrices X('!'), 1 5 k 5 m/b, 1 5 t 5 n/b. (BA2) Each submatrix X('7') is transformed into X,("') by the DCT or the SVD. (BA3) The matrix X,('")is rearranged into an (mlb)x (n/b)matrix X:"). (BA4) The Xii") matrices are put in the (i,j) position t o produce the m x n matrix X, which contains b2 parts and is similar to the matrix obtained by the DWT. The Kakarala-Ogunbona algorithm [6] is a kind of multiresolution algorithm. We explain here the two-dimensional algorithm for level 1.
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(K01) Each b x b submatrix X('1') of a given matrix X E Rmx" is reshaped into a b2 x 1 column vector. (K02) These column vectors are collected into a b2 x (rnn/b2) matrix T . (K03) T is factored into its reduced singular value decomposition in the form T = U S V T , where U E Rb2x(m"/"2) and V E R(m"/b2)xb2 have orthonormal columns, and S E R4x4is diagonal. (K04) Calculate the b2 x (mn/b2)matrix A = U T T = SVT. (K05) Each column vector of A is reshaped into a b x b matrix X!""). (K06) All the matrices Xi"")are rearranged into an rn x n matrix XI. Figure 1illustrates the algorithm for level-1 SVD MRA on a 32 x 32 matrix. Figure 2 illustrates the difference between SVD and 9/7 wavelet multiresolution at level 1 for the octagon figure. One notices that the four isolated diagonal segments appear in the lower-left and lower-right detail parts of the SVD and wavelet multiresolution, respectively. The singular values and left singular vectors for the level-1 SVD MRA of the octagon image are in the vector S and the columns of U , respectively, S =
4554.4
3524.2 3524.2 2024.0
U
= 0.5000
0.5000 0.5000 0.5000
-0.0000 0.7071 -0.7071 0.0000
-0.7071 -0.0000 0.0000 0.7071
-0.5000 0.5000 0.5000 -0.5000
One sees that the first column of U is a lowpass filter. The norm of the nth row of A is equal to the nth singular value of T because the columns of the matrix V are orthonormal. The b2 x b2 orthogonal matrix U and the singular values are needed for the inverse transform.
3 SPIHT The SPIHT [7] algorithm is based on the following two observations: OBSERVATION 1. The pixels of the analyzed image having large absolute values are concentrated in the upper-left corner. OBSERVATION 2. SPIHT encodes zerotrees based o n the principle that when a wavelet coeficient has small absolute ualue, then points at other levels corresponding to this coeficient also have small absolute values. SPIHT has three ordered lists: 0
0 0
the list of significant pixels (LSP), the list of insignificant pixels (LIP), the list of insignificant sets (LIS).
LIP and LIS are searching areas. LSP lists the pixels whose absolute values are greater than 2 N , thus requiring more than N bits. Each pixel of LIP is
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Fig. 1. Level 1 SVD MRA for a 32 x 32 matrix.
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log2(wavelet analyzed fig.)
Iog2(svd analyzed fig.)
Fig. 2. Negative image of level-1 approximation and detail subimages of the octagon figure produced with SVD and 9/7 wavelet MR, respectively. The level-1 approximation is in the top left subimages.
Bit row sign msb 6 5
4 3 2 1
lsb 0 Fig. 3. Left: hierarchical structure. Right: binary representation of the magnitudeordered coefficients tested as to whether its absolute value is less than 2N or not. Each pixel of LIS is tested as to whether all absolute values of its descendants are less than 2 N . At the first step, all the pixels of LIS are type ‘A’. Some pixels of LIS will be changed from type ‘A’ t o type ‘B’ in the following SP procedure:
(SP1) LSP is taken as an empty list and LIP is the set of top level coefficients. LIS is the set of top level wavelet coefficients and all the pixels of LIS are type ‘A’. N is set to the most significant bit of all coefficients. (SP2) Check each pixel of LSP and output 0 if its N t h bit is 0, and output 1 otherwise. (SP3) Check each pixel of LIP and output 0 if its absolute value is less than 2 N . Otherwise, output 1 and, moreover, output 0 when the value of this pixel is negative and 1 if positive, and move this pixel t o LSP. (SP4) Check each pixel of LIS.
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a) When the pixel is of type ‘A’, output 0 if the absolute values of all its descendants are less than 2N. Otherwise, output 1 and do the following: (i) Check all four children. (ii) When the absolute value of a child is greater than or equal to 2N, output 1 and, moreover, output 0 or 1 according to the sign of this child and add this child to LSP. (iii) When the absolute value of this child is less than 2N, add this child to the end of LIP. (iv) When this pixel has grandchildren, move it to the end of LIS as a pixel of type ‘B’. b) When a pixel is of type ‘B’, output 0 if the absolute values of all descendants, apart from the children, are less than 2N. Otherwise, output 1 and add each child to the end of LIS as type ‘A’ and delete this pixel from LIS. (SP5) Set N to N - 1 and go to step (SP2). (SP6) When the number of output bits exceeds the threshold (which is decided by user’s bpp), then stop this procedure. The SPIHT algorithm is very efficient for high compression rate when N is large but does not minimize memory nor bandwidth and is not designed to look at regions of interest, as opposed to JPEG 2000. The run-length and Huffmann coding can quantize the analyzed image economically by the following HU procedure:
(HU1) Divide each block of the analyzed image by some integer which depends on the image and the block location. Each pixel of this divided image is rounded to an integer. This quantized analyzed image has many 0 entries. (HU2) Reshape this image into a long row vector. In this step, use the following two methods: a) Reshape each block into a vector and stack these vectors together. b) Use the hierarchical tree (0 tree) algorithm. (HU3) Compress the 0 entries of this long row vector by the run-length coding. (HU4) Compress the run-length coded image by gzip.
4 Hybrid Wavelet-SVD Method We propose a hybrid method which combines wavelet and singular value decompositions. The analysis procedure consists in the following three steps: (AN1) Transform the m x n image X into the analyzed image X I by the level-two DWT using the 9/7 biorthogonal wavelets. (AN2) Decompose X I into 2 x 2-block SVD MRA up to level six to get X2. (AN3) Compress X2 by SPIHT and compress the resulting image with gzip. The synthesis procedure consists in the following three steps:
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Uncompress the gzip image with gunzip and decode the compressed code to x z . Obtain the synthesized image 2, by the inverse 2 x 2-block SVD transform. Obtain the reconstructed image 2 from X I by the inverse DWT. We have the following observation:
OBSERVATION 3. Our hybrid wavelet-SVD method i s better than SVD alone, It i s better than biorthogonal wavelet f o r the f p i and barb images. This observation leads to the following conclusions: (Cl) The SVD decomposition depends on the data and cannot deal with data in time-frequency domain. Because our hybrid method contains wavelet analysis, which is a kind of time-frequency analysis, our hybrid method performs better . (C2) The blocking effect in our hybrid method is weaker than with SVD, because we use long-filter wavelets in the last synthesis step.
5 Numerical Experiments Eight bit-per-pixel (bpp) images have been compressed by the following methods. 0
0
0 0
0
bior4.4 is the biorthogonal wavelet filter with 9/7 taps of [2]. db2 is Daubechies’ compactly supported wavelet filter with N = 2. 2by2SVDMR and 4by4SVDMR are the SVD multiresolution with block size 2 and 4, respectively. JPEG is MATLAB’S imwrite function. 2by2KLTMR and 4by4KLTMR are the KLT multiresolutions with block size 2 and 4, respectively. bior4.4+SVD consists of the following two steps. In the first step, the image is transformed by bior4.4 wavelet to level 2. In second step, the transformed image is decomposed by 2by2SVDMR to level 6.
The SPIHT algorithm [7] is used for coding the MRA methods. Six well-known images, 512 x 512 Lena, Boats, Barb, and Yogi, 512 x 640 Goldhill, and 768 x 768 fpl, shown in Fig. 4 have been tested. The fpl image is a sample of the FBI WSQ FINGERPRINT COMPRESSION DEMOS 4.2.5. Four objective measures, PSNR, MSE, MaxErr, and SNR, defined below, were applied t o rn x n original and reconstructed images, X and 2 .
Definition 1 Peak Signal to Noise Ratio (PSNR) and Signal to Noise Ratio (SNR) are:
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Fig. 4. The six original figures.
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where the square of the Frobenius norm. of a n m x n matrix A i s
The mean square error and the maximum error are
1 MSE = -JJX mn
- 211$,
MaxError = IJX- 211,.
In this work, bpp is the number of bits in the gzip image divided by the number of bits in the original image. Peak Signal to Noise Ratio (PSNR) with the bior4.4 method is generally higher except for the Yogi image at 1 and 0.5 bpp where 2by2SVDMR and 2by2KLTMR are superior. The numerical results listed in Tables 1 and 2 lead to the following conclusions: (C3) At high compression ratio, that is, low bpp, block effects appeared for SVD, KLT, and JPEG, especially remarkable for SVD2by2 and KLT2by2. On the other hand, in case of wavelet with long filters, images were out of focus. Our hybrid method using 9/7 wavelet with SVD lies between these two opposite cases. (C4) For the fingerprint, our hybrid method using 9/7 wavelet with SVD was superior to the other methods. ( C 5 ) Better performance was obtained with short-filter SVD2by2 and KLT2by2 for Yogi as it uses fewer grey levels, (C6) For other images, our hybrid method performed a little bit inferior to wavelet bior4.4, but superior to SVD, KLT, and JPEG.
Every experiment was run four times successively under the same conditions, and the cpu time, measured with the MATLABprofile function, was taken to be the mean value of the last three runs. The computations were done on a portable PC with the following specifications: Pentium I11 866 Mhz, 512 MB memory, Microsoft Windows 2000 and MATLABR13. Partial results are listed in Table 1 for the first five figures. Fuller results are in [3].
6 Visual Inspection of the Fingerprint Image at 0.15 bPP. The six compression methods, bior4.4, db2,2by2SVDMR,4by4SVDMR,4by4KLTMR, and bior4.4+SVD, applied to the 768 x 768 fpl image produce very similar synthesized images at 0.15 bpp on the screen and in 40%-reduced print form. However, at high compression ratio, that is, low bit per pixel, visual inspection is necessary to ascertain the quality of synthesized images.
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Table 1. Results for Lena, Boats and Goldhill at 0.25 bpp, Barb at 1.5 bpp, and Yogi at 0.5 bpp. The bpp used by JPEG is indicated in the first column. Image1 Lena
Boats
0.25
Gold
0.26
Barb
1.51
Method bior4.4 db2
ILevell PSNR I MSE I MaxErr I SNR ICPU 6 33.4193 29.5901 41.9485 27.7383 4.96 6 32.0355 40.6943 44.8188 26.3544 4.64
I
bior4.4 I 6 db2 I 6 2by2SVDMR 6 4by4SVDMR 4 JPEG 2by2KLTMR 6 4by4KLTMR 4 bior4.4+SVD 2+6 bior4.4 6 db2 I 6 2by2SVDMR 6 4by4SVDMR 3 JPEG 2by2KLTMR 6 4by4KLTMR 3 bior4.4+SVD 2+6 bior4.4 6 db2 6 2by2SVDMR 6 4by4SVDMR 4 JPEG 12by2KLTMRI 6 14bv4KLTMRI 4 I " I Ibior4.4+SVDI 2+6 I
129.49051 73.1191 I 80.172 124.14791 4.98 128.69231 87.8713 175.3258 123.34971 4.53 I 27.5786 113.5583 81.147 22.236 3.27 27.8278 107.2269 87.8259 22.4852 3.23 27.3174 120.5969 109 21.9748 0.99 27.658 111.5017 82.5188 22.3154 34.10 27.9562 104.1012 87.8324 22.6136 9.17 28.5882 90.0041 78.3846 23.2456 5.43 30.5292 57.5658 I 51.2446 I 23.5659 I 6.33 I I 29.77 168.5611 164.5146 122.80681 5.34 29.3633 75.2917 73.4238 22.4001 3.74 29.6741 70.0923 66.1833 22.7108 3.65 29.6083 71.1619 70 22.6451 1.15 29.5023 72.9213 74.7445 22.539 43.05 29.8132 67.8827 60.2749 22.8499 11.07 29.9528 65.7348 68.479 22.9896 6.24 30.3506 59.9822 34.2253 24.0747 5.94 30.0205 64.7188 37.389 23.7446 5.56 29.4611 73.6165 39.915 23.1852 4.14 29.8336 67.565 37.9863 23.5577 4.11 28.2041 98.3269 I 51 21.9282 1.01 129.41671 74.3729 II 40.5208 123.1408134.47 I I 129.93071 66.0707 II 39.6838 - - 123.65481 - - .- - - 9.76 I I 130.40381 59.252 38.8405 124.12791 6.51 I
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It is seen in Table 2 for f p i that PSNR is below 30 db at bpp
467 = 0.15
so that some visual deterioration of the synthesized images may be expected. Blocking effects (BE) and blurring of the fingerprint image at 0.15 bpp can be observed at 200% and 300% magnification with Adobe Illustrator. The following list goes from low to high performance. jpeg: strong BE at 200% 2by2SVDMR: weak BE at 200%, strong at 300% 2by2KLTMR: weak BE at 200%, moderate at 300% 4by4SVDMR weak BE at 200%, slightly strong at 300% 4by4KLTMR: weak BE at 200%, moderate at 300% db2: weak BE at 200% with a little blurring bior4.4+SVD: weak BE at 600% with some blurring in parts bior4.4: weak BE at 600% with some blurring in parts Visual inspection corroborates the PSNR. Figures 5 and 6 show a magnified part of the fingerprint image at 0.15 bpp. Again magnification is by Adobe Illustrator. It is seen that apart from bior4.4+SVD and bior4.4, the other methods introduce blocking effects. The curves in Figs. 7 show that the new hybrid method, bior4.4+SVD, has higher PSNR against bpp than other methods for the fingerprint image.
References 1. Aase, S. O., Husoy, J. H., Waldemar, P. (1999) A critique of SVD-based image coding systems, Proceedings of the 1999 IEEE International Symposium on Circuits and Systems VLSL, 4 pp. 13-16, IEEE Press, Piscataway, NJ. 2. Antonini, M., Barlaud, M., Mathieu, P., Daubechies, I. (April 1992) Image codi n g using wavelet transform, IEEE Trans. Image Processing, 1 205-220. 3. Ashino, R., Morimoto, A., Nagase, M., Vaillancourt, R. Image compression with multiresolution singular ualue decomposition and other methods, Mathematical and Computer Modelling, to appear 4. Chen, J. (2000) Image compression with SVD, ECS 289K Scientific Computation, Dec. 13, 2000 13 pages. http://graphics.cs.ucdavis.edu/"jchen007/UCD/ECS28QK/Project.html
5. Gerbrands, J. J. (1981) O n the relationships between SVD, KLT, and PCA, Pattern Recognition, 14 375-381 6. Kakarala, R., Ogunbona, P. 0. (May 2001) Signal analysis using a multiresolution form of the singular value decomposition, IEEE Trans. on Image Processing, 10, NO. 5, 724-735. 7. Said, A,, Pearlman, W. A. (June 1996) A new fast and eficient image codec based on set partitioning in hierarchical trees, IEEE Trans. on Circuits and Systems for Video Technology, 6, 243-250. 8. Unser, M. (1993) An extension of the Karhunen-Lo&e transform for wavelets and perfect reconstruction filterbanks SPIE, 2034 Mathematical Imaging, 45-56. 9. Waldemar P., Ramstad, T. A. (1997) Hybrid KLT-SVD image compression, in 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, 4 pp. 2713-2716, IEEE Comput. SOC. Press, Los Alamitos, CA.
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Table 2. Numerical results for the 768 x 768 f p l image bppl
Method
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MSE IMaxErr1 SNR lcpu
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bior4.4+SVD PSNR=29.906
bior4.4 PSNR=29.877
db2 PSNR=28.418
P E G PSNR=24.248
SVD4by4 PSNR=26.784
SVD2by2 PSNR=26.517
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Fig. 5. Compressed fingerprint image at 0.15 bpp for bior4.4+SVD, bior4.4, db2, jpeg, 4by4SVDMR and 2by2SVDMR.
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KLT2by2 PSNR=26.417
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Fig. 6. Compressed fingerprint image at 0.15 bpp for 4by4KLTMFt and 2by2KLTMR. fpl BPP-PSNR 1 45 I
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Non-Translat ion-Invariance in Principal Shift-Invariant Spaces J.A. Hogan’ and J.D. Lakey2 Department of Mathematical Sciences University of Arkansas Fayetteville AR 72701 USA j ef f [email protected] Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003-8001 USA j 1akeyQnmsu.edu
Summary. This paper is concerned with the difficulties caused by the non-translationinvariance of principal shift-invariant and multiresolution spaces, particularly in sampling theory. A quantitative measure of translation-invariance is introduced and computed for a wide class of examples. An algorithm is developed which provides appropriate initialization of sampling procedures by determining the times at which oversampled data was acquired. The effectiveness of the algorithm is proved and its application demonstrated on a synthetic data set.
1 Introduction The traditional model space of signal analysis is the Paley-Wiener space PW, of L2-signals f bandlimited to the interval [-6’/2,6’/2] in the sense that their Fourier transforms f” satisfy f”(<)= 0 for It‘\ > Q/2. Here we normalize the Fourier transform so that f(<)= J-”, f(t)e-2kitcdt whenever the integral converges. The Paley-Wiener spaces provide a good model for many phenomena. O n the one hand, many natural and synthetic signal generators can only output slowly varying signals. On the other, many natural and synthetic processors can only process bandlimited signals. For example the human ear cannot process/amplify frequencies beyond, say, 20 kHz. Signals in PW, enjoy a particularly simple sampling property [15], [14], [18]:
Theorem 1 (Classical Sampling Theorem) Iff E PW, and TQ > 1, then
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sinrt . where sinc(t) = - zs the cardinal sinc function and the convergence is 7rt both pointwise and in L2-norm. The Classical Sampling Theorem is of fundamental importance in digital signal processing since it provides a means of converting an analog signal f to a digital signal { f ($,)}g-,and of reconstructing f uniquely and continuously within PW, via (1). However, for many applications, different signal models are required - for example, when modelling the effect of real acquisition (sampling) devices, or for improved numerical performance of discrete processing algorithms. While the sinc functions are well-localized in frequency, they are poorly localized in time. In fact, their Heisenberg product f.1l 11211(f1112 is infinite, making them poor candidates for basis functions with which to implement post-sampling digital processing,cf. [2].In lieu of the sinc functions, we might consider as a basic signal model linear combinations of integer shifts of a function cp having better joint time-frequency localization, or with compact support, or any number of useful properties. Then, in analogy with PWn one considers as a model space
V(cp)= {f = C C k c p ( . - k); C k
E
P ( Z ) } c L2(R).
(4
k
The Paley-Wiener space PW, is an example of such a space with 'p = sinc. One of the main advantages of moving from PW, to more general spaces V('p)is that 'p may be chosen to have superior time-frequency localization. However, if a space V(cp)is to seriously compete with the Paley-Wiener spaces as a venue for signal processing, it is important that sampling theorems analogous to the Classical Sampling Theorem be established for this wider class of signals. Questions such as these have, in recent years, been the catalyst for a very active area of research; (see [l],[ 6 ] ,[9]and references therein). The focus in this paper, however, is elsewhere. What has been largely ignored in the literature on sampling in V(cp)are the many difficulties caused by the non-translationinvariance of V(cp).Notice that if f ( t ) E PW,, then f ( t a ) E PW, for all translations a. This is not the case for general V(cp),and this seemingly innocuous deficiency creates significant problems for the application of, among other things, sampling algorithms. This paper is organized as follows. In Section 2 we introduce the notions of principal shift-invariant (PSI) spaces, multiresolution analysis (MRA) and the Zak transform. In section 3 we discuss measures of non-translation-invariance in PSI spaces and investigate the degree of non-translation-invarianceexperienced in these spaces. Section 4 contains a brief description of a handful of sampling algorithms valid in PSI and MRA spaces, giving specific examples in the case where cp is the Daubechies scaling function 2cp. Finally, in section 5 we describe in detail the difficulties non-translation-invariancecauses for sampling and provide algorithms which help cope with them. The efficacy of the
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algorithm is also checked, and an example in which the algorithm is applied to synthetic data is also provided.
2 Preliminaries 2.1 Principal Shift-Invariant Spaces A principal shift-invariant (PSI) space is a pair (V, 'p) with V a closed subspace of L2(R) and 'p E V with the property that the collection {'p(. - k)}k is an orthonormal basis for V. The Shannon PSI space is the pair (VS,'ps)where VS = PW1 and ps(t) = sin .rrt/rt = sinc (t).The Classical Sampling Theorem expresses each f E V , as f(t) = Ck f ( k ) ' p s ( tAs a second example, consider the space of functions VH of signals f E L2(R) with the property that f is constant on each interval [Ic, k 1) (k E Z) with ' p ~ ( t=) x p l ) ( t ) . The pair (VH,' p ~ is) called the Haar PSI space. Since each f E VHis continuous except at the integers, it is clear that if 0 < t o < 1, then samples taken at t o + Z are well-defined and determine f via f ( t ) = C kf ( t 0 k ) p H ( t - Ic), a sampling result for signals in VH. This is an extreme pair of examples, each illustrating different features of sampling in PSI spaces. In the Shannon case, f is recovered from its integer samples, while in the Haar case f is recovered from its values at integer shifts of any nonintegral to. We are interested in intermediate examples. Except in these extreme cases, however, the V-coefficients of f E V, that is, the coefficients ck in the expansion f ( t ) = ck'p(t - Ic), will not be integer or translated integer samples of f .
v.
+
+
2.2 Multiresolution Analysis A multiresolution analysis (MRA) of L2(R) is a nested sequence of closed subspaces . - . c V-1 c VO c V1 c . . . c L2(R) and a function 'p E VOsuch that
(Vo,p) is a PSI space,
Since 'p generates Vo in the sense that {'p(t- k ) } k is an orthonormal basis for Vo, the second of the properties above implies that (2jl2p(2jt - k ) } k is an orthonormal basis for 4. Also, since V-1 c VO,'p must satisfy a scaling relationship of the form
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El
for some sequence {hk}k. Let H ( ( ) = hke21rikEbe the scaling filter associated with cp. The density condition on MRA’s imply @(O) = H ( 0 ) = 1. In the Haar case, c p H ( t ) = X [ o , l ) ( t ) , ho = hl = !j and hk = 0 if k # 0,1, so that H ( J ) = eKiEcosnJ. In t h e Shannon case @s(J) = x[-+,+]([) and H ( [ ) = xpal(() ~ [ q , ~ l ( (on ) [0,1] is periodic with period 1. There are several important conditions we impose on the (low-pass) filter H . The first of these is the quadrature mirror filter (QMF) condition
+
2.3 Zak Transform The main tool used in the analysis of uniform and staggered sampling is the Zak transform. We outline its relevant properties here. For a more detailed discussion, including some history and applications to signal processing, the reader is referred to Janssen’s excellent tutorial [ll]. Given f E S(R), the Schwarz space of rapidly decreasing functions, and t , ( E R, let the Zak transform Zf o f f at the point ( t , J )in phase space be given by k
The connection with the Poisson summation formula is apparent from the definition. Indeed, iff E S(R),the Poisson summation formula may be written Z f ( t ,() = e-2nitEZf(-J,t ) . The Zak transform is quasiperiodic in the sense that Z f ( t + k,( I ) = e-2Ki1tZf(t, (), ( I , k E Z).Consequently, the values of Zf on the square Q = [0,1) x [0,1) determine the values of f on all of phase space and we think of Q as the domain of Z f. A simple consequence of quasiperiodicity, and of crucial importance to us, is the fact that if Z f ( t ,€)‘ is continuous in both variables, then Zf has a zero in Q [lo]. The Zak transform is unitary from L2(R) to its range. The inversion forZ f ( t ,() d( = f ( t ) whenever the integral conmula is particularly simple: verges. It will certainly do so for f E S(R), and a limiting argument is required to deal with f E L2(R). It will be important for us, particularly when dealing with oversampled data, to extend the definition of the Zak transform from phase space R x JR to R x @. Hence, we define the complexified Zak transform Zcf(z, z ) for 2 E R, z E @ as the Laurent series Z c f ( t ,z ) = Ckf ( t k)zk whenever the sum converges. It is clear that, for fixed t , Z c f ( t ,2 ) is just the z-transform of the sequence of samples { f ( t k)}k. When z is restricted to the unit circle, we recover the previous definition of the Zak transform, that is, Z c f ( t ,eaniE)= Z f ( t ,(). Also, when f is supported on [0, M ] (Ad a positive integer) then for fixed 0 5 t < 1, Z c f ( t ,z ) is a polynomial in z of degree at most M - 1. Of crucial importance in what follows is the observation [12] that the Zak transform of a scaling function cp with scaling filter H satisfies
+
Jt
+
+
Non-Translation-Invariancein Principal Shift-Invariant Spaces
475
which is easily obtained by taking the Zak transforms of both sides of (3). Further, $I E L2(W) has orthogonal integer shifts $(t k) (k E 9)if and only if r~>12= J : I Z $ ( ~r>12 , d t = 1 for almost all E.
zk@(r +
+
3 Non-Translat ion-Invariance For each f E L2(lR) and a E R, let 7,f be the translation of f by a , i.e., ~ , f ( t ) = f ( t - a ) . A subspace V of L2(R) is said to be translation-invariant when f E V + r, f E V for all CY E R. The Paley-Wiener spaces are translation invariant, so it makes little difference whether one samples a bandlimited function exactly at (suitable multiples of) the integers or at an arbitrary translate thereof. In the case of sampling in general PSI spaces this is no longer true. It is well-known that the only closed translation-invariant (TI) subspaces of L2(R) are the subspaces VEof L2-signals bandlimited to a measurable subset E E R (i.e., signals whose Fourier transforms are essentially supported on E ) . In [13] it is shown that the only TI MRA’s (V,,cp) are those for which I@\ = X E for some measurable tile E in the real line of finite measure 1 (the so-called minimal frequency supported scaling functions). In what follows we quantify the non-translation-invarianceof PSI and MRA spaces and consider its consequences for sampling procedures. 3.1 Translation Discrepancy
For each a E R and f E V(p) we measure the degree to which the energy of f leaks out of V(p) when translated by a by the quantity 11~,f--p,T,fll~/llf 1; where P, is the orthogonal projection onto V(cp).Hence we define the translation discrepancy of V(cp)with respect to a by
It is natural then to consider quantities such as d, = ~ u p ~ ~ ~ < ~and d,(a) d;p = d,(cy) d a as measures of the non-translation-invarianceof V(cp).For the purpose of this paper, we consider the maximum discrepancy d, only. Related quantities are considered by Bastys in [3] and [4].Discrepancy is important in the analysis of signals via the wavelet spectrogram. A typical signal f E L2 has energy spread across the wavelet subspaces Wj (the orthogonal complement of V, in V,+1). The portion of its energy in Wj is Ej = IIPw,fll;. The (discrete) wavelet spectrogramof f is the sequence E ( f ) = {Ej(f)}&, and the cumulant is defined by F J ( ~=) Cizl,Ej(f) = llPv,Il; ( J E Z). Here PQ and Pw, represent the orthogonal projections onto PQ and Pwj respectively. As a technical device, we introduce the Wiener amalgam space
Jt
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J.A. Hogan and J.D. Lakey
W of functions f on the line satisfying llfllw =: C ksupo
fact we have the following rather surprising result. Theorem 2 Let cp E W be an orthogonal generator for a PSI space V(cp). Then there exists a E ( 0 , l ) with d,(a) = 1, i e . , d, = 1.
Observe that if cp E W , then according to Theorem 2, for all E > 0 there exists = 1 and a E ( 0 , l ) such that IIP,~,,fll2 < E - translating f by a almost removes it from V(cp)!The interpretation when V(p) = Vo is the base space for an MRA of L2(rW) is especially disturbing. The wavelet subspaces Wj are usually thought of as spaces of details of order j (or 2j). Then VJ = @$&,Wj is the accumulation of these details - signals in VJ are thought of as "blurred" signals where the level if blurriness is quantified by the level J . Theorem 2 states that there exists f E Vo and a E ( 0 , l ) such that the translated signal f ( t - a ) has details of higher order than those of f !
f E V(cp),Ilfll2
Proof. Let f(t) =
Ck ckcp(t - k) E V(p). Then 1
k
Taking the Fourier transform of both sides of this equation yields
c)
It will be enough to show that &(a, has a zero at some (ao,to) E [O,l)'. We then choose C to be concentrated about 6. By increasing the level of concentration of C about t o we may force the integral on the right-hand side of (6) to be as small as desired. The continuity of Zcp in the second variable means that B,(a, t )is continuous in both variables. Further, a simple calculation shows that B, is quasiperi, odic, i.e., BV(a+ j , k) = e-2"ijcB,(a, E). Hence B, E 2 = Z ( L 2 ( R ) )the range of the Zak transform. The continuity and quasiperiodicity of B, now imply that it has a zero at some point in [0, 1)2and the proof is complete.
+
Non-Translation-Invariancein Principal Shift-Invariant Spaces
477
It is worth noting that B, is, in fact, the Zak transform of the function g(a)= J-", 'p(t-a)p(t)d t , the time-inversion of the auto-correlation function of 'p. As indicated by the proof of the Theorem 2, locating the zeroes of B, is the key to constructing examples of f E V('p)with poor discrepancy, or equivalently, poor spectrogram invariance. In fact, d,(a) = 1 if and only if B,(a, .) has a zero. In the case where 'p is an orthogonal scaling function for an MRA of L2(R), we have the following result. Theorem 3 Suppose 'p is a n orthogonal scaling function for a n MRA with associated scaling filter H . If H is continuous, then B,(1/2, &) = 0 (0 5 (* < 1) if and only i f IH(-&/2>l2 = 1/2. The theorem may be phrased in the following terms: if 'p is an orthogonal scaling function with continuous scaling filter, then d , ( i ) = 1. Of course, this does not preclude the possibility of translations a other than having d,(a) = 1.
Proof. Notice that an application of ( 5 ) and the quasiperiodicity of Z'p gives
+ IH(-S/2 + 1/2)12/l
(Zq(2s,4
0
However, Further
2
+ 1/2)I2ds
(7)
JiI Z p ( 2 s , [ ) l 2 d s = 1, by the orthonormality of the shifts of
'p.
But by the quasiperiodicity of Z'p and a change of variable, the second integral on the right-hand-side is easily seen to be the negative of the first. Hence Z9(2s, - ( / 2 ) 5 ( 2 s , 4 / 2 + 1 / 2 ) d s = 0 and by (7) and the QMF condition
Jt
(4)7 ~,,,,(1/2,5)
-
= e-""[IH(-~/2)1~ - IH(-t/2 = e-"iE[21H(-E/2)12 - 11 = 0
and the proof is complete.
IH(-E/2)12 = 1/2
+ 1/2)12]
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J.A. Hogan and J.D. Lakey
Notice that for the Haar scaling filter, H(<) = eTiEcos7rE and IH(-c/2>l2= 1/2 cos2(7r(/2) = 1/2 so that B,(1/2,1/2) = 0. To construct an c ~ + ( t - k) with C(<) example with llPv71pf 112/11 f (12 small, let f (t) = concentrated around E = 1/2. As a concrete example, let c c = (-l)kxlqgv if IlcJ 5 N and f ~ ( t = ) xkcFcp(t - k) = xr=-N(-l)kcp(t - k). Then fN is supported on [ - N , N 11 so r1pf is supported on [ - N 1/2, N 3/21 1/2 ; while and ( 7 1 p f ~ , c p (-. Z)) = ( - 1 ) N / 2 ( 6 ~ 6 - ( ~ + 1 ) . So I I P , T ~ ~=~ ~ llfll; = 2N 1 and, since N can be arbitrarily large, we have d , ( i ) = 1. Notice that in this case,
+
+
+
+
+
is centred around E = 1/2, the zero of B,(1/2, As a second illustration of the computability of the QMF condition for the vanishing of B,, consider the Daubechies scaling functions cp,, supported on [0,3], parameterized by -1 < v < 0 [5]. The associated scaling filters are given by a).
H ( ( ) = H, (6) = [v(v-1)+ (1-v)e2ffiE+(v+I ) ~ +v(v+ ~ ~ ~ 1 E) e ~ ~ ~ t ] / [ 2v2)] (1+ from which we easily compute
JHv(()12= [v2(v2 - 1)cos67rE + (3v2 + 1)cos27r( + (v2+ 1)2]/[2(1+ v 2 ) 2 ] which equals 1 / 2 when E = 1/4, 3/4. There are no other real values of E with this property. The zeroes of IH(<)l2- $ at C = 1/4, 3/4 are characteristic of QMF’s with real coefficients.
4 Sampling Theorems for PSI Spaces There is now a substantial literature on sampling in PSI and MRA spaces. Iterative techniques for dealing with non-uniform samples have been studied by Aldroubi and Grochenig [l].Janssen [12] and Walter [17] proved results analogous to the Classical Sampling Theorem in that the sampling was uniform and at the “critical” (Nyquist) rate called for by the Classical Sampling Theorem. Djokovic and Vaidyanathan [6] developed several techniques for reconstruction from oversampled and “bunched” data. The application of these techniques was justified by the authors in [7], [9]. In this paper we are interested only in some of the more primitive sampling techniques as these are sufficient to demonstrate the translation determination algorithm of section 5.
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4.1 Uniform Critical Sampling Janssen [12] introduced the notion of translated integer sampling - determining a signal f from its translated integer samples ( f ( t o + k ) } k for some fixed to. If f(t) = ckcp(t - k) E V(cp),then taking the Zak transform of both sides gives Z f ( t ,[) = C([)Zcp(t, If there exists t o such that infg IZcp(to, # 0, then (Zcp(t0,[))-' E L"([O, 11) and we may write C(E)= Zf(to, [)/zcp(to, E) from which we may obtain the coefficients Ck. Define a sampling function St, bv
Ck
r).
I)[
and a sampling operator St, by & , f ( t ) = E l f ( t o be reconstructed from uniform samples by
+ Z)Sto(t- 1). Then f may
f(t) = C C k [ P ( t - k) = &,f(t).
(8)
k
+
S,, is to-cardinal in the sense that S t o ( t ~ k) = b k . Janssen [12] computes these sampling functions for several examples. This scheme also provides flexibility in that the shift t o may be thought of as a parameter against which the aliasing performance of the scheme can be optimized [12], [7], [9]. Several inadequacies of this scheme are studied in [9]. Principal among these is the fact the cardinality of the scaling function cp and sampling function St, are competing requirements which, at least in the case t o = 0, are incompatible. Also, if Zcp(t0, has a zero, the scheme becomes unstable. Since every continuous Zak transform has a zero (a consequence of quasiperiodicity) this is a genuine concern. a)
4.2 Oversampling
One technique for overcoming the problems associated with integer sampling or translated integer sampling is to oversample, that is, to sample at a rate higher than once per shift. The cost of a higher sampling rate is often compensated by improved properties of the sampling functions. For example, they can inherit compactness of support from the generator cp. Furthermore, added flexibility can be exploited to improve performance of implementations, e.g. [7], [8]. The most primitive of the oversampling schemes is as follows. We assume that cp generates a PSIV(p), L 2 2 is an integer and 0 I t o < tl < . . . < t ~ - 1< 1 are such that L-1 m=O
for all [. Let t be the oversampling vector t = ( t o , t i , . . . ,t~,-i).If f(t) = C kckcp(t - k) E V(p) then Z f ( t ,[) = C([)Zcp(t,[). Putting t = tz, multiplying both sides by F(tl, [) and summing over Z gives zf(tz,S)%(ti, =
cf=il
r)
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J.A. Hogan and J.D. Lakey
C ( ( )Cf't (Z'p(tj,E)I2. Dividing both sides by the non-vanishing Fourier series on the left-hand-side of (9) gives
from which the coefficients Ck may be obtained. Define sampling functions Sl(t) (0 5 15 L - 1) by
and an oversampling operator Ot by O t f ( t )= Cf=&'C , f(tz Then f may be reconstructed from its samples via
+ m)Sz(t- m).
The assumption (9) which ensures the stability of the scheme, is justified for sufficiently large L. Estimates of lower bounds on L are contained in [7], [9]. In general, equation (10) which defines the sampling functions does not ensure the compactness of their supports, even when 'p is compactly supported. More sophisticated oversampling techniques which give rise to compactly supported sampling functions are included in [6] and justified in [9].
5 Determining the Translation of Oversampled Data Now we turn to the problem of determining the translation of oversampled data. Preliminary results of this work appear in [8]. Recall that if infg IG(()I > 0 then the integer sampling operator SOis well-defined. Suppose f E V('p)and let f a ( t ) = x k f ( k a)So(t- k). The cardinality of SOgives f a ( l ) = f ( l + a ) for all integers 1 and it is clear that integer sampling, or even translated integer sampling, is insufficient for the calculation of the translation of sampled data since for every translation a (except, of course, those for which infg I@(()I = 0) and every f E V('p),there exists fa E V(p) with f a ( Z ) = f(1 + a ) (1 E Z). The redundancy of oversampling schemes may be used to determine a. With consideration of the oversampling scheme, a signal f E V(p) may be recovered from its samples f(a tz k) with 0 5 a < 1 and 0 5 t o < tl < - - t ~ - 1< 1 fixed and Ic E Z. Transmission of the signal is equivalent to the less costly transmission of the samples f(a tl k) (0 5 1 5 L - 1,k E Z). Suppose that the receiver knows that the data stream she is receiving consists of samples of a signal from a known PSI space V('p),and that the sampling nodes t l are also known, but the shift a is unknown. Before using either of the sampling schemes above (or more generally, those of [6] or [9]) to reconstruct
+
+ +
+ +
Non-Translation-Invariancein Principal Shift-Invariant Spaces
481
the signal, the receiver must first determine the value of a. One way this might be achieved is as follows. With the vector t = ( t o , t l , . . . ,t ~ - l E) [0, l)L,recall that the sampling operator Op+t may be written as Op+tf(t) = Ckl', El f(P+tm+Z)Si(t-Z)
+
Of course we require that infc Cf=-'IZp(P t l , t)I2> 0. The receiver takes - ~ , it into a doubly-indexed sequence b: = a data stream { U ~ } Ztransforms a l + k L (0 5 Z 5 L - 1,k E Z)and generates a signal fa E V(p) via L-1
m=O
1
Notice that the notation has been slightly abused. The operator Op+t is acting here on a sequence, whereas it usually acts on continuous functions. After sampling, the two notations agree. Given that b; = f(a+ ti k) for some f = Ckckp(t- k) E V(p), we attempt to determine a. With this assumption on the sequence b: we have
+
L-1
Op+tb(t) =
C 1f(a+ t m + l)si(t 1). -
m=O
(12)
1
When a = 0, O:$,b(,8+tj + k ) = f ( a + t j + k ) for all 0 5 j 5 L - 1 and all k E Z.Computing the Fourier series of both sides of this equation and using (12) gives
c
L-1
0=
Izcp(P+tl,t)lZ[ZOp+tb(P+tj,E)- Zf(a++tj,E)I
l=O
c
L- 1
=
c(<> [z~p(a + t m , t)zcP(~+ t m , OZp(P + t j , E ) m=O
- z ~ ( a + t j , t ) I Z ( ~ +tm,E>121 (P or equivalently,
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J.A. Hogan and J.D. Lakey
for all E supp (C). Consequently, for each 0 _< j 5 L - 1, there exists a function A j on [0,1] such that
on supp (C). Equating the second entries of the vectors on the left and righthand sides of (13) gives Zp(P t j , f ) = Aj(#) Cf;;=',JZcp(P t,, #)I2, hence
+
+
IZcp(P+tj,#)I2 =
c
: ; c 14(o12(c::
L-1
IAj(E>I2 =
j=O
Izcp(P+tm,E)12)2,i.e.,
c (:::
)
lZ4P+tj,#)l2
.
(14)
-l
Similarly, applying the Cauchy-Schwarz inequality to the first entries of the vectors on the left and right-hand sides of (13) gives L-1
/L-1
L- 1
/L-1
zk;:
\
+
or equivalently, 1 5 C f l t IAj(<)12 IZcp(P tm,E)l2= 1 where the last equality is just (14). Hence, the inequality in (15) is in fact an equality and in the application of Cauchy-Schwarz, there must be equality, ie., Z p ( a t,, t )= p(()Zcp(P t,, () for some function p on [O, 11. Clearly,
+
+
so that for all m, n we may write
and consequently
for all m, n. The question is then how we can use this information to produce an effective algorithm for the shift associated with sampled data. These equations hold on the support of C , a set of positive measure. If cp is compactly
Non-Translation-Invariancein Principal Shift-Invariant Spaces
483
supported, then the left-hand-side of (16) is a trigonometric polynomial and the fact that it vanishes on a set of positive measure implies that it will vanish everywhere. Multiplying both sides of (16) by C ( t )gives Zf(a+&,t)Z(p(p+ tm,J)-Zf(a+t,,J)Z'p(P+tn,J) = 0, and computing the Fourier coefficients of both sides yields 0 = F ( n ,m;p)= ~ l [ b ~ ' p ( P + t , + p - Z ) - b ~ ( p ( P t t , f p - l ) ] . The quantity on the right-hand-side will be zero when a = p, so as a practical device we compute the function
C [ ~ Y+V tm + ( PP 1) - ~ T Y ( P+ t n + P
F ( P )=
-
( m
n
P
I
I')
- 1>1
ll2
)
(17) and as our estimate of a , choose a value of p which minimizes F(P). Unfortunately this does not necessarily provide a unique estimate of Q since there is no reason t o believe that F has a unique zero. To force the zero of F t o be unique, we need to impose more conditions on 'p.
Fig. 1. Synthetic signal in V(p) with cp the Daubechies Dq scaling function. On the right is the shift determination function F for this data.
4 3
41 0
ID
D
rn
a
ca
m
m
e4
rn
Irn
Theorem 4 Let 'p be a compactly supported orthogonal scaling function in L z p ( a ) (a > 0 ) and 0 5 t o < tl <
F(P)=
(c,zncP
1x1 f(a
+ t n + ~ ) C P (+P tm + P - 1) 1/2
-
T h e n F( P) = 0
f(a+trn+ l > V ( P + t n + P -1 ) l2 )
.
a = p.
Proof. By extending equation (16) t o the complex plane we have
+
+
+
ZCC~(C. tn, z ) ~ c v ( P t m , 2) = Z C C P ( tm, ~ ~
) ~ c v (+Ptn, 2).
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J.A. Hogan and J.D. Lakey
The co-primality condition and the fact that Z,cp(z,1) = 1 for all x gives Zccp(cr t,, z ) = Z~cp(p t,, z ) and hence that cp(a t , Ic) = cp(P t , k) for all Ic E Z and 0 5 n 5 L - 1. By the dilation equation (3) satisfied by 'p we have cp = cp (,-)for all Ic and all n. The Lipschitz continuity of cp implies that cp is constant, but since cp is compactly supported, we see that $0 = 0.
+
+
+ +
+ +
(w)
The conditions of Theorem 4 can be checked for particular examples of pairs (cp, t). This is easily done when, for example, cp is the Daubechies 2cp scaling function supported on [0,3], L = 2 and t o = 0, tl = 1/2. On the lefthand side of figure 5 , we consider a signal f in the multiresolution space V(p) supported on [0,100]. The signal was translated by 1/4 then sampled at the integers. On the right the function F of Theorem 4 is computed for this data using :-integer sampling. As expected, F has a well-defined zero at E = 1/4, thus demonstrating the efficacy of the algorithm for the determination of the translation of sampled data and the initialization of sampling algorithms.
References 1. A. Aldroubi and K. Grochenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Review 43 (2001), 585-620. 2. N. Atreas, J.J. Benedetto and C. Karanikas, Local sampling for regular wavelet and Gabor expansions, preprint (2002). 3. A. J. Bastys, Orthogonal and biorthogonal scaling functions with good translation invariance characteristic, Proc. SampTA, Aveiro (1997), 239-244. 4. A. J. Bastys, Translation invariance of orthogonal multiresolution analyses of L2(R), Appl. Comp. Harm. Anal. 9 (2000), 128-145. 5. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Ser. Applied Math., vol. 61, SIAM, Philadelphia 1992. 6. I. Djokovic and P.P. Vaidyanathan, Generalized sampling theorems in multiresolution subspaces, IEEE Trans. Sig. Proc. 45 (1997), 583-599. 7. J.A. Hogan and J.D. Lakey, Sampling f o r shift-invariant and wavelet subspaces, Proc. SPIE, San Diego, .(2000), 36-47. 8. J.A. Hogan and J.D. Lakey, Sampling and aliasing without translationinvariance, Proc. SampTA, Orlando, (2001), 61-66. 9. J.A. Hogan and J.D. Lakey, Sampling and oversampling in shift-invariant and multiresolution spaces I: validation of sampling schemes , preprint. 10. C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1998), 628-666. 11. A.J.E.M. Janssen, The Zak transform: a signal transform for sampled timecontinuous signals, Philips J. Res. 39 (1998), 23-69. 12. A.J.E.M. Janssen, The Zak transform and sampling theorems for wavelet subspaces, IEEE Trans. Sig. Proc. 41 (1993), 3360-3364. 13. W.R. Madych, Some elementary properties of multiresolution, in Wavelets: A Tutorial in Theory and Applications, ed C.K. Chui, Academic press, San Diego (1992), 259-294.
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14. R.J. Marks 11, “Introduction to Shannon Sampling and Interpolation Theory,” Springer, Berlin, 1991. 15. C. Shannon, “The Mathematical Theory of Communication,” University of Illinois Press, 1949. 16. P.P. Vaidyanathan, Sampling theorems f o r n o n bandlimited signals: theoretical i m p a c t and practical applications, Proc. SampTA, Orlando, (2001), 17-26. 17. G. Walter, A sampling theorem f o r wavelet subspaces, IEEE Trans. Info. Th. 38 (1992), 881-884. 18. A. Zayed, “Advances in Shannon’s Sampling Theory,” CRC Press, Boca Raton, 1993.
Time-F’requency Spectra of Music James S. Walker’ and Amanda J . Potts’ Department of Mathematics University of Wisconsin Eau Claire, WI 54702-4004 USA walkerjsQuwec.edu Department of Mathematics University of Nebraska Lincoln, NE 68588-0130
USA Summary. We analyze some musical passages using hanning windowed spectrograms. Preliminary results indicate that the highest peaks in the spectra for these musical passages do not obey a l / f power law, contrary to results reported by some other workers. Further study is needed to determine the exponents for the spectra for a variety of musical passages from around the world.
Introduction Initial work on analyzing musical signals was begun in [l],[3], [5], and [ 6 ] . The mathematical model for a musical signal Y ( t )proposed in [l]is the finite sum
where h(.) is a taper function of duration oj. Formula (1) has constants (amplitude), 7-j (offset in time), and X j (frequency). The Fourier transform p(X)of Y ( t )is then
V,
where 6[oj(X - A j ) ] is a narrow function with peak centered on frequency X j . When 1 (3) %2 aj2 cx , all j
4
we say that the signal Y has a l/f spectrum.
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James S. Walker and Amanda J. Potts
In [l],statistical data are gathered to establish the claim that musical signals have l/f spectra. In this paper we shalI analyze several musical passages using hanning spectrograms to obtain time-frequency portraits of those signals. Our preliminary results cast doubt o n the validity of (3) f o r these musical passages. Our musical signals include examples from non-Western music (whereas [l]examined only Western styles of music). The sound recordings we used are all available for downloading at the following website:
http://www.uwec.edu/walkerjs/TimeFrSpMusic """
""
Chopin
Bosavi rainforest
Gyuto monks
Fern
(4)
""
Fig. 1. Dynamic power spectra for four musical passages.
1 Hanning Spectrograms The basic tool we used to obtain time-frequency spectra for a musical signal is hanning spectrograms. For such spectrograms, a Hann windowing function
Tirne-Frequency Spectra of Music
489
Fig. 2. Complete dynamic power spectra for the Fern musical passage. w ( . ) is defined as follows: For more details on Hann windows, see [4]or [7]. w(t) =
{
g;5
+ 0.5 cos(nt),
for It\ 5 1.0 for It1 > 1.0
(5)
A hanning spectrogram is then obtained by the following three-step process. This process is discussed in more detail in [8]. Step 1: Discretely sample the signal Y ( t ) at the time values ( t k } , with fixed spacing At = t k - t k - 1 , to obtain the (digital) signal { Y ( t k ) ) . Step 2: Multiply the signal { Y ( t k ) } by M slidinn windows v
{ (%)} , with ij - ij-1 = 2564t. Step 3: Apply a 512-point FFT to each one of these windowed signals (centered on ij):
obtaining the hanning spectrogram,
With band-limited signals, if the time-values { t k > are close enough to each other, Steps 1 to 3 are invertible. With discretely sampled data {Y(tk)} to begin with (as in a .wav audio file), Steps 2 and 3 are reversible, i.e. the map
is invertible. See [8] for more details on the inversion of (9). The map in (9) is a Gabor transform. See [2] for a thorough treatment of Gabor transforms
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James S. Walker and Amanda J. Potts
and their inverses. When displaying these spectrograms, one typically takes the moduli-squared of all values, { 1% (X,)I2}, which are called dynamic power spectra. In plotting these dynamic power spectra, one displays values on a logarithmic intensity scale with larger values plotted as darker pixels. See Fig. 1 for dynamic power spectra for four musical passages [the ones available for downloading at website (4)]. In Fig. 1, the Chopin signal is a recording of a Chopin piano passage; the Bosavi rainforest signal is a recording of some members of a South American indigenous tribe singing; the Gyuto monks signal is a recording of Buddhist monks singing; and the Fern signal is a computer generated fractal-music signal. See Fig. 2 for a longer dynamic spectra for this last passage.
Chopin
Bosavi Rainforest
Fern
Gyuto
Fig. 3. Global transform amplitudes.
In Fig. 3 we graph the FFT magnitudes IP(X,)l for each of these four test signals. By inspection it is easily seen that, for each signal as a whole, the FFT magnitude is not proportional to l/X0.5, hence the power spectrum IP(x,)~~is not l / j . In Fig. 4(a) we show the dynamic spectra for the Chopin passage. In Figures 4(b) to (d) we show the magnitudes {IE(X,)I} for the times ij = 1.0,
Time-Frequency Spectra of Music
491
1.5, and 2.0, respectively. The least-square curves for the log-log values of
Dynamic Spectra
t M 1.0 sec, ~ / f ’ . ~
t M 1.5 sec, l p 4
t
M
2.0 sec, i/f3.6
Fig. 4. Analysis of Chopin passage.
the 5 highest peaks from the FFT magnitudes {IE(X,)I} are also plotted in Figures 4(b) to (d). The exponents at ij = 1.0, 1.5, and 2.0 are 2.6, 1.4, and 3.6, respectively. These exponents are not close to 0.5 as they would be if there were a l/f power law in effect. In Table 1 we show the estimated exponents for {Iq(Xm)[} at times fj = 0.5, 1.0, 1.5, 2.0, and 2.5. These exponents are widely at variance from the l/f-exponent of 0.5. This casts doubt on the proposed l/f power law for spectra of musical signals. 6
Table 1. Estimated Exponents y for
1810: l/f’.
Time (sec) Chopin Fern Bosavi Gyuto 0.5 1.8 0.0 2.8 1.6 1.o 2.6 1.8 3.5 0.5 1.5 1.4 2.1 0.4 0.6 2.0 3.6 0.2 13.9 2.6 0.5 3.4 2.5 2.4 0.6
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James S. Walker and Amanda J. Potts
2 Synthesized Sounds In the previous section we attempted to calculate exponents for a power law l / f Y and found that values for y are widely dispersed for four test signals. In this section we take a different tack. Suppose that a commonly recognized musical passage, such as the Chopin piano passage considered above, is subjected to a transformation which surely distorts it away from a l/f power law (assuming it has one to begin with). Given that the spectrogram has the form we subject it to two transformations: { 1% A(),
and
We then apply the spectrogram inversion procedure to each of the data in (10) and (11).This produces two sampled signals, denoted {Yz(tk)} and {Y&k)}, respectively. The signal {Y2(tk)} was saved as PowerTwo .wav, and the signal { Y a ( t k ) }was saved as Sqrt .wav, at the website listed in (4). The reader is invited to listen to these two .wav files. If they sound like music (maybe not great music, but still music), then this casts further doubt on the l/f hypothesis: Assuming that the Chopin passage Chopin.wav is l / f , then neither PowerTwo .wav nor Sqrt .wav are l/f signals.
Conclusion We have reported some preliminary results on determining exponents y, for a spectral power law l / f Y , for various musical passages. Further study is in order, such as increasing the number of peak values used in the spectrogram analysis, and obtaining further data on subjective impressions of synthesized sounds. Preliminary results cast some doubt on the l/f power law proposed by other workers.
References 1. Brillinger, D.R., and R.A. Irizarry. (1998). “An investigation of the higher-order spectra of music.” Signal Processing, 65, 161-79. 2. Grochenig, K. (2001). Foundations of Time-Frequency Analysis. Boston, MA: Birkhauser. 3. Lowen, S.B., and M.C. Teich. (1993). “Fractal renewal processes generate l/f noise.” Physical Rev., 47, 992-1001. 4. Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. (1992). NZLmerical Recipes, 2nd Edition. Cambridge, England: Cambridge University Press.
Time-Frequency Spectra of Music
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5. Voss, R.F., and J. Clarke. (1975). “‘l/f Noise’ in music and speech.” Nature, 258, 317-1 8. 6. Voss, R.F., and J. Clarke. (1978). “‘l/f Noise’ in music: music from l / f noise.” J. Acous. SOC.Am., 63,258-63. 7. Walker, J.S. (1996). Fast Fourier Transforms, 2nd Edition. Boca Raton, FL: CRC Press. 8. Walker, J.S., and J.F. Alm. (2002). “Time-Frequency Analysis of Musical Instruments.” SIAM Review, 44, 457-76.
Dynamics of Spectra of Toeplitz Operators S. Grudsky and N. Vasilevski Departamento de Matem6ticas CINVESTAV del I.P.N. Apartado Postal 14-740 07000 M6xico, D.F. MQxico [email protected], [email protected]
Summary. We consider Toeplitz operators Tix',for the certain classes of symbols, acting on a scale of the weighted Bergman spaces A!, X E [0, m). Motivated by the Berezin quantization procedure we study the dependence of the spectral properties of such operators on the parameter of the weight X and, in particular, under the limit procedure X --+ 00.
1 Introduction Toeplitz operators with smooth (or continuous) symbols acting on weighted Bergman spaces over the unit disk naturally appear in the context of problems in mathematical physics. Given a smooth symbol a = a ( z ) , the family of Toeplitz operators T, = { T i h ) } ,with h E ( O , l ) , is considered under the Berezin quantization procedure [l,21. For a fixed h the Toeplitz operator Tih' acts on the weighted Bergman space A:@), where h is the parameter characterizing the weight on A",lID). In the special quantization procedure each Toeplitz operator Tih' is represented by its Wick symbol &, and the correspondence principle says that for smooth symbols one has lim iih = a.
h-0
The same weighted Bergman spaces as in a quantization procedure appear naturally in many questions of complex analysis and operator theory. In these cases a weight parameter is normally denoted by X and runs through (-1, +m). In the sequel we will consider weighted Bergman spaces A; parameterized by X E (-1, +m) which is connected with h E (0, l ) , used as the parameter in the quantization procedure, by the rule X 2 = One of the important problems here is to study of the behavior of different properties of Tix) in dependence on A, and especially their limit behavior under X -+ 00.
+
i.
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S. Grudsky and N. Vasilevski
In the paper we survey the results from [3, 4, 51 describing the behaviour of the spectra under X + 00. For further results, details, and proofs consult ~3~4~51. The authors would like to express their deep gratitude to 0. Grudskaia for her help in preparation of the figures in the paper.
2 Representations of the Toeplitz Operators It seems to be quite impossible to get a reasonably complete answer to the problem mentioned for general symbols. At the same time the recently discovered classes of commutative C*-algebras of Toeplitz operators on the unit disk suggest the classes of symbols for which a satisfactory complete answer can be given. Recall in this connection (for details see [6, 71) that all known cases of commutative C*-algebras of Toeplitz operators on the unit disk are classified by pencils of (hyperbolic) geodesics of the following three possible types: geodesics intersecting in a single point (elliptic pencil), parallel geodesics (parabolic pencil) and disjoint geodesics, i.e., all geodesics orthogonal to a given one (hyperbolic pencil). Symbols which are constant on the cycles, the orthogonal trajectories to the geodesics forming a pencil, generate in each case a commutative C*-algebra of Toeplitz operators. Each pencil of geodesics can be reduced to a certain model case by mean of a Mobius transformation. In the sequel we will consider only these model cases and Toeplitz operators with symbols constant on the corresponding cycles, which for elliptic, parabolic, and hyperbolic pencils are as follows: Toeplitz operators on the unit disk with radial symbols, Toeplitz operators on the upper half plane with symbols depending on y = Im z , and Toeplitz operators on the upper half plane with symbols depending on 0 = arg z , respectively. The key feature of symbols constant on cycles, permitting us to get much more complete information that is obtained studying general symbols, is as follows. In each case of a commutative C*-algebra generated by Toeplitz operators the Toeplitz operators admit the spectral type representation, they are unitarily equivalent to multiplication operators. We pass now to the exact definitions and statements. Let D be either the unit disk D,or the upper half-plane IT in @. Introduce the weighted Hilbert space L a ( D , d p D , x ) , where the measure dpD,x is given correspondingly by X+1
dPD,A(Z) = (1 7r
or
dPn,x(z)= 7r -k (21m 2)’ d z d y .
Further, let A:(D) denote the weighted Bergman space defined to consist of functions analytic in D belonging to & ( D , d p o , ~ )The . orthogonal Bergman projection B g ) of L2(D,d p o , ~onto ) the weighted Bergman space A i ( D ) has in either case the form
Dynamics of Spectra of Toeplitz Operators
497
or
For a symbol a = a ( z ) , z E is defined as follows
D,the Toeplitz operators TiA’ acting on d”,D)
- B(Wa TiA) 9 D 9,
‘PEdW).
Theorem 1 Given any model pencil and a symbol a E L,(D), constant o n corresponding cycles, the Toeplitz operator TiA’is unitarily equivalent t o the multiplication operator y a ,I ~, where: I n the parabolic case: a = a ( y ) , y E R+, Ta,AI : L2(R+)+ Lz(R+),and
In the elliptic case: a
= a ( r ) , r E [0, l),
ya,a,xI: 12
4
12,
and
In the hyperbolic case: a = a ( 8 ) , 8 E (O,n), rya,J : Lz(R) 4 L2(W), and
3 Continuous Symbols Let E be a subset of R having +oo as a limit point, and suppose t,hat, for each X E E , there is a set MAc @. Define the set M, as the set of all z E @ for which there exists a sequence of complex numbers ( Z , ) , ~ N such that (i) for each n E W there exists A, E E such that zn E MA,, (ii) limn+, A, = +m,
(iii)z = limn+, z,. We will write
M,
= lim
A-++m
MA,
E X 4 +a. and call M , the (partial) limit set of a family { M A } A ~when The a priori spectral information for general L,-symbols (see, for example, [l],[2]) says that for each a € L,(D) and each X 0
>
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S. Grudsky and N. Vasilevski
sp T,(’) Given a symbol a E L,(D),
c conv(ess-Range a ) . constant on cycles, the Toeplitz operator T,(’1
is unitarily equivalent to the multiplication operator y a , ~ 1where , ~ ~ is given , x in Theorem 1. Thus we have obviously SPT,“’ = M A(U ) ,
where Mx(a) = Rangey,,’.
Theorem 2 Let a be a continuous symbol constant o n cycles. Then X++W lim
spT,(’) = M,(a)
= Rangea.
(2)
Note, that Rangea coincides with the spectrum s p a 1 of the operator of multiplication by a = a(y), thus the another form of (2) is lim sp~,(’) = spa1.
A++,
We illustrate the theorem on two continuous symbol (both are hypocycloids)
presenting the images of the sequence yal,’(n) and the function y,,,x(() the following values of A: 0, 5 , 12, and 200.
The images of T ~ ~ and , x ya2,x for X = 0.
for
Dynamics of Spectra of Toeplitz Operators
4.5
0
0.5
1
1.5
T h e images of
0
0.5
1
T h e images of
4.5
0
-yal,xand - y a 2 , ~for X
1.5
4.5
0
0.5
1
1.5
1
1.5
= 5.
0.5
for X = 12.
-yal,xand -ya2,x
. . . . . . . . ,. . . . . . . . . . .
. . . . . . . . . ,. . . . . . . . . . .
........................
4.5
0
0.5
1
T h e images of
1.5
4.5
-yal,x and -ya2,x
0
0.5
for X = 200.
1
1.5
499
500
S. Grudsky and N. Vasilevski
4 Piecewise Continuous Symbols Let a be a piecewise continuous symbol constant on cycles and having a finite ) union of the straight number m of jump points. Denote by U ~ l I j ( athe line segments connecting the one-sided limit values of a at the jump points. Introduce
g ( a ) = Rangea U
(6
'.(a)) .
j=1
Theorem 3 Let a be a piecewise continuous symbol constant on cycles. Then lim spAT,") = M , ( u )= E ( a ) .
A-00
We illustrate the theorem on the following piecewise continuous symbol which has six jump points
a(8)=
I
expi [-: + Ir . E l , 6 E [o, $) + e x p i [ : + T&. ( -=7e -~)], 8 E [ E 2") expi (7; 73 - 2 ) ] , 8 E [8;T 73,,T ) 37r 4a expi [-$ $ . - 3 ) ] , 8 E [7, T ). 4a 57r expi [-: :( 8 E [T, T) 1 3 expi [-: :( - 5 ) ] , 8 E exp (-i:) ,
[-a + % . +
+ (z 41, +9.
We present the image of the symbol a = a ( 8 ) , the image of ya,x for the following values of A: 1, 10, 70, and 500, as well as the limit set Adm(.).
4 2
0
02
04
The symbol a ( 8 ) and the function ' y a , for ~ X = 1.
06
08
1
Dynamics of Spectra of Toeplitz Operators
501
The function 'ya,x for X = 10 and X = 70.
The function ya,x for X = 500 and the limit set M w ( a ) .
For general L,-symbols, apart from the a priori information (l),we have obviously lim spATiA) = M,(a) C conv(essRangea). A+,
At the same time the collocation of M,(a) inside conv(ess Range a ) may essentially vary. We give a number of examples illustrating possible interrelations between these sets. The examples are given for the parabolic case, but the results are the same for the either case, parabolic, elliptic, or hyperbolic. Example 1. Let a ( y ) E C[O,+m].Then according to Theorem 2,
M,(a)
= Range a
(= ess Range a).
where a1, E C and a1 # a2. Then according to Theorem 3 M,(a) coincides with the straight line segment [cq,a21 joining the points ~1 and a2, whence
M,(a)
= conv(ess Range a ) (= conv(Range a ) ) .
S. Grudsky and N. Vasilevski
502
Example 3. Let a(y/2) = where
{
Q l ,t
E W ) ,
t E [1,2), Q3, t E [ 2 , 4 , Q2,
are different points from @. Then by Theorem 3 we have " M,(a) = [ a l a , 21 u [QZ, a31 c aconv(Rangea).
a1, a2, a3
Example 4. Let
al,a2,a3 be
as above, and
By Theorem 3 the set Mm(a) coincides with the triangle having vertices Q1, Q 2 , a3
MW (a)= [QI ,~
2U 1 [QZ, a31 U [a3,a41
a
= conv(Range a).
5 Oscillating Symbols We consider here a discontinuity of the second kind, the oscillating symbols. To be more precise, the following two model situation will be considered: a strong oscillation and a slow oscillation in the parabolic case. In spite of their qualitative identity, an oscillation type discontinuity, the results differ drastically.
Theorem 4 (Strong oscillation). Let a(y) = e2iY, then Rangea = T and
M&)
= D.
We note that for each fked A, the image of -ya,x looks like a spiral leaving the point z = 1 and tending to z = 0 as 2 tends to 0. Moreover, when X is growing the branches of a spiral became closer to each other.
Theorem 5 (Slow oscillation). Let a(y) = ( 2 ~ then ) ~ ~Rangea = 11' and M&) = T. Theorems 4 and 5 can be generalized to a wide class of strong and slowly oscillating symbols. For example, if al(y) = (2y l)i,then M m ( a l ) = 'IT. For a fixed X the image of -yal,x is a spiral leaving the point z = 1 and tending to the limit circle with the radius ( r ( X 1 i ) / r ( X 1)1and centered at origin. We illustrate the above presenting the images of the function -ya,x for two oscillating symbols
+ +
+
a l ( y ) = (1 2 ~ =)ei'n(1+2y) ~ and
+ +
a a ( y ) = eiZy, y E [0,co),
Dynamics of Spectra of Toeplitz Operators
503
and for the following values of A: 0, 10, and 1000. Both of these symbols are continuous at the point y = 0 and have an oscillation type discontinuity at infinity, both of them are of the same form ak(y) = e i v k ( Y ) ,
IC = 1,2,
where the corresponding functions 'pk (y) are continuous and growing in [O,+m) with ' p k ( 0 ) = 0 and 'pk(+m) = +m. The only difference between them is in speed of growth at infinity. And this difference leads to a drastic difference between the spectrum behaviour of the corresponding Toeplitz operators. 1
08 06
04 02 0
4 2 4 4
4 8
4 6
-1 1
4 8
4 8
4 4
4 2
0
02
04
08
01
1
The functions -yal,x(z) and yaz,~(z) for X = 0.
The functions -yal,x(z)and yaz,x(z)for X = 10.
S. Grudsky and N. Vasilevski
504
. 0 . ...,. :.
-1
-0.8
.
.
.. .
. .. .
. .. .
. ..
.
.
.
.
.
. I. . . . .:. ....j . . . ...;... ... .;.......~.. .. .. .
.....I....
-0.8
.
-0.4
4.2
0
0.2
0.4
0.8
. .
. ...{........
0.8
I
The functions -yal,x(x) and -ya,,x(x)for X = 1000.
Acknowledgments This work was partially supported by CONACYT Project U40654-F, MQxico. The first author was partially supported by CONACYT Grant, CQtedraPatrimonial, Nivel 11, No 010186.
References 1. F. A. Berezin. Quantization. Math. USSR Izvestzja, 8:1109-1165, 1974. 2. F. A. Berezin. General concept of quantization. Commun. Math. Phys., 40:135174, 1975. 3. S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators with radial symbols. Integr. Equat. Oper. Th., 20:217-253, 2004. 4. S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators on the upper half-plane: Hyperbolic case. Bol. SOC. Mat. Mezcicana (3), 10:119-138, 2004. 5. S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case. J. Operator Theory, 52:185-204, 2004. 6. N. L. Vasilevski. Toeplitz operators on the Bergman spaces: Inside-the-domain effects. Contemp. Math., 289:79-146, 2001. 7. N. L. Vasilevski. Bergman space structure, commutative algebras of Toeplitz operators and hyperbolic geometry. Integr. Equat. Oper. Th., 46:235-251, 2003.
Operator Equalities for Singular Integral Operators and Their Applications Oleksandr Karelin Universidad Politecnica de Pachuca Hidalgo MQxico skare1inQueah.reduaeh.mx Summary. The article consists of two parts. In the first, we establish a relation between the singular integral operators on the unit circle with a model orientationreversing shift operator and the singular integral operators on the unit circle with a model orientation-preservingshift operator. In the second part, a Riemann boundary value problem with shift is considered. We reduce this problem to a matrix characteristic singular integral operator without shift.
1 Introduction We denote the Cauchy singular integral operator along a contour
C by
In the paper [l]we constructed a similarity transformation
F ’ A F = D, (1) between the singular integral operators A with the rotation operator WT through the angle 27r/m on the unit circle If, acting on the space &(If), and a certain matrix characteristic singular integral operator without shifts acting on the space Ly(If). For m = 2, ( W ~ p ) ( = t ) cp(-t), A = a0.h
+ bosT + ~ I W +T ~ I S T W T ,A E [~52(If)], D = U I T + VST, D E [Li(T)]
the coefficients ao(t),bo(t),a l ( t ) ,bl ( t )are bounded measurabIe functions. We denote by [ H I ,H2] the set of all bounded linear operators mapping the Banach space H I into the Banach space H2, [HI]= [HI,H I ] .
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Oleksandr Karelin
Now we describe the operator F . W e d e f i n e T + = { z : IzI=l, O < a r g z < r } , T - =T\T+. Let L and 7 be contours, L c 7. The extension of a function f(t), tEL, to ;?.c, by the value zero, will be denoted by ( J q , f ) ( t ) , t ~ 7The . restriction of a function cp(t),t E 7, to L will be denoted by (C,cp)(t), t E L. The operator F E [L;(T+), L2(T)] is given by F=MT,ZGNT+ where
MT+
(F:)
+
= J ~ - c p i W<1J~-cp2,
( N T + W )=
MT+E [L%+),Lz(T)],
w2>,(NF;o(t) = r(t9,
NT+ E [ L m ,G(T+)I,
N-l E [L;(T+),L m I *
The following equalities hold: ker AJL2(T)= ker DF-lJL;(T) = F ker DJL#L"). The vertical bar denotes the spaces on which we consider the operators. We denote by LY(,C,e) the Banach space of all vector functions, such that (e(z))'/2f(z)belongs to L?(L). The norm in LY(L,e) is given by I1f IIL T ( L , e ) = II P1l2f IILT(Lc)* Consider now the operator
B
=a h
+ bQw + CSR+ dQwSw,
acting on the space Lz(R), where R is the real axis, the coefficientsa(z),b(z),c ( x ) ,d(z) are bounded measurable functions,
6, p are real numbers, S2+p > 0. The function a(%)is a Carleman (a(a(2)) = shift. In the paper [l]acting by invertible operators from the right hand and lefthand side we reduced the singular integral with linear fractional involution B to a matrix characteristic singular integral operator without shift, X) orientation-reversing
Applications of Operator Equalities
507
Nw+E[L;(R+t- 4 ),L;(R+)I, Mw+E[G(R+),Lz(R)I. The connection between the coefficients of the operators B and Dw, are
QEL52(R)I,PEP%+)],
7
given by the formulas: 1 2
( w(t) = (
u(t)= -
( c + d)(C+(tN - ( c + d ) ( C - W ( a ( c d)(C+(t)) (c d)(C-(t)) (a
+
1 2 (a - b)(C+(t))- ( a - b)(C-(t)) ( c - d)(C+(t))
where
I+@)=
+
x2fi 21 &+l
-x2fi 7
C-(t> =
1
+ b)(C+(t)) - ( a + b)(C-(t))
+ + + b)K+(t)) + (a + b)(C-(t)) (a - b)(C+(t))+ (a - b)(C-(t)) ( c - d)(C+(t)) + ( c - d)(C-(t))
+ x1 ,
- (c-
'
)?
d)(C--(t))
tER+.
The following equalities hold ker BI L2 (R) = ker Dw,F-' IL i (R+,2-3 ) = F ker Dw+ILz (R+, x- ) . We will call the formulas (1) and (2) operator equalities. The above results allow us (in section 2) to establish a relation between singular integral operators on the unit circle with a model shift operator (Vcp)(t)= c p ( i ) and singular integral operators on the unit circle with a model shift operator (Wcp)(t)= cp(-t) and (in section 3) t o transform a Riemann boundary value problem with shift to a matrix characteristic singular integral equation without shift.
2 Relation between Singular Integral Operators with Orientation-Reversing and Orientation-Preserving Shifts Consider the singular integral operator
with bounded measurable coefficients a ~ ( t )b ,T ( t ) , m(t),d T ( t ) . Using the operators
Oleksandr Karelin
508
AE
[ J 5 2 ( W ) ,L2(1")1, A-1 E
[L2(T),L2(W)1
we reduce the operator Bp. to an operator on the real a x i s
B~ = A - ~ B ~=Aa d R
+ bRww+ asR+ dRwRs~,BR E [ L ~ ( R ) ]
By the operator equality (2) we reduce the operator BR to a matrix characteristic singular integral operator
773~3 = DR+,DR+= UR+ IR+ + Q+SR+, DR+E [L;(R+, 2-t )] Using the operators
where
hw, = 2-3,pa+ ( t )= (t + 1)1/2(t - 1)-1/2,
+
(to avoid possible confusion, we write the lower index in the notation of these operators: A+ and AT1 ) we reduce the operator Dw+ to the following operator
Dlr,
+
= A+DR+AT1 = m+h+ W,ST+,
DT+ E [L3T+,plr,)I.
Then we extend DT+by the identity operator to the operator D i E [Li(T,p l r ) ] , m = m+ on the whole unit circle
Di = JT+ Car- IT + Jar- DT+CT+= ii~Iar+ %ST, where GT = XT-
+ JT- UT+ GT = JT- 2$+ 7
and X T - is the characteristic function of the contour T-. By the operator equality (1) we reduce the matrix characteristic operator 0: to a scalar operator with the rotation operator WT
We have
Theorem 2.1 The following equalities hold
A,
=F
+
( J ~ + c ~ - IJ ~ - A + ~ - ~ A - ~ B ~ A F A F ;- ~~ c, ~ + )
B~ = ~ 3 - l - 1 ~( c~*1+ F - ~ A ~ F J ~A- +) F A - ~ , kerAT = FJT-A+F-'A-'kerBp.,
kerBT = FJT_A+F-lAjlkerz?T,
dim kerBT = dim kerAp..
Applications of Operator Equalities
509
3 Riemann Boundary Value Problem with Shift We consider the problem of determining an analytic function F ( z ) in the strip z : -1 5 3 z I +1 subject to the functional relation
+
A ( x ) F ( z i) + B ( x ) F ( z- i)
+ C ( x ) F ( x )= H ( x ) ,
(1)
where x E R, the coefficients A ( x ) ,B ( x ) ,C ( x ) are bounded measurable functions, and H ( x ) E L2(R);we assume as well that F ( x i) E L2(R),F ( x - i) E L2 (R). Following [3], we introduce the functions
+
f(x) = (v-’F)(x),
[ e x ~ ( x+>e x ~ ( - x ) ] f ( x ) , Y(x) = (Vfe>(x),
fe(x) =
where V is the Fourier transform:
From 1
v-cosh x
=
/-*
*
2 . [cosh -42
l
,
it follows that
and taking into account that
+
V - ~ F ( Z iy) = f(x)exp(-yx),
we have
-1 < Y < 1,
00
F ( x + Zy) =
/
Y (t)dt cash 5 (t - z ) , z = x + i y .
-W
The limit values of the function F ( z ) are given by the formulas [3]: 1 2
F ( x f 2) = -Y(z)
-W
Thus, the problem (1) is reduced to the integral equation
510
Oleksandr Karelin
Changing the variables in (2) 1
t = -1nT
1+7 , x = - l1n - 1 + E 1-7n 1-J’
we obtain that the problem (1) is equivalent to the integral equation
where
-1
is the singular integral operator with local singularities at the endpoints. The operator (4) is defined on j = (-1,l). We extend it to the whole real axis R. We denote by r ( Q , g [ H )the set of all solutions from the Banach space H of the equation Qncp = g , 0 E [ H ] , gE H and let XL:be the characteristic function of the contour C.
Theorem 3.1 1. T h e operator U, can be represented in the form
Applications of Operator Equalities
where
KA = cilrpIw
511
+ h S w + &TwSw,
6w = (XIw\, + Jw\,a,), G2 = (JR\,CJ), JR = (Jw\,d,). 3. T h e operator K3 i s invertible o n the space L z ( j ) i f and only i f the operator KA is invertible on the space Lz(IW) . Proof. Representation ( 5 ) is obtained from 1 1-st
--
1 1
z t - ;1 '
Formula ( 6 ) follows from the fact that if (p, is a solution of the equation K,cp = 0, then the function J~\,cp, is a solution of the equation KA+ = 0. On the other hand, if the function cpw is a solution of the equation KA+ = 0, then it possesses the property C q 3 ' p= ~ 0, and hence the function (p, = C,cp, is the solution of the equation K,cp = 0. Formulas (7) and (8) are proved analogously. Item 3 can be proved if we indicate the type of inverse operators . Thus, if K;' is inverse to K,, then the operator 1-1-
(KR)
- XR\JR
+ Jw\,K;lc,(lw
- KAXW\,IW)
is inverse to the operator KA. If (KA)-l is inverse to KA, then the operator
K;'
= C,(KA)-l Jw\,
is inverse to the operator K,. Indeed,
and
K,-lK3 = c3(KA)-1Jw\3 . C,KAJw\, = C,(K,)1 - 1 =wG)-lK;x,
Jw\,
=
c,x,Jw\,
x, K w ~w\gJ
= 13.
The relations KA(KA)-' = Iw K3KF1= I3 are proved analogously. Note that x~\,(K;)-l Jw\, = 0. This follows from the fact that xw\,KAcp = 0 if and only if q , ( p = 0. Consider now the case of operators acting on R+ = (0,+m). Assume that the operator
512
Oleksandr Karelin
acts on the weighted space LF(R+, QR+), where
the operators SR+and UR+belong to [L,”(R+,a,)]; the coefficients aR+ (x),a+(x),dR+ (x)are measurable and bounded; the operator
Aik = (XR\R+ belongs to
+ JR\R+aR+)IR + (JR\R+Cw+)SR + (JW\R+dR+)WR~LQ
[LF(R,~R)] and the weight functions are connected as follows:
C R + e R = eR+;gR, E
L;(R+,m+).
Theorem 3.2 The operator UR+ can be represented in the form
UR+
= C R + W R S R JR\R+ 7
where (Ww’p)(x)= cp(-x); the following relations hold r(AR+,gR+lL;(Rt, QR+> = cR+r(Aik, JR\R+swlL;(~,
ew)L
JR\R+r(AR+,sw+lL;(R+,e R + > = r ( 4 JR\R+swIL,(~, , a); the operator AR, is invertible o n the space L,“(R+,elw,) if and only if the operator A& is invertible o n the space L,”(R,a). This theorem is proved analogously to Theorem 3.1. Applying the operator equality (2) to the equation ( K i ~ p ) ( x=) Jnp\JgJwe have
Theorem 3.3 The problem (1) of determining an analytic function F ( z ) in the strip { z : -i 5 3.z 5 +i} subject t o the functional relation
+ + B ( x ) F ( x- i) + C ( Z ) F ( X=) H ( x ) ,
A ( x ) F ( x i)
is equivalent t o the matrix characteristic singular integral equation B R + $ = j R + , B R + = X K i F = GR+IR+ + ~ , S R , , D R ,
where
ow,
and
= XJR\,S,,
h+E L;(R+,t-i)
E
[L;(R+,t-i)] (9)
Applications of Operator Equalities
513
The connection between resolutions of the problem (1) and the equation (9)
h+(4 = F’-l%(.>, is one-to-one. We mention now an application of the theorems obtained. There are many results in the theory of solvability for matrix characteristic singular integral equations: conditions for the existence and uniqueness of the solution, applicability of approximate methods, factorization of matrix function, solution of special cases. Hence, all this we can use under the study of the Riemann boundary value problem (1).In fact we have obtained one more tool for the investigation of the problems. As an example, conditions of invertibility of the operator R(Ga) = P$ GwPG, where Pwf = (Iw f Sw) , with a piecewise constant matrix GR having three values were found [4]. Let us consider the boundary value problem (1) with the piecewise constant coefficients
+
A(z) = A-XR-
(4+ A+XE%+ (4, B ( z )= B-xw- (4+ B+XR+ (4, C(4 = c-xw(4+ C+xw+(z),
where A&,B*, C* are constants. By Theorem 3.3 this problem is reduced to the matrix characteristic singular integral equation without shift
h+h+ = Jw+,Dw+= h + I w + +GQ+sw+,
D R + E [L;(R+,t-$)].
As we can see, the coefficients iiw+ and &+ are piecewise constant matrix functions having two values and the point of discontinuity z = 1. By Theorem 3.2 the operator Dw+ is extended on R- = ( - m , O )
hf = Jw-sw,,
= (xw-
Assuming that det(&
+ CQ)
+ Jw-fiw+)Iw + Jw-GQSw, Dw E [L;(R, t - i ) ] .
# 0 we rewrite the operator & in the form R(Gw) = PL
where
GR = (XW-
+ GwPG,
+ JW-&+ + GR+)-~(XR+ JR-GIR+
- GR+).
The matrix Gw(z),z E R is a piecewise constant matrix function having three values and the points of discontinuity 2 = 0,z = 1. To obtain a condition for the existence and uniqueness of the solution of the boundary value problem (1) it remains to apply Corollary 2 from [4] to the operator R(Gw).
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Oleksandr Karelin
References 1. Karelin, A.A.: 2001, ‘On a Relation between Singular Integral Operators with a Carlemann Linear-Fractional Shift and Matrix Characteristic Operators without Shift’, Boletin SOC.Mat. Mexcicana Vol. 7,no. 3,pp. 235-246 2. Karelin, A.A.: 2002, ‘On the Operator Equality and Some of Its Application’, Proceedings of A . Razmadze Math. Inst. Vol. 128,pp. 105-116 3. Vasilevski, N.L., Karelin, A.A., Kerekesha, P.V., Litvinchuk G.S.: 1977, ‘A Class of Singular Integral Equations with Involution, Application to Boundary Value Problems for Partial Differential Equation’, Diff. Equations Vol. 13,pp. 1430143% 4. Spitkovski, I.M., Tashbaev, A.M.: 1991, ‘Factorization of Certain Piecewise Constant Matrix Functions and Its Application’, Math. Nachr. Vol. 151,,pp. 241-261
On Sets of Range Uniqueness for Entire Functions Maria Teresa Alzugaray Departamento de Matemstica Universidade do Algarve 8000- 117 Far0
Portugal mtrodrigQua1g.pt
Summary. The set E c C is called a set of range uniqueness if for entire functions f and g the condition f ( E ) = g ( E ) implies f z g. It is proved that exp(T), where T is the unit circle, is a set of range uniqueness. Results about the range uniqueness property of P ( T ) ,where P is a polynomial are discussed.
Sets of range uniqueness were first introduced by H.G. Diamond, C. Pomerance and L. Rube1 in 1981 in [2]. They gave the following definition:
Definition The set E c C is called a set of range uniqueness (sru) if for entire functions f and g the condition f(E)= g ( E ) implies f 3 g . The authors of [a] established several properties of the srus and provided many examples of srus and non-srus. All of these examples are countable sets of the type {zn}c R and some of them are referred in the table below: Non-srus
I
srus
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Maria Teresa Alzugaray
The srus were later studied by several authors. We will mention only some of the results obtained. In 1983, E. Thomas constructed a conformally rigid Jordan curve such that its interior is an sru (see [6]). Also in 1983, E.H. Johnston constructed a C1 arc which is an sru (see [4]).In 1985, A.A. Gol'dberg and A.E. Eremenko indicated some classes of domains and curves that are not srus (see [3]). They also constructed an infinitely smooth Jordan curve that is an sru. We consider sets of the type H(T), where H is an entire function and T is the unit circle. Our main results are the following:
Theorem 1 The set exp(T) is an s m .
+
+ +
Theorem 2 Let P ( z ) = u,zn un-@-l . . . uo. (i) If n 5 2 , then P(T) is not an sm. (ii) I f n 2 3 and a,-l # 0 , then P(T) is a n sru. The proofs of the theorems above are based on the following fact that can be found in the paper [5] by K. Stephenson, although the idea of its proof originated from [l]by I.N. Baker, J.A. Deddens and J.L. Ullman.
Theorem 3 I f f and g are entire functions and f(T) = g(T), then f(z) = h(z") and g(z) = h(Xzn),where h is an entire function, m and n are natural numbers and 1x1 = 1. Ideas used to prove Theorem 1 It follows from Theorem 3 that if the entire functions f and g satisfy the condition f(exp(T)) = g(exp(T)), then f(exp(z)) = h(z") and g(exp(z)) = h(Xzn), where h is entire, m , n E N and 1x1 = 1. Setting F ( z ) := f(exp(z)) = h(z"), it is easy to show that if m > 2, then F is a function possessing two non-parallel periods. Therefore, F is identically constant which yields f = g. The cases m = n = 1 and m = n = 2 can be handled in a similar fashion leading to f = g. The case m = 1 , n = 2 is also impossible if F is not constant. Indeed, if p is a polynomial of degree 2 and F is an entire non-constant periodic function, then F o p cannot be periodic which contradicts our assumptions. Therefore, exp T is an sru.
Observation Although the proof of Theorem 1 is heavily based on the periodicity of the exponential, we cannot prove that the image of 1" by any periodic entire function is an sru. Let f ( z ) = z and g ( z ) = -2. Then f(sin(T)) = g(sin(T)) but f # g.
+
Ideas used to prove Theorem 2 If d e g P = 1 or P ( z ) = u2z2 ao, it is easy to see that P(T) being a circle is not an sru. Let P ( z ) = ( z b)2. In this case setting h ( z ) := s i n s 2 we can easily check that h(-z - 2b) = h(-z 2b) = h ( z ) . It means that h ( z ) must be of
+
+
On Sets of Range Uniqueness for Entire Functions
517
the types f ( P ( z ) )and g(P(-2)) for certain entire functions f and g such that f # 9. This functions satisfy f(P(T)) = g(P(T)). Thus P(T) is not an sru. It becomes obvious now that if deg P 5 2, then P(T) is not an sru and assertion ( i )of Theorem 2 is proved. Assertion (ii)of Theorem 2 is proved in a way analogous to the proof of for f and g entire, then Theorem 1. By Theorem 3, if f(P(T)) = g(P(T))
f ( P ( z ) )= h(z") and g(P(z)) = h(Xz"), where h is entire, m, n E N and 1x1 = 1. On the other hand, we have the following lemmas (the second one can be proved with the help of results related to the theory of iteration of analytic functions).
+
Lemma 1 If P ( z ) = z 1 +al-lzl-' . . . +a0 and deg P 2 2, t h e n there exists a function cp(z) = E Z + co % ... holomorphic in a neighborhood of infinity such that P o cp = P, E = e2ni/1 and co # 0 , if az-1 # 0 .
+ +
+
Lemma 2 If F i s a n entire f u n c t i o n and cpl(z) = z a1 + % + . . . and cpz(.z) = z + a2 + % + . . . are meromorphic functions in certain neighborhood of infinity with $ Iw such that F o cp1 = F o cp2 = F , t h e n F i s constant.
2
Note that in this case by Lemma 1 the function cp plays with respect to the polynomial the same role the period plays with respect to the exponential function in the case of Theorem 1. Lemma 2 suggests that using reasoning similar to that above the function F ( z ) = h ( z " ) = F ( P ( z ) )must be constant leading us to f f g and showing that P(T) is an sru in case (ii).
Acknowledgements I am greatly indebted to A.E. Eremenko for suggesting this investigation and fruitful discussions on this subject.
References 1. Baker, I. N.; Deddens, J. A.; Ullman, J. L.: A theorem on entire functions with applications to Toeplitz operators. Duke Math. J. 41 (1974), 739-745. 2. Diamond, H. G.; Pomerance, C.; Rubel, L.: Sets on which an entire function is determined by its range. Math. 2. 176 (1981), no. 3, 383-398. 3. Gol'dberg, A. A.; Eremenko, A. E.: Generalized sets of uniqueness for entire functions. (Russian.) Akad. Nauk Annyan. SSR Dokl. 81 (1985), no. 4, 159-161 (1986). 4. Johnston, E.H.: On sets of range uniqueness, Math. Z., 184 (1984), 533-547. 5. Stephenson, K.: Analytic functions of finite valence, with applications to Toeplitz operators. Michigan Math. J. 32 (1985), no. 1, 5-19. 6. Thomas, E: Two examples of sets of range uniqueness. Illinois J. Math. 27 (1983), no. 1, 110-114.
Analytic Mappings in the Tree M u Z t ( K [ z ] ) Kamal Boussaf' , Alain Escassut2, and Nicolas Ma'inetti3 Laboratoire de Mathkmatiques Universitk Blaise Pascal Les Ckzeaux 63177 Aubiere Cedex F'rance Kama1.BoussafQmath.univ-bpclermont.fr Laboratoire de Mathkmatiques Universitk Blaise Pascal Les Ckzeaux 63177 Aubiere Cedex France A1ain.EscassutQmath.univ-bpclermont.fr LLAIC, IUT DQpartement Informatique Universitk d'Auvergne Complexe Scientifique des Ckzeaux 63177 Aubiere Cedex F'rance mainettiQl1aic.u-clermontl.fr Summary. Let K be an algebraically closed complete ultrametric field, let D c K be closed and bounded, and let H ( D ) be the Banach K-algebra of analytic elements on D. Let Mult(K[z]) (resp. M u l t ( H ( D ) ,11 . llo)) be the set of multiplicative seminorms on K [ z ] )(resp. of continuous multiplicative semi-norms on H ( D ) ) which are known to be characterized by circular filters. (Circular filters were also considered by B. Dwork as generic disks and by J.C. Yoccoz and J. Rivera-Letelier, as points of a hyperbolic space). Mult(K[z]) is provided with the topology of simple convergence, and with a metric topology based upon a tree structure for which it is complete. Given a bounded closed infraconnected set D c K, the boundary of M u l t ( H ( D ) ,(1 . 11) inside Mult(K[z])with respect to the topology of simple convergence, is equal to the Shilov boundary for ( H ( D ) ,(1 . 11). If D is affinoid (particularly), ) respect to this is also the boundary of M u l t ( H ( D ) ,11 . 11) inside M u l t ( K [ s ] with the metric topology. We show that every element f E H ( D ) has continuation to a mapping f * from M u l t ( H ( D ) ,11 . 11) to Mult(K[z])which is continuous for both topologies and uniformly continuous for the metric topology. The family of functions 2~ from H ( D ) to MuZt(K[z]) defined as 2 ~ ( f =)f *(F) (where F is a circular filter intersecting with D ) is uniformly equicontinuous with respect to the metric topology. If the field K is separable, f' is uniformly continuous for both topologies. The results also apply to meromorphic functions in K . A meromorphic function in K
520
Karnal Boussaf, Alain Escassut, and Nicolas Mainetti
defines an increasing function f* (with respect to the order of M u l t ( K [ z ] ) )if and only if it is an entire function. In a Krasner-Tate algebra H ( D ) = K { t } [ z ]where , K { t } is a topologically pure extension of dimension 1 and z is the identical function on D integral over K { t } ,we can precisely show that the Gauss norm on K { t } admits a number of extensions to K { t } [ z which ] is equal to the cardinal of the Shilov boundary of M u l t ( H ( D ) ,11 . 110).
1 Introduction and Tree Structures Throughout the paper, K is an algebraically closed complete ultrametric field and we denote by IKI the set { 121 I J: E K } . Given a K-algebra A , we denote by MuZt(A) the set of non identically zero K-algebra multiplicative semi-norms. We shall denote by d ( K ) the K-algebra of entire functions in K i.e. the power series with coefficients in K whose convergence radius is infinite. And we denote by M ( K ) the field of meromorphic functions in K , i.e. the field of fractions of d ( K ) . Let D be a closed bounded subset of K . We denote by R ( D ) the K-algebra of rational functions without poles in D. It is provided with the K-algebra norm of uniform convergence on D that we denote by (1 . (ID. We then denote by H ( D ) the completion of R ( D ) for the topology of uniform convergence on D: H ( D ) is a Banach K-algebra whose elements are called the analytic elements on D [lo]. The multiplicative semi-norms on R ( D ) which are continuous with respect to the norm )) . ) I D is denoted by M u l t ( R ( D ) ,11 . 110). All elements of MuZt(R(D),(1 . 1 1 ~ )have continuation to H ( D ) ,and similarly, the set of multiplicative semi-norms on H ( D ) which are continuous with respect to the norm 11 . 1 1 is~ denoted by M u l t ( H ( D ) ,11 . 110) . Such a set is seen as . semi-norms on K [ z ]are characterized a subset of M u l t ( K [ z ] )Multiplicative by certain filters on K named circular filters. The set Mult(K[z]) is then provided with the topology of simple convergence for which it is locally compact (101, [ll],[12], and every compact is sequentially compact [14]. But it is also provided with a metric topology which is strictly stronger. This metric comes from a partial order which gives M u l t ( K [ z ] a) tree structure [4].However the set is also called Berkovich one dimensional affine space, denoted by A', and then the topology used by V. Berkovich is the simple convergence topology. The set A' is seen like a set of points extending the field K [l].On the other hand, B. Dwork defined generic disks which have been much used in differential equations, and actually represent the same notion [ 6 ] ,[17]. The Banach K-algebra H ( D ) is called a Krasner-Tate algebra if it is isomorphic to a K-algebra topologically of finite type, also called K-affinoid algebra [l]. Then a Krasner-Tate algebra is of the form K { h } [ z ]where h E K ( z ) is of the form P(J:) with P, Q E K[J:], deg(P) > deg(Q) [7]. Q(4' Here, following and generalizing a first study by K. Boussaf [3], we will study mappings from MuZt(H(D),11 . 110) to MuZt(K[z])which are defined
Analytic Mappings in the Tree MuZt(K[z])
521
by elements of H ( D ) . In particular, we will show that such mappings are continuous for both topologies, and uniformly continuous for the metric topology.
Definition Let E be a set provided with an order relation 5. Then E will be called a tree if i) For every a , b E E there exists sup(a, b) E E . ii) For every a E E and b, c E E satisfying a 5 b, a 5 c , then b and c are comparable with respect to the order 2. The following basic theorem is easily obtained, and mostly known: Theorem 1.a Let ( E , 5 ) be a tree and let f be a strictly increasing function from E to IR. For all a , b E E we put a l ( a ,b) = (f(sup(a, b ) ) min( f ( a ) ,f ( b ) ) and ~ ( ab) ,= 2 f (sup(a, b ) ) - f ( a ) - f ( b ) . Then a1 and a2 are two equivalent distances such that a1 5 a2 5 2a1. Moreover, if b, c E E satisfy a 5 b, a _< c and if uj ( a ,b) = uj ( a ,c ) f o r some j , then b = c. Definitions Let E be a tree and let f be a strictly increasing mapping from E into IR. We will use the term metric topology associated to f on E for the topology defined by the distances introduced in Theorem 1.a. The distance denoted by a1 in Theorem 1.a will be called the supremum distance associated to f and that denoted by 0 2 in Theorem 1.a will be called whole distance associated to f . A Hausdorff space E is said to be pathwise connected or arcwise connected if for every a , b E E there exists a continuous mapping q5 from an interval [a,p] into E such that +(a)= a, r#@) = b. Theorem 1.b is classic in basic topology: Let ( E , 5 ) be a tree and let f be a strictly increasing Theorem 1.b function from E to R.Let a , b E E be such that a 5 b and {f(z) a 5 z 5 b } = [ f ( a ) f, ( b ) ] . Then there exists a mapping from [ f ( a ) , f ( b ) ]into E , bicontinuous with respect to the metric topology of E associated to f . Let ( E , 5 ) be a tree and let f be a strictly increasing Theorem 1.c function from E to IR. If for all a , b E E such that a _< b, the equality {f(z) I a 5 z 5 b} = [ f ( a ) f(b)] , holds, then E is arcwise connected with respect to its metric topology associated to f .
I
2 Tree of Circular Filters Circular filters were introduced in order to characterize multiplicative seminorms on K [ z ]in , particular absolute values on K [ z ]and K ( z ) continuing this of K [lo], [ll],[12].The set of circular filters is provided with a tree structure, and of two topologies: the topology induced by this of simple convergence [lo], [12] on the set of multiplicative semi-norms, and a metric defined by the positions and the diameters of filters [4].
522
Kamal Boussaf, Alain Escassut, and Nicolas Maynetti
Certain results from part 2 were published in [4]and are recalled in order to making the paper self contained. First we shall recall the definition and basic properties of circular filters. Definitions and Notation Let D be a subset of K . We denote by ID1 the set (1x1 I z E D}. Let a, b E K and r, s E IKI such that r < s, we denote by d ( a ,r ) the disk {x E K 1 1z-al 5 r } , by d ( a , r - ) the disk {z E K I Iz-aI < r } and by r ( a ,r, s) the annulus d ( a , s-) \d(a, r ) . We call class of d ( a , r ) each disk d ( b , r - ) included in d ( a , r ) . We put 6(a,b) = Ib-a[, and given subsets F, G of K , we put 6 ( a , F ) = inf{lz-al I z E F } , S(F,G) = inf{lz-yI I z E F , y E G}. Given a bounded subset A of K , we denote by the smallest disk d ( a , r ) containing A , called the envelope of A. We use the term circular filter of center a and diameter r o n K for the filter 3 which admits as a generating system the family of sets F ( a ,T I , r") with a E d ( a ,r ) ,r' < r < r", i.e. 9 is
A
4
r:, r;)
the filter which admits for base the family of sets of the form a== 1
with ai E d(a,r),ra < r < r; (1 I i I g , g E IN). For reasons that will appear when characterizing the absolute values of K ( z ) ,we use circular filter with n o center, of diameter r and canonical base (Dn)nc nv for a filter admitting for base a sequence ( D n ) n Ewhere ~ each D, M
d ( a n , r n ) = 0 and lim r, = r. n+Oo n=l Finally the filter of neighbourhoods of a point a E K is called the circular filter of the neighbourhoods of a. It will also be named circular filter of center a and diameter 0 or Cauchy circular filter of limit a. A circular filter is said to be large if it has diameter different from 0, and to be punctual if it is a Cauchy circular filter. Given a circular filter F,its diameter will be denoted by d i a m ( 3 ) and the set of its centers is denoted by Q(3). So, if F is a circular filter of center a and diameter r , then Q(3) = d(a,r). The set of circular filters on K intersecting with a subset D of K will be denoted by @(D)and the set of large circular filters will be denoted by @'(D). Given two circular filters F and G, 3 is said to surround G if either 0 is intersecting with Q(3), or if 3 = G. We will denote by 5 the relation on the set of circular filters defined as 3 4 G if 6 surrounds 3 and by 4 the relation defined as F 4 G if F 5 4 and
is a disk d ( a n , r n ) , such that
32G. Let D be a subset of K . Then D is said to be infruconnected if for every a E D ,the mapping I , from D to R+ defined by l,(z) = Ix - a1 has an image whose closure in IR+ is an interval. In other words, a set D is not infraconnected if and only if there exist a and b E D and an annulus r ( a ,r1,r2) with 0 < rl < 7-2 < la-bl such that r ( a , r l , r 2 ) n D= 0. If D is infraconnected,
( ud ( b j , r ; ) ) , n
it is said to be ufinoid if it is of the form d(a,ro) \
j=1
with
Analytic Mappings in the Tree M u l t ( K [ z ] )
523
rj E I KI V j = 0 , ...,n. More generally, a set D is said to be afinoid if it is a finite union of infraconnected affinoid sets. Given a circular filter 3,an affinoid subset B of K will be called 3 - a f i n o i d if it belongs to 3.In other words, if 3 has center a and diameter r , an affinoid subset B of K will be called 3-affinoid if either it is a disk d ( a , s ) , with s E IKI,
u Q
or it is of the form d ( a ,r”) \
i=l
d(ai,r’-), with r’, r” E (KI, r‘
< r < r”, and
lai - ajl = r V i # j. Theorem 2.a [4] The relation 5 is an order relation on @ ( K )and 4 is the strict order associated to this order relation. Moreover, if 3,8,‘Hsatisfy 3 5 (i and 3 5 ‘H then 8 and 7-1 are comparable. Theorem 2.b [4] Let D be infraconnected and let 3 E @ ( D ) .Then 3 is a minimal element in @(D)i f and only if: either it is punctual, or it has no center, or Q ( 3 ) n D = 8. Theorem 2.c is given in [lo] as Lemma 41.2: Theorem 2.c Let 3 be a circular filter on K and let s > d i a m ( 3 ) . There exists a unique circular filter on K , of diameter s , surrounding 3. Theorem 2.d Let D be infraconnected, let 3 be a circular filter on K intersecting with D and let r ~ ] d i a m ( 3d) i,a m ( D ) [ . The unique circular filter of diameter r surrounding .F is intersecting with D. Proposition 2.e is proven in [4]: Proposition 2.e Let 3, g be two circular filters on K which are not comparable for the relation 5. There exist disks F E 3 and G E 8 such that F n G = 8. Moreover, given F’ E 3, G’ E 8 such that F‘ n G‘ = 8, we have b(F,G ) = b(F’, G’) > m a x ( d i a m ( F ) ,d i a m ( ( i ) ) . Notation Let F ,8 be two circular filters which are not comparable for 5. By Proposition 2.e, S(F,G) does not depend on the choice of disks F E 3 and G E 8, so we can put X ( F , ( i ) = S(F,G) with F, G disks such that F E F , G E G ,F n G = 8 . Theorem 2.f [4] Let D be an infraconnected subset of K and 3,B E @(D). There exists sup(3, 8) E @(D)and it is the unique circular filter of diameter A(F,g ) which surrounds both F, E. Theorem 2.g If D is infraconnected, @ ( D )is a tree with respect to the order 5. Notation Let 3, 8 be two circular filters and let S = sup(3,G). We put 6 ( 3 ,8) = m a x ( d i a m ( S ) - d i a m ( 3 ) , d i a m ( S ) - d i a m ( 8 ) ) and S’(3, 8) = 2diam(S)- d i a m ( 3 )- diam(8). Corollary 2.h 6 is the supremum distance associated to the mapping d i a m and 6‘ is the whole distance associated to the mapping d i a m defined on the tree @(K).
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Kamal Boussaf, Alain Escassut, and Nicolas Mainetti
Definition We will call the topology defined by the distances 6, 6’ the S-topology on @(K). 1 Remark In particular, if F 5 6 , we have 6(F,9 ) = 56’(F, G) = diam(G)diam(F). Theorem 2.i Every bounded monotone sequence of @ ( K )has a limit with respect to the 6-topology. Theorem 2.j @ ( K )is complete with respect to the 6-topology.
Remark
The distance introduced in [18]is log
(
diam(f)diam(G) it defines another distance whose topology is equivalent to that defined by 6 and 6’ when we restrict the space-to the set of large circular filters. The completion of this restricted space for that distance is showed in Proposition 2.12 of [18].However, we notice that every point of the field K is at an “infinite distance” from large circular filters, if we try to extend this distance to the whole space @ ( K ) .
3 Analytic Elements and Meromorphic Functions on
MuZt(K [4 )
We must now recall how circular filters characterize multiplicative semi-norms on K [ z ] on , R ( D ) and continuous multiplicative semi-norms on H ( D ) . This characterization also holds in the field M ( K ) of meromorphic functions.
Theorem 3.a (G. Garandel) [lo],[ll] For every circu1arfilter.F o n K , for every polynomial P ( x ) E K [ z ]IP(z)I , has a limit cp,(P) along the filter
3. Theorem 3.b (Garandel-Guennebaud) [lo],[ll] The mapping 9 f r o m @ ( K )into Mult(K[z])defined by !P(F)= cp7 is a bijection. Moreover, the restriction !PI of !P to @’(K)is a bijection from @’(K)onto the set of multiplicative norms on K [ x ] ,and therefore onto the set of multiplicative norms on K ( x ) . Corollary 3.c Let D be a closed bounded subset of K . The restriction of !P to @(D)is a bijection f r o m @ ( D )onto M u l t ( H ( D ) ,11 . 110). Corollary 3.d The restriction of 9 to @’(K)is a bijection from @’(K)onto Mult ( M ( K ) ). Henceforth, according to Theorem 3.b, we will consider that both M u l t ( K [ x ] ) and @ ( K )are provided with the topology of simple convergence defined on M u l t ( K [ z ] )and , the metric 6 defined on @ ( K ) Moreover, . we will confound a Cauchy circular filter F with its limit in K. From [4]and [14]we recall Theorems 3.e and 3.f:
Theorem 3.e M u l t ( K [ x ] )is locally compact and every compact is sequen) respect to the topology of tially compact. K is dense inside M u l t ( K [ x ] with
Analytic Mappings in the Tree M u l t ( K [ z ] )
525
simple convergence. For any closed bounded subset D of K, D is dense inside M u l t ( H ( D ) ,11 . 110) with respect to the topology of simple convergence. Given 3 E @(IT),the family of subsets M u l t ( H ( B ) ,IJ . 1 1 ~ ) where B is a 7 - a f i n o i d , forms a base of neighborhoods of 93 in M u l t ( K ( [ x ] ) .
Theorem 3.f The 6-topology is stronger than the topology of simple convergence on MuZt(K[z]) and on M u l t ( H ( D ) ,11 . 110) for any closed bounded subset D of K . Moreover, the S-topology is equivalent to the topology of simple convergence i f and only if there is no circular filter of K intersecting with D . Theorem 3.g Let D be a closed bounded subset of K and let f E H ( D ) . Then f is uniformly continuous on D . Definition and Notation A filter 3 is said to be thinner than a filter C; if every element of 8 belongs to 3. In the same way, a sequence (u,),~IN is is said to be thinner than a filter G if for every A E G, there exists q E IN such that u, E A V n 1 q. Let h = (',), with P, Q E K[z] relatively prime. We will call the number
Q (4
max(deg(P), deg(Q)) the absolute degree o f h and we will denote it by deg,(h).
A circular filter F will be said to be D-bordering if it is intersecting with both D and K \B. The set of 93 E M u l t ( K [ z ] such ) that F is D-bordering is denoted by C ( D ) . Let ( A ,11 . 11) be a normed K-algebra whose norm is semi-multiplicative (i.e. power multiplicative). We use the term boundary for (A, (1 . 11) for a subset S of Mult(A,11 . 11) such that for every z E A, there exists q5 E S such that 4(z) = And we call a closed boundary for ( A , 11 . (1) which is the smallest of all closed boundary with respect to inclusion a Shilov boundary for (A, )I . 11) 191, ~ 1 1 . Let D be closed and bounded. In I21 it was shown that there exists a Shilov boundary for ( H ( D ) ,11 . 110) which is the set of 93 E M u l t ( H ( D ) ,11 . 110) such that .F is D-bordering. By [2], we have the following Theorem: Theorem 3.h Let D be infraconnected closed bounded. The following three sets are equal:
((XI(.
~
L
J
W),
The boundary of MuZt(H(D),(1 . 110) inside MuZt(K[z])with respect to the topology of simple convergence, The Shilov boundary for ( H ( D ) ,11 . 110) . Theorem 3.i Let D be infraconnected closed bounded having finitely many D-bordering filters. The following four sets are equal:
aD),
The boundary of M u l t ( H ( D ) ,1) the topology of simple convergence,
. 110) inside
Mult(KIz]) with respect to
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Kamal Boussaf, Alain Escassut, and Nicolas Maihetti
The boundary of M u l t ( H ( D ) ,11 . 110) inside Mult(K[z])with respect to the 6-topology, The Shilov boundary for ( H ( D ) ,11 . [ I D ) . Here we will define a continuous mapping from M u l t ( K [ z ] to ) itself which associates to any $ E M u l t ( K [ z ] another ) $' E M u l t ( K [ z ] )Similar . studies were made in [3], [6] and [18].We show that this mapping is continuous for the two topologies defined on M u l t ( K [ z ] ) . Theorem 3.j Let D be a closed bounded subset of K , let f E H ( D ) and let F E @(D).There exists a unique circular filter f*(.F)less thin than the filter admitting for base f ( F n 0 ) .If D' is a closed bounded set containing f ( D ) , then 'pf*(F) E Mult(H(D'),(1 . 1 1 ~ ) and for all h E H(D'), it satisfies 'pf*(F)(h) = V F ( h 0 f). Let 9 E H ( D ) . If f*(.F')# g*(.F'>, then PF(f - 9 ) = diam(sup(f*(F) ,9* (3)))) therefore 6(f* 9* (3)) I PF(f -9 ) I Il f -9110. Remark In [3] it is shown that if .F is large, when two elements f and g of H ( D ) are such that 11 f - 9110 is small enough, then f , ( F ) = g*(F). Notation Let D be a closed bounded subset of K . Given f E H ( D ) , we ) will denote by f * the mapping from M u l t ( H ( D ) ,11 . 110) to M u l t ( K [ z ] by f * ( c p ~ ) = 'pf*(F). For every $ E M u l t ( H ( D ) ,)I . /lo), we will denote by 2, the mapping from H ( D ) to M u l t ( K [ z ]defined ) by Z@(f) = f*($). Corollary 3.k Let D be a closed bounded subset of K . For every $ E Mult(H(D),II . /ID), 2, is continuous with respect to the norm on H ( D ) and to both topologies on M u l t ( K [ x ] )and it is uniformly continuous with respect to the 6-topology on M u l t ( K [ x ] ) Moreover, . the family of mappings 24,(4 E M u l t ( H ( D ) ,11 . l l ~ ) )is uniformly equicontinuous with respect to the 6-topology on M u l t ( K [ z ] ) . Theorem 3.1 Let D be a closed bounded subset of K and let f E H ( D ) . The mapping f* from M u l t ( H ( D ) ,I( . 110) to Mult(K[z])defined as f * ( c p ~ ) = 'pf*(F) is continuous with respect to the topology of simple convergence and is uniformly continuous with respect to the 6-topology. Remark When the field K is topologically separable (i.e. contains a dense countable subset), it is shown in [15]that the topology of simple convergence is also metrizable, though it is a topology weaker than this defined by 6. Particularly, this applies to CP.Thus, we can easily obtain Theorem 3.m: Theorem 3.m Assume K is topologically separable, let D be a closed bounded subset of K and let f E H ( D ) . The mapping f* from M u l t ( H ( D ) ,11 . 110) to M u l t ( K [ z ] )defined b y f * ( ' p ~ )= ' p f * ( ~ )is uniformly continuous with respect to any metric defining the topology of simple convergence. Remark On the other hand, when the field K is topologically separable, we can also ask whether mappings 2 4 are uniformly continuous, and whether the family of mappings 2 4 , (4 E M u l t ( H ( D ) ,11 . llo)) is uniformly equicontinuous with respect to the metric of simple convergence. Since each 24 is uniformly continuous with respect to the metric 6, it is likely uniformly continuous with respect to the metric of simple convergence, and similarly,
m,
Analytic Mappings in the Tree MuZt(K[z])
527
the family of mappings Z b , (4 E MuZt(H(D),(1 . 110)) should be uniformly equicontinuous with respect to this metric. But a proof is not easily written, due to the complexity of defining distances defining the topology of simple convergence. And yet, the answer might depend on the distance we consider to define this topology. However, we can answer the question on a bounded subset of H ( D ) :
Theorem 3.n Assume K is separable. The family of functions Zb, 4 E M u l t ( H ( D ) ,I( . 110) is uniformly equi-continuous on every ball o f H ( D ) with respect to any metric defining the topology of simple convergence. Notation Let f E M ( K ) .We will denote by r ( f )the set of poles o f f . Now applying Theorem 3.j to meromorphic functions, we obtain Theorem 3.0: Theorem 3.0 Let f E M(K), let 3 be a circular filter o n K which i s not the filter of neighborhoods of a pole o f f and let 3' be its intersection with K \ r(f ) . Then f (7')is a base of a unique circular filter f,(3) such = V F ( Po f ) V P E K [ s ] .Moreover the function f * f r o m that 'pf*(~)(P) M u l t ( K [ x ] ) \ . r r ( f ) to M u l t ( K [ s ]defined ) as f*('pF)= ' p f * ( ~is ) a continuous function with respect to both topologies and i s open with respect to the topology of simple convergence. Theorem 3.p Let f E M ( K ) . Then f* is increasing from @ ( K )\ r ( f )to @ ( K ) ,with respect to 5 , i f and only i f f E d ( K ) . Theorem 3.q Let E be a closed bounded subset of K and f E K(x) and let D = f - ' ( E ) . Then C ( E )= f * ( C ( D ) )and C(D)= ( f * > - l ( C ( E ) ) . Corollary 3.r Let G be a circular filter of center a and diameter r E 1K1, let h E K ( x ) and let D = h - l ( d ( u , r ) ) . Then h*-'({G}) = C(D). Definition and Notation: We use the term K-affinoid algebra (or Tate algebra) for a Banach K-algebra which is a quotient of a topologically pure extension of K , i.e. a K-algebra of restricted power series in n variables provided with the gauss norm [19].And we call Krasner-Tate algebra on Banach K-algebra which is isomorphic to both a K-affinoid algebra and an algebra H P ) ~71. Let U be the disk d ( 0 , l ) and let K { s } be the topologically pure extension of K of dimension 1, i.e. the K-algebra of restricted power series in one variable also isomorphic to H ( U ) . We will denote by 11 . 11 the Gauss norm on K{z} 00
Theorem 3.s Let H ( D ) be a Krasner-Tate algebra of the f o r m K { h } [ z ] , with h E K ( z ) , deg,(h) > 0, and D = h - l ( U ) . Let S be the set of Dbordering filters. Then S is finite and the restriction of @ to S i s a bijection from S onto the set of absolute values expanding the Gauss n o r m on K { h} to K{h}[Xl.
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Kamal Boussaf, Alain Escassut, and Nicolas Maynetti
In other words, there is a natural bijection between the Shilov boundary of the Krasner-Tate alebra K { h } and the expansions of the Gauss norm on K {h } to absolute values on K { h}[z]. a Example Let h ( z )= X+ -, with a E K , such that la1 < 1, let r = IQI, and
[XI
X
let D = {X I r 5 1x1 5 1). Then we check that D = h-l(U), and there are two D-bordering circular filters: the circular filter 3' of center 0 and diameter r , and the circular filter 3" of center 0 and diameter 1. Then, the Krasner-Tate algebra H ( D ) is obviously equal to K { h } [ z ] ,and the Gauss norm on K { h } admits two extensions to K { h } [ z ] :( p 3 ' and ( p 3 ~ " . Now, if we take la1 = 1, then 3' = 3":the Gauss norm on K { h } has a unique extension to H ( D ) .
References [l]Berkovich, V. Spectral Theory and Analytic Geometry over Non-archimedean Fields. AMS Surveys and Monographs 33, (1990). [2] Boussaf, K. Shilov Boundary of a Krasner Banach algebra H ( D ) Italian Journal of pure and applied Mathematics, N.8, p. 75-82 (2000). [3]Boussaf, K. Image of circular filters International Journal of Mathematics, Game Theory and Algebra, Volume 10, N. 5, pp. 365-372. [4]Boussaf, K., Hemdahoui, M., and Mainetti, N. Tree structure on the set of multiplicative semi-norms of Krasner algebras H ( D ) . Revista Matematica Complutense, vol XIII, N. 1, p. 85-109 (2000). [5] Boussaf, K. An interpretation of analytic functions. p-adic functional analysis, Lecture Notes in Pure and Applied Mathematics n. 222, Marcel Dekker, (2001). [6] Dwork, B. Lectures o n p-adic differential equations. Springer-Verlag, (1982). [7]Escassut, A. Algkbres de Banach ultrame'triques et algkbres de KrasnerTute, Asterisque n. 10, p.1-107, (1973). [8]Escassut, A. Algkbres d 'e'le'ments analytiques en analyse n o n archime'dienne, Indag. math.,t.36, p. 339-351 (1974). [9] Escassut, A. and Mainetti, N. Shilov boundary for ultrametric algebras, Preprint. [lo] Escassut, A. Analytic Elements in p-adic Analysis, World Scientific Publishing Inc., Singpore (1995). [ll] Garandel, G. Les semi-normes multiplicatives sur les algkbres d'e'lkments analytiques au sens de Krasner, Indag. Math., 37, n4, p.327-341, (1975). [12] Guennebaud, B. Sur une notion de spectre pour les algkbres norme'es ultrame'triques, thkse Universith de Poitiers, (1973). [13]Mainetti, N. Spectral properties of p-adic Banach algebras. Lecture Notes in Pure and Applied Mathematics n. 207 p. 189-210, (1999).
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[14] Mainetti, N. Sequential Compactness of some Analytic Spaces, Journal of Analysis, 8, p. 39-54 (2000). [15]Mainetti, N. Metrizability of some analytic a f i n e spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, (2001). [16] Motzkin, E.L ’arbre d ’un quasi-connexe: un invariant conforme p-adique, Groupe d’ktude d’Analyse Ultramdtrique de I’IHP, 9e annke, 1981-82, n.3. [17]Robba, Ph. and Christol, G. Equations diffhentielles p-adiques, applications aux sommes eqonentielles, Hermann, Paris, (1994). [18] Rivera-Letelier, J.Espace hyperbolique p-adique et dynamique des fonctions rationnelles, to appear in Compositio. Math. [19] Tate, J. Rigid analytic spaces, Inventiones Mathematicae 12, p. 257-289 (1971).
Using Level Curves to Count Non-Real Zeros of f t t Stephanie Edwards* Department of Mathematics University of Dayton Dayton, OH 45469-2316 USA sedwardsQudayton.edu
Summary. In the late 1980’s T. Sheil-Small verified A. Wiman’s 1915 conjecture, “Iff E lJzp then f ” has at least 2p non-real zeros”, by using a level curve argument. He looked at the geometry of the components of I m M > 0 and Im(t - #) >0 in H’. Since then, A. Eremenko, S. Hellerstein, A. Hinkkanen, the author, and others have used extensions of the level curves technique to prove various theorems. We discuss the level curves technique, including how the geometry of the components changes as we look at functions in various classes. We also look at various applications of the technique.
Introduction In the late 1980’s,T.B. Sheil-Small verified a 1915 conjecture of A. Wiman by
a)
using a level curves argument. This technique involved looking at the geometry of the components of Im% > 0 and Im(z > 0 in H+ where f is in a certain class of functions. We begin by defining relevant classes of functions and discuss the history of the conjecture.
Definition 1 The function f E
V2,
f ( 4= e
i f and only i f -az2p+2
9(4
where a 2 0 and
*This research was supported by AWM-NSF Mentoring Travel Grant, UD Research Council SEED Grant, UD Women’s Center Grant
532
Stephanie Edwards
and ho i s a real polynomial with deg ho I 2p is a non-negative integer, and A # 0 E R.
+ 1, m I 2p + 1, a, E R 'd n, b
Definition 2 The class Uz, is defined as follows:
Uo = v, u2p
= v2p - h p - 2
f o r p 2 1.
We note that Uo is often referred to as the Laguerre-P6lya class and denoted by LP.
Definition 3 The function f E Ugp i f and only i f f ( z ) = c ( z ) g ( z ) where g E UzP and c ( z ) is a real polynomial with no real zeros. We remark that i f f is a real entire function of finite order with only finitely many non-real zeros then f E Uzp for a unique p.
Definition 4 Let f E U . p .A n y entire function f has Bore1 exceptional zeros i f the order of the canonacal product of its zeros is less than the order of f. The following theorem, first given in part by Laguerre and then completed by Pblya, illustrates the special nature of the class Uo.
Theorem 5 ([lo] and [12]) Let f be a real entire function. Then f E UO i f and only if there exists a sequence {p,} of real polynomials with only real zeros such that p , t f uniformly o n all compacts in C .
If p , + f uniformly on compacts in C , the same holds for the convergence of pk + f', and if p , (non-constant) has only real zeros so does pk by Rolle's theorem and the fundamental theorem of algebra. Thus, Theorem 5 implies Theorem 6 If f E UO then f (k) E UOf o r k 2 1. In particular, f ('1 has only real zeros. In 1914 P6lya asked whether the converse was true, i.e., whether it was true that if f is real entire and f(') has only real zeros for all k 2 0 then
f
E
uo.
Conjecture 1 (/13]). Suppose that f is a real entire function. If f , f', f", . . . have only real zeros, then f E UO.
In [13] P6lya affirmed this conjecture for real entire functions of the form Peh where P and h are real polynomials. M. Alander, 111, [2] showed that if f E U2 or U4 then either f' or f" has some non-real zeros. The first major result concerning Conjecture 1 was due to B. Ja. Levin and I.V. Ostrovskii in the early 1960's [ll].They showed that i f f is a real entire function with only real zeros of high infinite order, then f" has an infinite number of non-real zeros. The finite order case was settled in 1977 by S. Hellerstein and J. Williamson who proved
Using Level Curves to Count Non-Real Zeros of f”
533
Theorem 7 ([7])Let f E Uz,. I f f ’ has only real zeros, then f ” has exactly 2p non-real zeros.
The same year, Hellerstein and Williamson also proved the analogue of Theorem 7 for entire functions of low infinite order [8]. Their result together with that of Levin and Ostrovskii mentioned above completed the proof of PMya’s conjecture and yielded the following stronger result.
Theorem 8 Let f be a real entire function. Suppose f and f’ have only real zeros. I f f E Uz,, then f “ has exactly 2p non-real zeros. I f f is of infinite order, then f ” has infinitely many non-real zeros. Levin and Ostrovskii’s result, which does not require the reality of the zeros of f’, is an infinite order analogue of A. Wiman’s 1915 conjecture:
Conjecture 2 (Wiman’s conjecture). If f E real zeros.
U2,,
then f “ has at least 2p non-
We note that the added assumption that f’ has only real zeros enable Hellerstein and Williamson to determine the exact number of non-real zeros o f f ”. They were unable to use their purely analytic methods to prove Wiman’s conjecture. In 1989, Sheil-Small was able to avoid the hypothesis on f’ to prove the Wiman conjecture by using the auxiliary functions
and by studying the zero level curves of the imaginary part of these functions. The choice of Q is motivated by noting
and, therefore, the non-real zeros of f ” are critical points of &. In order to establish the existence of non-real critical points of Q and to obtain a lower bound on their number, Sheil-Small studied the components of
A = { z E Hf : ImL(z) > 0)
K
= { z E Hf: ImQ(z)
> 0)
(3)
where H+ = { z E C : Imz > 0). In particular, he counted the number of critical points of Q in the components of K by using a result of RiemannHurwitz. This Riemann-Hurwitz result, the “Counting Lemma,” is based on a well known connection between the degree of a map and the number of critical points of the map.
Definition 9 Let U be a component of K . Then a hole in U is a bounded component of C - U .
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Stephanie Edwards
-
Lemma 1 (The Counting Lemma [15]).Suppose that U is a component of K such that U has exactly m holes. Suppose further that Q' = 0 exactly k times (counting multiplicity) in U. Then Q maps U onto H+ exactly u times where v = k 1 - m.
+
In order t o count the number of critical points of Q, the geometry of K and K - (where K - = { z E H+ : ImQ(z) < 0)) must be studied. In SheilSmall's situation, there are no bounded components of K and it is unknown whether unbounded components of K - exist. We note that the HellersteinWilliamson result, which gives the exact number of non-real zeros of f" has not been recovered using the level curves technique. This is because the exact number of unbounded components of K - must be known in order to get a precise count. In the case that f has Borel exceptional zeros, there are no unbounded components of K - , and there is a single unbounded components of K [ 5 ] . Therefore, with the added hypothesis on f of having Borel exceptional zeros, the Hellerstein-Williamson result can be recovered using the level curves technique.
1 Other Applications The level curves technique has since been used by many others to prove various results. A. Hinkkanen, while investigating the analogue of P6lya's conjecture 1 for meromorphic functions followed Sheil-Small's approach and employed the auxiliary functions of (1) and the components of (3). He also studied the components of
A- = { z E H+ : ImL(z) < 0}
K - = { z E H+ : ImQ(z) < 0).
(4)
Theorem 10 ([9]) Suppose that f is a meromorphic non-entire function in the plane, and suppose for every k 2 0, the derivative f ( k ) has only real zeros. Then f (az + b) = P ( z ) / Q ( z ) for a , b E R, a # 0 , where & ( z ) = Z" or Q ( z ) = ( z 2 + l)", n is a positive integer, and P is a polynomial with only real zeros such that deg(P) <deg(Q) 1; or f ( a z b) = C ( z - 2)-" or f ( a z + b) = C ( z - a)/(.- i ) where a E R, C is a non-zero complex constant.
+
+
The geometry in Hinkkanen's situation was a bit more complicated than the case when f was in the class U2p. Hinkkanen was unable to rule out the existence of bounded components of K (the possible bounded component of K would have to contain the non-real pole of the function f ) . Expanding the techniques of Hinkkanen and Sheil-Small, S. Hellerstein and the author proved:
Theorem 11 ( [ 5 ] ) Iff E U;p, then f ( k ) has at least 2p non-real zeros for k 2 2.
Using Level Curves to Count Non-Real Zeros of f“
535
Theorem 12 ( [ 5 ] ) I f f E Uzp,f has Bore1 exceptional zeros, and f ’ has only real zeros, then f “ has exactly 2p non-real zeros for k 2 2. When one allows the function f to have non-real zeros, the geometry of the components K and K - remains the same as when f E U Q ~ . In 2003 A. Eremenko, W. Bergweiler, and J. Langley, while verifying Wiman’s conjecture for functions of infinite order, used the zero level curves of both A and K .
Theorem 13 ( [ 6 ] ) For every real entire function of infinite order with only real zeros, the second derivative has infinitely many non-real zeros. Sheil-Small writes a chapter about the level curves technique in his book [14] and mentions that he believes that a level curves argument will be useful in solving a long standing conjecture originally attributed t o Gauss. The conjecture is also known as the Hawaii Conjecture and can be found in [3].
Conjecture 3 [Gauss, [3]). real zeros of
($)’
Let f be a real polynomial. Then the number of
does not exceed the number of non-real zeros of f .
2 Extreme Curvature Another application involves a conjecture by R. Gordon regarding extreme curvature.
Conjecture 4 ((41). If f is a real polynomial of degree n > 1,then the curvature of f has at most n - 1 relative extreme values. The curvature of a twice differentiable function is given by
f”
= (1+ ( f ’ ) 2 ) 3 / 2 ‘
fc
And, if f “‘ is continuous, /
K =
+
(1
(f’)2)f’’’
(1
+
- 3fyf’’)2
(f’)2)5/2
Now, if f is a real polynomial of degree n, let g = f‘
Then
9
and h =
dGi7
+ (f’)2)f”’ - 3f’(f”)2 (1+
h” = (1 + g 2 ) g ” - 3 g ( g ’ ) 2 - (1 (1 9 2 ) 5 / 2
+
(f’)”5/2
Stephanie Edwards
536
So, h” = 0 exactly when K’ = 0. Since the points of extreme curvature can be expressed as the zeros of the second derivative of a function and since the resulting auxiliary functions L and Q are rational, the level curves technique can be applied. However, the geometry of the components K and K - changes significantly. Because of the poles of the function h, the existence of bounded components of K can not yet be ruled out. Also, unbounded components of K - do indeed exist because the function Q has a pole at infinity of order 2 n 1 where n 1 is the degree of the polynomial f , Using the level curve technique, the author was able to show the following partial result.
+
+
Theorem 14 Iff i s a real polynomial of degree n > 1 and f’ has only real zeros, then f has exactly n - 1 points of extreme curvature. However, a stronger version can be shown using analytical techniques.
Theorem 15 ([4]) Iff i s a real polynomial of degree n > 1 and f ” has only real zeros, then f has exactly n - 1 points of extreme curvature.
3 Conclusion The use of level curves to count non-real zeros of the second derivative of a function is dependent on the geometry of the components of K , K - , A, and A - . To get an accurate count, one must know the number of unbounded components of K - and the number of bounded components of K . Iff E U&, then there are no bounded components of K , however, if f has poles, then the existence of bounded components of K can not be ruled out - and the number of possible bounded components of K must be less than or equal to the number of poles in H+. This technique will be even more powerful when the questions about bounded components K and the unbounded components of K - are answered.
References 1. .&lander,M. (1914). Sur le dkplacement des ekros des fonctions enti6res par leur d4rivation. ThBse, Uppsala. 2. .&lander,M. (1916). Sur les zCros extramdinaires des d6rivkes des fonctions entikres r6elles. Ark. Math. Astronom. Fys., 15:ll. 3. Craven, T., Csordas, G., and Smith, W. (1987). The zeros of derivatives of entire functions and the P6lya - Wiman conjecture. Annals of Mathematics, 125:405-431. 4. Edwards, S. and Gordon, R. (2004). Extreme curvature of polynomials. mer. Amer. Math. Monthly, 111:890-899. 5. Edwards, S. and Hellerstein, S. (2002). Non-real zeros of derivatives of real entire functions and the P6lya - Wiman conjectures. Complex Variables,47:25-57.
Using Level Curves to Count Non-Real Zeros of f"
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6. Eremenko, A., Bergweiler, W., and Langley, J. (2003). Real entire functions of infinite order and a conjecture of Wiman. GAFA, 13. 7. Hellerstein, S. and Williamson, J. (1977a). Derivatives of entire functions and a question of P6lya. Transactions of the A.M.S., 227227-249. 8. Hellerstein, S. and Williamson, J. (1977b). Derivatives of entire functions and a question of P6lya 11. Transactions of the A.M.S., 234:497-503. 9. Hinkkanen, A. (1998). Iteration, level sets, and zeros of derivatives of meromorphic functions. Ann. Acad. Sci. Fenn. Math, 23:317-388. 10. Laguerre, E. (1898). Sur les fonctions du genre zkro et du genre un. C.R. Acad. Sci., 98 (1882), Oeuvres I:174-177. 11. Levin, B. J. and Ostrovskii, I. (1963). The dependence of the growth of an entire function on the distribution of the zeros of its derivatives. Amer. Math.Soc. Transl. (.), 32:322-357. 12. Pblya, G. (1913). Uber annaherung durch polynome mit lauter reellen wurzeln. Rend. Circ. Mat. Palermo, 361279-295. 13. P6lya, G . (1914). Sur une question concernant les fonctions entibres. C.R. Acad. Sci. Paris, 158:330-333. 14. Sheil-Small, T. (2002). Complex Polynomials. Cambridge University Press, New York. 15. Sheil-Small, T. B. (1989). On the zeros of the derivatives of real entire functions and Wiman's conjecture. Annals of Mathematics, 129:179-193.
The Functional Equation P ( f ) = Q ( g ) in a p-Adic Field Alain Escassut’ and Chung Chun Yang2 Dkpartement de Mathkmatiques Universitk Blaise Pascal Les Ckzeaux 63177 Aubiere Cedex France A1ain.EscassutQmath.univ-bpclermont.fr Department of Mathematics Hong Kong University of Science and Technology Kowloon, Hong Kong China mayang(9ust.hk
Summary. Let K be a complete ultrametric algebraically closed field of characteristic x . Let P, Q be in K [ x ]with P’Q’ not identically 0. Consider two different functions f , g analytic or meromorphic inside a disk Ix - a1 < T (resp. in all K ) , satisfying P(f) = Q(g). By applying the Nevanlinna value distribution theory in characteristic T ,we give sufficient conditions on the zeros of P’, Q’ to assure that both f , g are “bounded” in the disk (resp. are constant). If 7r # 2 and deg(P) = 4, we examine the particular case when Q = XP (A E K ) and we derive several sets of conditions characterizing the existence of two distinct functions f , g meromorphic in K such that P(f)= XP(g).
1 Introduction Let K be a complete ultrametric algebraically closed field of characteristic T and let K* = K \ ( 0 ) . Given a E K and r > 0 we denote by d(a, r - ) the disk {x E K 15 - a1 < r }. Let M ( K ) (resp. M ( d ( a ,r - ) ) be the field of meromorphic functions in K (resp. in d(a, r - ) ) and let d ( K ) (resp. d(d(a,r - ) ) be the ring of analytic functions in K (resp. in d(a, r - ) ) . In d ( d ( a , r - ) ) we denote by d b ( d ( a , r - ) ) the subring of functions f E d(d(a,r-)) which are bounded in d(a,r-) and by d u ( d ( u , r - ) ) the set d(d(a,r-))\db(d(a, r - ) ) . Similarly, in M ( d ( a ,r - ) ) we denote by Mb(d(a,r - ) )
I
2000 Mathematics Subject Classification: Primary 12E05 Secondary llC08 llS80 30D25
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4 .JI
the subfield of functions f E M ( d ( a , r - ) ) which are of the form - with
4, $ E d b ( d ( a ,r - ) ) and by M , ( d ( a , r - ) ) the set M ( d ( a ,r - ) ) \ M b ( d ( a , r - ) ) . Many applications of the Nevanlinna values distribution theory were made in padic analysis as in complex analysis, to study uniqueness problems [15] and the parametrization of curves or functional equations ([5], [6], 171). In the present paper we examine whether a meromorphic function may admit two different decompositions of the form P o f and Q o g, where P, Q are two nonlinear polynomials. The main property we shall use consists of assuming that P’ admits zeros a such that P ( a ) # Q ( b ) for all zeros b of Q’. This problem was motivated by some previous work in the complex plane C,made by H.K. Ha and C.C. Yang ([14]) as well as by C.C. Yang and P. Li ([lS]). After the general results, we shall consider the particular case when Q is proportional to P, while P is of degree 4, a particular study after previous works by Julie Wang, Ta Thi Hoai An and Pit-Mann Wong [l],[17]. In order to introduce our main results, we first state some basic results on the topic and recall the definition of the ramification index for an analytic function.
Lemma 1.1 ([12]) Let P, Q be in K [ x ] ,where P’Q’ is not identically 0. A point ( a ,b) E K 2 is a singular point of the affine algebraic curve whose equation is P ( x ) = Q ( y ) , i f and only i f P and Q satisfy P(a) = Q ( b ) , P’(a) = Q’(b) = 0. Lemma 1.2 Let P E K [ x ]satisfy P’ # 0, deg(P) = 3 and be such that P’ has two distinct zeros c1,cz. Then P(Q) # P(c2). Lemma 1.3 ( [ 6 ] ,[7])Let P, Q be in K [ x ]with 2 5 deg(P) I: deg(Q) = 3 and P’Q’ not identically zero. There exist nonconstant meromorphic functions f, g E M ( K ) such that P ( f ) = Q(g) if and only i f P’ has a zero a and Q’ has a zero b satisfying P ( a ) = Q ( b ) . As a consequence, the following Theorem 1.4 is easily shown by classical methods which do not require the Nevanlinna theory:
Theorem 1.4 Let P, Q be in K [ z ]with deg(P) = deg(Q) = 2 and P’Q’ not identically zero and let a (resp. b) be the zero of P’ (resp. Q’). Let a lie in K and r in R+*. The following three statements are equivalent: (i) There exist f, g E d ( K )\ K satisfying P ( f ) = Q(g); (zi) There exist f, g E d U ( d ( a , r - ) ) satisfying P(f)= Q(g); (iii) P ( a ) = Q(b). Henceforth, we shall look for more general results linked to the property P ( a ) # Q(b), where a and b are zeros of P’ and Q’, respectively. In order to state and prove our main results (Theorems 2.1 and 2.9), we need to recall the
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notion of the so-called Nevanlinna characteristic function and its associated functions. However, here, for the sake of convenience we will define counting functions of zeros and poles without multiplicities which, in characteristic T # 0, will be the same as in characteristic 0, contrary to some other results that require a more accurate form ([6],[7]).Moreover, in characteristic T # 0, we shall use the ramification index for analytic functions ([6],[7]).
Definition and Notation We shall denote by x the characteristic exponent of K , i.e. if 7r = 0, then x = 1 and if 7r # 0, then x = 7 r . Given h E M ( d ( a , r - ) ) (resp. h E M ( K ) )we use the term ramification index of h for the unique integer t such that belongs to M(d(a,r-)) (resp. belongs to M ( K ) )and h(t)# 0 ([6], [7]). The ramification index of any function h E M ( d ( a ,r - ) ) (resp. h E M ( K ) )will be denoted by v(h). Remark If 7r = 0, then every function in M ( d ( a , r - ) ) or M ( K ) has a ramification index equal to 0. Notation Given m, n E lN*,we shall denote by g.c.d(m, n ) the greatest common divisor of m and n. Let f E M ( d ( 0 , r - ) ) and Q E d(O,r-). If f has a zero (resp. a pole) of order n at Q, we put w,(f) = n (resp. w,(f) = -n). If f(a)# 0 and 00, we put wa(f)= 0. Let f E M ( d ( 0 ,r - ) ) with f(0) # 0, 00. We denote by Z ( p , f ) the counting function of zeros of f in d(O,r-):
Next,we put
We shall also consider the counting functions of poles of f in d(O,r-): 1 1 W P , f) = Z(P, and @, f) = Z ( P , The Nevanlinna function T ( p , f ) is defined by T ( p , f ) = max[Z(p,f)
7)
7).
+
Remarks 1) By definition, we have z ( p , f ) 5 Z ( p , f ) 5 T ( p , f )+ 0(1),s ( p , f ) L N ( p , f ) i T ( p , f ) in ]0,+00[ whenever f E M ( K ) (resp. in 10, r [ whenever f E M(d(0,r - ) ) ) . 2) In a field of characteristic 0, the above functions Z ( p , f ) and f l ( p , f ) are just those also denoted by z ( p , f) and R ( p , f ) respectively [ 5 ] , [6],[7].
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Theorems 1.5 and 1.6 were proved in a field of characteristic 0 [4], but actually all their proofs hold in any characteristic. In order to use the properties of the Nevanlinna functions defined above, we must recall a few basic results on analytic functions:
Theorem 1.5 ([4]) Let f be in M ( d ( 0 , r - ) ) with f(0) # 0 , 00. Then f belongs t o M b ( d ( 0 ,r - ) ) i f and only ifT(p,f) is bounded in [0,r [ . Theorem 1.6 ([4]) Let f be in M ( d ( 0 , r - ) ) and let P be of degree n. T h e n T(P,P(fN = n T ( p , f) O(1)
+
Corollary 1.7 Let f be in M ( d ( 0 ,r - ) ) and let P be in K [ x ] .T h e n f belongs to M b ( d ( 0 ,r - ) ) i f and only i f so does P ( f ) . Corollary 1.8 Let f, g be in M ( d ( 0 , r - ) ) , let P, Q be in K [ x ]and assume that f and g satisfy P ( f ) = Q(g). Then f belongs to M b ( d ( 0 ,r - ) ) i f and only i f so does g.
2 Main Results in the General Case Theorem 2.1 Let P, Q be in K [ x ]with P'Q' not identically zero and let p = deg(P), q = deg(Q) satisfy 2 5 min(p,q). Assume that there exist k distinct zeros c1, ..., c k of P' such that P(ci) # P(cj), V i # j, and P(ci) # Q ( d ) for every zero d of Q' (i = 1,...,k). Assume that there exist two functions f, g E M u ( d ( a , r - ) ) such that P(f)= Q ( g ) and let t = v(f). T h e n 2 ( p , f) 2 p # q,
+ O(1). Moreover, suppose f
< q. T h e n k 5 2. Further, if then k = 1, c1 is a simple zero of P' and either q < p or g.c.d.(p,q) =
T(p'f)(kq-p) Xt9
4-P.
Remark In Corollaries 2.2 to 2.7, when we assume that one of the functions f and g belongs to M u ( d ( u , r - ) ) , then by Corollary 1.8, both f and g belong to M , ( d ( a , r-)). We shall first apply Proposition 2.1 t o analytic functions inside d(a, r-). Corollary 2.2 Let P, Q be in K [ x ]with P'Q' not identically 0 and such that deg(P) = deg(Q). Assume that there ezist two distinct zeros c1, c2 of P' such that P(c1) # P ( c 2 ) and P(ci) # Q ( d ) for every zero d of Q'. If two functions f, g E d ( d ( a ,r - ) ) satisfy P ( f )= Q ( g ) , then f, g E d b ( d ( u ,r - ) ) . Proof Let q = deg(P) = deg(Q) and assume that the conclusion is wrong, hence by Corollary 1.8, f and g are not bounded. By Proposition 2.1 we have
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hence lim fi(p,f ) = +oo, because f is unbounded and so is T(r,f ) in ]O,r[. P-++W
But since f E d ( d ( a , r - ) ) , this is absurd.
Corollary 2.3 Let P, Q be in K [ x ]where P'Q' is not identically zero, and 3 5 deg(P) and deg(Q) 2 min(2,g). Assume that there exist two distinct zeros C I , c2 of P' such that P(ci) # Q(d) for every zero d of Q' (i = 1,2). Moreover, i f p 2 4, we assume that P(c1) # P(c2). If two functions f , g E A(d(a,r - ) ) satisfy p ( f )= Q(g), then f, E &(d(a, r - ) ) . Proof If p = 3, by Lemma 1.2 we have P(c1) # P ( c ~ hence ), that inequality is satisfied anyway. So, we can apply Proposition 2.1 with k = 2 which shows that $ ( p , f ) is unbounded (because 29 - p > 0), a contradiction when f is
unbounded.
Corollary 2.4 Let P, Q be in K [ x ] with P'Q' not identically zero and deg(P) < deg(Q). Assume that there exists a zero c of P' such that P(c) # Q ( d ) for every zero d of Q'. If two functions f , g E d ( d ( a , r - ) ) satisfy P ( f )= Qb),then f , 9 E d b ( d ( a , r - ) ) . Proof Let p = deg(P), q = deg(Q) and assume that f or g is not bounded. By Corollary 1.8 both f and g are not bounded. We can apply Theorem 2.1 with k = 1, p < q, which shows that fi(p,f ) is unbounded, a contradiction when f is unbounded. We shall now apply Theorem 2.1 to meromorphic functions in d(a, r-). Corollary 2.5 Let P, Q be in K [ x ]with P'Q' not identically zero and p = deg(P), q = deg(Q) satisfying p # q and 2p < 39. Assume that there exist two. distinct zeros c1 and c2 of P' such that P(c1) # P(c2) and P(ci) # Q(d) for every zero d of Q' (i = 1, 2). I f f , g E M ( d ( a , r - ) ) satisfy P ( f )= Q(g), then f , g E M b ( d ( a , r - ) ) . Proof Let k be the number of zeros ci of P such that P ( c i ) # P(cj) Vi # j and P ( c i ) # Q ( d ) for every zero d of Q' (i = 1, 2). By Corollary 1.8, if one of the functions f and g belongs to M u ( d ( a , r - ) ) , then both belong to M U ( d ( a , r - ) ) .So, if f and g belong to M u ( d ( a , r - ) ) , then by Theorem 2.1 and the assumption that 2p < 39, we have k = 1, a contradiction.
Corollary 2.6 Let P , Q be in K [ x ]with P'Q' not identically zero and 3 5 deg(P) 5 4, deg(Q) 2. 3, deg(P) # deg(Q). Assume that there exist two distinct zeros c1 and c2 of P' such that P ( c i ) # Q ( d ) for every zero d of Q' (i = 1, 2). Moreover, ifdeg(P) = 4, we assume that P(c1) # P(c2). If f , g E M ( d ( a , r - ) ) satisfy p ( f )= Qb),then f, g E M b ( d ( a , r - ) ) . Proof Let p = deg(P), q = deg(Q). We notice that the inequality P(c1) # P(c2) is assured by Lemma 1.2 when p = 3 and is assumed when p = 4, hence hold in all cases. Assume that one of the two functions f, g belongs to
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M u ( d ( a , r - ) ) , hence by Corollary 1.8 both f and g belong to M u ( d ( a , r - ) ) . By Theorem 2.1 we have k = 1, a contradiction to the hypothesis. Corollary 2.7 Let P, Q be in K [ x ]with P'Q' not identically zero and p = deg(P), q = deg(Q), satisfying p < q and q - p # g.c.d.(p, 4). Assume that there exists a zero c of P' such that P(c) # Q ( d ) for every zero d of Q'. If f, g E M ( d ( a , r - ) ) satisfy P(f) = Q(g), then f , E M b ( d ( a , r - ) ) . Proof Assume that one of the two functions f, g belongs to M u ( d ( a , r - ) ) , hence by Corollary 1.8 both f and g belong to M , ( d ( a , r - ) ) . Since p # q, by Theorem 2.1 we have that either q < p or g.c.d.(p,g ) = q - p , both cases that are excluded from the hypotheses of the corollary. Hence both f and g belong to M d d ( a , r - ) ) Corollary 2.8 Let P, Q be in K [ x ] with P'Q' not identically zero and deg(P) = deg(Q) 2 4. Assume that there exist three distinct zeros c1, c2, c3 of P' such that P(c,) # P ( c j ) V i # j and P(ci) # Q ( d ) f o r every zero d of Q' (i = 1, 2, 3). Iff, g E M ( d ( a , r - ) ) satisfy f' # 0 and P ( f ) = Q(g), then f, E M b ( d ( a ,.-))* Proof Let q = deg(P) = deg(Q) and assume that f or g belong to M u ( d ( a ,r - ) ) , hence by Corollary 1.8 both f and g belong to M , ( d ( a , r - ) ) . Since f' # 0, we have v(f) = 0, hence by Theorem 2.1 f l ( p , f ) 2 T ( p , f)
(T) +
O(l), therefore lim N ( p , f ) - T ( p ,f) = +oo, which is absurd. p++w
Theorem 2.9 Let P, Q be in K [ x ]with P'Q' not identically zero and let p = deg(P), q = deg(Q) with 2 5 min(p, q ) . Assume that there exist k distinct zeros c1, ..., ck of P' such that P ( q ) # P ( c j ) V i # j and P(ci) # Q(d) for every zero d of Q' (i = 1,..., k). Assume that there exist two nonconstant functions f , g E M ( K ) such that P ( f ) = Q(g) and let t = v(f). Then q 5 p and f satisfies f l ( p , f) 17' ( ' f)('q -P) logp 0(1). Moreover, i f P- < q, 2 Xt4 then k = 1 and c1 is a simple zero of P'.
+
+
Corollary 2.10 Let P, Q be in K [ x ]with P'Q' not identically zero and deg(P) 5 deg(Q). Assume that there exists a zero c of P' such that P(c) # Q ( d ) f o r every zero d of Q'. If there exist f, g E d(K) satisfying P ( f ) = Q(g), then f and g are constant. Proof Let p = deg(P), q = deg(Q). Assume that f and g are not constant. Then by Theorem 2.9 we have E ( p , f) 1 T ( p , f) logp O(1) 2 logp
+ O(1), hence
(x)+
+
lim 8 ( p , f) = +oo, which is absurd. p-++w
Corollary 2.11 Let P, Q be in K [ x ]with P'Q' not identically zero and let p = deg(P), q = deg(Q) with 2 I min(p,q) and P- < q. Assume that there
2
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exist 2 distinct zeros c1, cg of P’ such that P(c1) # P(c2) and P(c,) # Q ( d ) for every zero d of Q‘ (i = 1 , 2 ) . Assume that there exist two functions f , g E M ( K ) such that P ( f ) = Q ( g ) . Then f and g are constant.
3 Case Q = XP, deg(P) 5 4. In order to specialize certain results from in [l]and [16] when dealing here with polynomials of degree 3 or 4, we must introduce new definitions and notice Lemmas 3.1 and 3.2 which are easily verified.
Definition and Notation A subset S of K is said to be afinely rigid if there exists no d n e transformation of K (i.e. of the form cp(z) = ax b) different from the identity, such that cp(S)= S [3], [9]. A polynomial P E K [ z ]is said to satisfy Condition (F) [12] if for any two distinct zeros a , b of PI, we have P ( a ) # P ( b ) (i.e. the restriction of P to the set of zeros of P‘ is injective).
+
Lemma 3.1 Let S be a subset of K with four elements. The following two conditions are equivalent. 1) The set S is not affinely rigid; 2) Either S is of the form {a,a u, a + w u , a + u 2 u } , with w 2 + w + 1 = 0, or S i s of the f o r m { a - h, a + h, a - k,a + k}, with h, k E K*.
+
Lemma 3.2 Assume 7r # 2 and let P E K [ x ] be monic of degree 4. The following three statements are equivalent: (i) P does not satisfy Condition (F). (ii) P is of the f o r m [(z - a 1)(z - a - 1)12 + A with A E K , 1 E K*. (iii) There exists an a f i n e change of variable transforming P into a n even function.
+
Corollary 3.3 Assume 7r # 2, let P E K [ x ]be of degree 4 such that its set of zeros is affinely rigid. Then P satisfies Condition (F). Now we have to recall a kind of characterization of uniqueness polynomials made in [17].The following Theorem 3.4 is extracted from [17]:
Theorem 3.4 Let P E K [ z ] be of degree n, with n o multiple zeros and 1
satisfying Condition (F). Let P’(x) = b n ( x - ~ j ) ~ with j , ci # C j V i # j j=1 and let S be the set of zeros of P . Moreover, i f r # 0 , we assume that the multiplicity of the factor (x - b j ) in P ( x )- P ( c j ) i s rnj 1 V j = 1,..., Z., and i f 7r divides n, we also assume that the coefficient of xn-’ in P is not 0.
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Then there do not exist X E K and nonconstant distinct functions f , g E M ( K ) such that P ( f )= X P ( g ) i f and only i f S is afinely rigid and P satisfies one of the following three conditions: i ) 1 2 2, K divides n; ii) 1 = 2, min(m1,ma) 2 2; 3) 1 2 3 and P does not satisfy: = Ekd = w andw2 + w + 1 = 0. n = 4, ml = m2 = m3 = 1, %.Ed P(c2) - P(c3) P(C1) Further, when X = 1, there do not exist nonconstant distinct functions f, g E M ( K ) such that P ( f ) = P ( g ) i f and only i f ( n - 2 ) ( n - 3) > mj(mj - 1).
Here we devote our efforts to examine particularly the cases n = 4 and 7r
# 2. In such a situation, we can characterize all polynomials of degree 4,
with four distinct zeros, such that there exist X E K and nonconstant distinct functions f, g E M ( K ) satisfying P ( f ) = XP(g). Moreover, we shall notice that when polynomials of degree 4 are concerned, one of the following two conditions is redundant: (1) the set of zeros of P is afinely rigid (2) Condition (F). Theorem 3.5 Assume K # 2, let P E K [ x ]be of degree 4 with four distinct zeros and let S be the set of its zeros. Then there do not exist X E K and nonconstant distinct functions f , g E M ( K ) such that P ( f ) = XP(g) i f and only if P satisfies the following three conditions: S is afinely rigid; PI has three distinct zeros c1, c2, c3; ‘(‘1) ‘(~2) ‘(‘3) = 8, with 82+8+ P does not satisfy the equality - -P ( C 2 ) - P(C3) - P ( C 1 ) 1 = 0. ~~
Theorem 3.6 Assume 7r # 2 and let P E K [ x ]be of degree 4 with four distinct zeros. Then there do not exist X E K and nonconstant distinct functions f, g E M ( K ) such that P ( f ) = XP(g) if and only i f P satisfies the following three conditions: P satisfies Condition (F); .~ PI has three distinct zeros c1, c2, c3; ‘(‘1) ‘(‘2) P does not satisfy the equality = -= ‘(‘3) - 8, with 02+e+
P(C2)
1 =o.
P(C3)
P(C1)
Remark According to Theorems 3.5 and 3.6 we might think that a set of four points is affinely rigid if and only if the polynomial P admitting these points for zeros satisfies Condition (F). Actually the equivalence only holds
P(Cd = P(C2) - P(C3) when the zeros cj of PI do not satisfy the equality P(c2) P(c3) = with e2 8 1 = 0. Indeed, let P ( x ) = x4 - 4 x and let 0 satisfy O2 0 1 = 0.
++
po
++
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The zeros of PI are 1, 8, 02,and '(') ='(02) = '(') - 8 hence Condition P(1) P(O) P P 2 ) (F) . . is obviously satisfied. However, the set of zeros of P is clearly preserved by the affine mapping h ( x ) = Ox and therefore is not affinely rigid. Now, we can give a complete characterization of uniqueness polynomials of degree 4: Theorem 3.7 Assume 7r # 2, let P E K [ x ] be of degree 4 and X E K . Then there exist nonconstant distinct meromorphic functions f , g E M ( K ) such that P ( f ) = X P ( g ) with X E K i f and only i f one of the following five conditions is satisfied: (i) P admits a zero of order 2 3, A E K*. (ii) X = 1 and there exists two different zeros c1, c2 of PI such that P(c1) = p(c2),
(iii) X = -1 and there exists three different zeros c1, c2, c3 of P' such that p ( c l ) = P(c3) = - p ( c 2 ) , (iv) X = 1 and PI admits a zero of order 2 2, (v) X2 + X + 1 = 0 and the three zeros c1, c2, c3 of PI, provided with a suitable
[a] we know that an algebraic curve over a nonarchimedean field admits a parametrization by meromorphic functions if and only if it is of genus zero, i.e. if and only if it admits a parametrization by rational functions. Thus, if one of the conditions (i),(iz), (iii), (iv), (v) of Theorem 3.7 is satisfied, then the curve of equation P ( x ) = X P ( y ) is of genus 0 and therefore there exist f , g E K ( t ) such that P ( f( t ) )= X Q (9( t )1.
Remark By the Picard-Berkovich Theorem
4 References [l] An Ta Thi Hoai, Wang J. T.-Y. and Wong P.-M. Uniqueness range sets and uniqueness polynomials in positive characteristic. Acta Arithmetica 193,3, p.259-280 (2003). [2] Berkovich, V. Spectral Theory and Analytic Geometry over non-Archimedean Fields. AMS Surveys and Monographs 33, (1990). [3] Boutabaa, A. Escassut, A. and Haddad, L. O n uniqueness of p-adic entire functions. Indagationes Mathematicae 8 p.145-155, (1997).
[4] Boutabaa, A. and Escassut, A. Urs and Ursims for P-adic Meromorphic Functions inside a Disk. Proceeding of the Edinburgh Mathematical Society, 44, 485-504 (2001).
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[5] Boutabaa, A. , Escassut, A. Applications of the p-adic Nevanlinna theory to functional equations Annales de I’Institut Fourier, T.50 (3),p751-766. (2000). [6] Boutabaa, A. and Escassut, A. Nevanlinna Theory in characteristic p and applications Preprint. [7] Boutabaa, A. and Escassut, A. Nevanlinna Theory in characteristic p and applications Analysis and Applications, 3rd ISAAC Congress, Kluwer, p.97-107 (2001). [8] Cherry, W. Non-Archimedean analytic curves in Abelian varieties, Mathematishe Annalen 300, p.396-404 (1994). 191 Cherry, W. and Yang C.-C. Uniqueness of non-Archimedean entire functions sharing sets of values counting multiplicities, Proceedings of the AMS, Vol 127, n.4, p. 967-971, (1998).
[lo] Escassut, A. Analytic Elements in p-adic Analysis. World Scientific Publishing Co. Pte. Ltd. (Singapore, 1995). [ll] Escassut, A. Haddad, L. Vidal, R. Urs, Ursim and nonurs f o r p-adic Functions and Polynomials. Journal of Number Theory 75, p.133-144 (1999).
[12] Fujimoto, H. O n uniqueness of Meromorphic Functions sharing finite sets, Amer. J. Math. 122 no. 6, 1175-1203 (2000). [13] F‘ulton, W. Algebraic Curves, An Introduction t o Algebraic Geometry, W. Benjamin Inc. New-York, (1969). [14] Ha, H.K. and Yang, C.C. O n the functional equation P ( f ) = Q ( g ) , Value Distribution Theory, Marcel Dekker (2003). [15] Hu, P.C. and Yang, C.C. Meromorphic Functions over non-Archimedean Fields. Kluwer Academic Publishers, (2000). [16] Hua, X.H. and Yang, C.C. Unique polynomials of entire and meromorphic functions. Matematicheskaia Fizika Analys Geometriye, v. 4, n.3, p. 391-398, (1997). [17] Wang, J. T.-Y. Uniqueness polynomials and bi-unique range sets f o r rational functions and non-Archimedean meromorphic functions. Acta Arithmetica, 104, p.183-200 (2002). [18] Yang, C.C. and Li, P. Some further results o n the functional equation P ( f ) = Q ( g ) , Preprint.
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Conjectures and Counterexamples in Dynamics of Rational Semigroups Rich Stankewitzl, Toshiyuki Sugawa2, and Hiroki Sumi3 Department of Mathematical Sciences Ball State University Muncie, IN 47306
USA rstankewitz(0bsu.edu Department of Mathematics Graduate School of Science
Hiroshima University Higashi-Hiroshima, 739-8526 Japan sugawa(0math.sci.hiroshima-u.ac.jp
Department of Mathematics Tokyo Institute of Technology 2-12-1 O-Okayama, Meguro-ku Tokyo 152-8551 Japan sumi(0math.titech.ac.jp
Summary. We will discuss the dynamics of rational semigroups, an extension of the Fatou-Julia theory of iteration of a rational map defined on the Riemann sphere. Specifically, we will give some counterexamples to some conjectures relating to completely invariant Julia sets and nearly Abelian polynomial semigroups. We then state the modified conjectures as open problems.
1 Introduction A rational semigroup G, as defined by A. Hinkkanen and G. Martin in [3] t o generalize the dynamics of the iteration of rational maps, is a semigroup of rational functions of degree greater than or equal t o two defined on the Riemann sphere with the semigroup operation being functional composition. When a semigroup G is generated by the functions { f ~ , f i , ., . ,fn,.. .}, we write this as G = ( f l , f2,.
. . ,fn,.. .).
(1) It should be noted that often one may allow some or all of the maps in G to be Mobius, for example, when one is considering Kleinian groups as in [8].
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Rich Stankewitz, Toshiyuki Sugawa, and Hiroki Sumi
Since the examples constructed here all contain maps of degree two or more, we will use our simplified definition to avoid any technical complications which are not necessary to the aims of this paper. For a rational semigroup G we define the set of normality of G (also called the Fatou set of G ) by
N(G)= { z E
: G is a normal family on some neighborhood of z }
(2)
and define the Julia set of G by
J(G)=C \ N(G).
(3)
Clearly from these definitions we see that N ( G ) is an open set and therefore its complement J ( G ) is a compact set. These definitions generalize the case of iteration of a single rational function and we write N ( ( h ) )= Nh and J ( ( h ) ) = Jh. Note that J ( G ) contains the Julia set of each element of G. Moreover, the following result due to Hinkkanen and Martin holds.
Theorem 1 ([3],Corollary 3.1) For rational semigroups G we have
J(G)=
U
Jj.
f€G For research on (semi-)hyperbolicity and Hausdorff dimension of Julia sets of rational semigroups, see [lo, 11, 12, 131.
2 Partial and Complete Invariance If h is a map of a set Y into itself, a subset X of Y is called: i) forward invariant under h if h ( X ) c X ; ii) backward invariant under h if h - l ( X ) c X ; iii) completely invariant under h if h ( X ) c X and h - ' ( X )
cX.
It is well known that for a rational function h the set of normality and the Julia set are completely invariant under h (see [2], p. 54), i.e., h(Nh) = N h = h-'(Nh)
and
h ( J h ) = Jh = h-l(Jh).
(4)
In fact, we have the following result which may be chosen as an alternate definition to the definition of Jh given in (3) in terms of normality.
Property 1. The set Jh is the smallest closed completely invariant (under h) set which contains three or more points (see [2], p. 67).
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551
F'rom the definition given in (2), we get that N ( G ) is forward invariant under each element of G and thus J ( G ) is backward invariant under each element of G (see [3], p. 360). The sets N ( G ) and J ( G ) are, however, not necessarily completely invariant under the elements of G (see Example 2). This is in contrast to the case of single function dynamics as noted in (4). The question then arises, what if we required the Julia set of the semigroup G to be completely invariant under each eIement of G? We consider in this paper the consequences of such an extension, given in the following definition. For a rational semigroup G we define the completely invariant Julia set E = E ( G ) to be
E = n{S : S is closed, completely invariant for each g E G , # S 2 3)
(5)
where # S denotes the cardinality of S , and we define the completely invariant set of normality of G to be
W ( G )= C \ E(G). We note that E(G)exists, is closed, is completely invariant under each element of G and contains the Julia set of each element of G by Property 1. Thus, by Theorem 1, it follows readily that J ( G ) C E(G).
3 Examples and Conjectures Example 1. Suppose that G = (f,g) and J f = Jg. Then E ( G ) = J f = Jg since J f is completely invariant for f and J g is completely invariant for g . Since E ( G ) 1J ( G ) 3 J f = Jg we see that E(G) = J ( G ) = J f = Jg. Example 2. Let a E C be such that Jal > 1 and set G = ( z 2 , z 2 / a ) . One can easily show that J ( G ) = { z : 1 5 IzI 5 )1.1 (see [3], p. 360). Since E ( G ) is forward invariant and E(G) 3 J ( G ) , we conclude that E ( G ) = by noting that the forward images of J ( G ) under the iterates of each generating map union to C \ ( 0 ) . Note that I n t ( J ( G ) ) # 0 yet J ( G ) # This departs from the iteration case where it is known that I n t ( J h ) # 0 for a rational map h implies that Jh must equal all of However, for the completely invariant Julia set we get the result suggested by the iteration case, namely, I n t ( E ( G ) )# 0 implies E ( G ) = (see [6], Lemma 2).
c.
c.
When polynomial semigroups G (rational semigroups consisting of only polynomials) are considered, Examples 1 and 2 are somewhat canonical, that is, the completely invariant Julia set E ( G ) will be trivial (in that it equals the Julia set of some map in G ) ,or E(G) = Furthermore we have the following result.
c.
Theorem 2 ( [ 6 ] ,Theorem 2) For a rational semigroup G which contains
two polynomials f and g , we have that J f
# Jg implies E ( G ) = c.
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Rich Stankewitz, Toshiyuki Sugawa, and Hiroki Sumi
The following examples show that the same classification of E ( G ) does not hold if we remove the restriction that the maps are polynomials.
Example 3. Suppose f and g are such that Jg is completely invariant under f, but J f 2 J g . Specifically, we may select f ( z ) = 22 - z-l and g ( z ) = ( z 2- 1)/(2z). It is known that J f is a Cantor subset of [-1,1] (see [2], p. 21) and Jg = R (since g ( z ) is the conjugate of z H z 2 by a Mobius map which takes the unit circle J,z to R). Then E ( ( f , g ) )= J ( ( f , g ) )= Jg # J f . The following example demonstrates that, unlike the above examples, it is not always that case that E ( G ) = J ( G ) or E ( G ) =
c.
+
Example 4. Let f ( z ) = 22 - z-l and g ( z ) = f ( z - 1) 1. In this case J f is a Cantor subset of the interval [-1, I] and Jg = J f 1. It is shown in [7] that J ( (f,9 ) ) = [-1,2] (by using iterated function system theory) and E ( (f , 9 ) ) =
+
R. In the above and other examples, the completely invariant Julia sets E(G) of rational semigroups are trivial (equal to the Julia set of one a single member of the semigroup G ) ,the entire Riemann sphere or the Mobius equivalent of a circle. This led Hinkkanen and Martin to put forth the following conjectures (see [71).
c,
Conjecture 3 (Hinkkanen and Martin) I f G i s a rational semigroup which then contains two maps f and g such that J f # Jg and E(G) # W ( G )= \ E(G) has exactly two components, each of which is simply connected, and E ( G ) is equal t o the boundary of each of these components.
c
c,
Conjecture 4 (Hinkkanen and Martin) If G is a rational semigroup which contains two maps f and g such that J f # Jg and E(G) # 6, then E(G) is a simple closed curve in 6. In the next section we give a method for constructing functions (as well as providing concrete functions) whose Julia sets are unequal, but which generate a semigroup whose completely invariant Julia set is a line segment. Hence the above conjectures do not hold. But since the only non trivial completely invariant Julia sets of rational semigroups which are known at this time are (see [6] and [7])or sets which are Mobius equivalent to a line segment or circle, the authors put forth the following conjecture, which is currently unresolved.
Conjecture 5 If G is a rational semigroup which contains two maps f and J f # Jg and E ( G ) is not the whole Riemann sphere, then E(G) is Mobius equivalent to a line segment or a circle.
g such that
Remark 1. We briefly explain some evidence that compels us to pose Conjecture 5 the way we did. Our example of a rational semigroup G with E ( G ) being a line segment is rigid since G contains a Tchebycheff polynomial, which is known to be postcritically finite (and hence, rigid). On the other hand, an
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553
example of a rational semigroup G with E ( G ) being a (unit) circle generated by rational functions f l , . . . ,f n with non-equal Julia sets is easily constructed by choosing finite Blaschke products as the fj's. However, it seems difficult to quasiconformally deform f i , . . . ,f n simultaneously so that the completely invariant Julia set of the resulting rational semigroup is not a circle.
4 Counterexamples to Conjectures 3 and 4
+
$ ( z ) = ( z 2 - 1)/(z2 1) and U = { z : 3 z C \ [-1,1] in a one-to-one onto fashion and C#I : one onto fashion.
Let -
> 0). Then q5
a
t
: U f2 = I = [-1,1] in a two-to---f
Lemma 1 (see [9]). Let f be a rational map such that f ( U ) = U . Then there exists a rational map f such that 4 o f = f o 4 i f and only i f f is odd.
See [9] for a classification of such maps f and their semi-conjugates
$.
Lemma 2 (see [9]). For (semi-conjugate) rational semigroups G = (gj : j E = (hj : j E 2)where there exists a rational function k satisfying the semi-conjugacy relation k o hj = gj o k f o r each j E Z, we have
1)and H
J(G)= k(J(H))
and
N(G)= k(N(H)).
If we also have that k - l ( k ( E ( H ) ) = E ( H ) , then E(G)= k ( E ( H ) )
and
W ( G )= k ( W ( H ) .
With these lemmas we can now construct our counterexamples.
Example 5 (Counterexamples t o Conjectures 3 and 4 ). Let f be an odd rational map such that f ( U ) = U.Then by Lemma 1 there exists a rational function f satisfying the semi-conjugacy relation 4 o f = f o 4. Similarly we let g be an odd rational map with g ( U ) = U and so there exists a rational map ij with 4 o g = ij o 4. By choosing f and g such that J f # R and Jg = R, we have that J f # I and J? = I by Lemma 2. Since is completely invariant under both f and g we have E ( G ) c R where G = ( f , g ) . Since E ( G ) 3 Jg = R, we conclude that E ( G ) = E.For G = ( f 1 , i j ) we see that since $ - l ( $ ( E ( G ) ) )=-+-l(+(R))= R = E ( G ) ,we must have E ( G ) = = I. Since J f # J?, G is a counterexample to Conjectures 3 and 4. Specifically we may select f (2) = 2z -2-l and g ( z ) = ( z 2- 1)/(2z). Hence J f is a Cantor subset of I (see [2], p.21). Since g is the conjugate of z H z 2 under z H we see that Jg = E. In this case one can calculate (via the proof of Lemma 1) that $(z) = (32 5z2)/(1 32 4z2)and i j ( z ) = 2.z2- 1.
+(a)
z e
+
+ +
Note that in the above example J ( G ) = E(G). In the next example, we construct a semigroup G that provides a counterexample to Conjectures 3 and 4 with the additional property that J ( G ) 5 E(G).
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Rich Stankewitz, Toshiyuki Sugawa, and Hiroki Sumi
Example 6. Consider f ( 2 ) = 22 - 2-l as in Example 5. Let cp(z) = 22, and z )22-4z-l. Note that Jg = cp(Jf)= 2Jf and that set - g(z) = (cpo f o ( ~ - ~ ) ( = W is completely invariant under g. Hence for G = (f,g ) , we have E(G) c 8. Suppose that E ( G ) # 8. Since 8 is completely invariant under both f and g, it follows from Lemma 3.2.5 in [5] that if E ( G ) contains a non-degenerate interval in the real line, then E ( G ) = 8.Hence we may select an open interval L = (z, y ) in W \ E ( G ) with both 2,y large. Since the length of the intervals f n ( L ) tends to +00, we may assume that y - z is large. By expanding the interval we may also assume that z, y E E ( G ) (note that we used here that 00 is a non-isolated point in E(G) which follows since 2 E Jg C E ( G ) and
fnP) 4. +
Since z is large, we can use the fact that f(z)is slightly greater than g(z) to see that g-l((f(z>}) contains a point slightly larger than z (and hence less than y). But by the complete invariance of the set E ( G ) under f and g , we get g - l ( { f ( z ) } )c E(G).This is a contradiction since the interval (z, y ) does not meet E ( G ) .We conclude that E ( G ) = 8. Since 00 is an attracting fixed point under both f and g, we see that small neighborhoods of 00 map inside themselves under each map in G. Hence 00 E N ( G ) and so J ( G ) # E. As in Example 5 we may semi-conjugate the odd rational maps f and g by 4 to get maps f(z) = (3.2 5z2)/(1 3z 4z2) and ij(z) = (5z2 40.2 29)/(3z2+40z-27). Hence for G = (f,ij)we have J ( G ) = d ( J ( G ) )S; $(8)= I and E ( e ) = 4 ( E ( G ) )= 4(8)= I . Since J j # Jg (otherwise one would have E ( e )= J ( e ) = J j = Ja), we see that 6' is a counterexample to Conjectures 3 and 4.
+
+ +
+
5 Polynomial Semigroups In [3], p. 366 Hinkkanen and Martin give the following definition.
Definition 6 A rational semigroup G is nearly abelian i f there is a compact family of Mobius transformations @ = {4} with the following properties: (i) 4 ( N ( G ) )= N ( G ) for all 4 E @, and (ii)f o r all f,g E G there is a 4 E @ such that f o g = 4 o g o f. Theorem 7 ([3],Theorem 4.1) Let G be a nearly abelian semigroup. Then f o r each g E G we have Jg = J ( G ) .
A natural question is to what extent does the converse to Theorem 7 hold. Using a result of A. Beardon (see [l],Theorem 1) Hinkkanen and Martin have proved the following result for polynomial semigroups. Theorem 8 ([3],Corollary 4.1) Let 3 be a family of polynomials of degree at least 2, and suppose that there is a set J such that Jg = J f o r all g E 3. T h e n G = (3)is a nearly abelian semigroup.
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555
Note that under the hypotheses of Theorem 8 we have Jh = J ( G ) for each generator h E F.The following conjecture due to Hinkkanen and Martin [4] suggests that if Jh = J(G) for just one h E G, then G is still nearly abelian. Conjecture 9 Let G be a polynomial semigroup such that Jh = J ( G ) f o r some h E G. Then J f = Jg for all f , g E G (and hence G i s nearly abelian by Theorem 8).
The following counterexample shows that this conjecture requires the allowance of a special case. Example 7 (Counterexample t o Conjecture 9). Define f (2) = z2 - 2,g(z) = 4z2 - 2 and G = (f,9). One can quickly verify that f is a conjugate of 22' - 1 by z H 22 and so J f = [-2,2] (see [2], p. 9). It can easily be seen that g maps [-1,1] onto [-2,2] in a two-to-one fashion. Since g-'([-l, 11) c g-'([-2,2]) = [-1,1] it follows that J g c [-1,1]. In particular Jg J f . We also note that C \ [-2,2] is forward invariant under both f and g and as such must lie in N ( G ) by Montel's Theorem. It follows that J(G) = I-2,2] = J f , yet J f # Jg. We remark that any map g that maps a proper sub-interval of [-2,2] onto [-2,2] in a deg(g)-to-one fashion would suffice in the above example and such functions can easily be obtained by constructing real polynomials with appropriate graphs. Also, f may be replaced by any Tchebycheff polynomial (see section 1.4 of [2]), normalized so that J f = [-2,2].
In our counterexample J(G) is a closed line segment. Since no other types of counterexamples are known, we modify this conjecture as follows and note that it remains unresolved. Conjecture 10 Let G be a polynomial semigroup such that Jh = J ( G ) f o r some h E G where J(G) i s not a line segment. T h e n J f = Jg f o r all f , g E G (and hence G i s nearly abelian by Theorem 8).
References 1. A. F. Beardon. Symmetries of Julia sets. Bull. London Math. SOC., 22:576-582, 1990. 2. A. F. Beardon. Iterations of Rational Functions. Springer-Verlag, New York, 1991. 3. A. Hinkkanen and G.J. Martin. The dynamics of semigroups of rational functions I. Proc. London Math. Soc., 3:358-384, 1996. 4. A. Hinkkanen and G.J. Martin. Personal communication. 1997. 5. R. Stankewitz. Completely invariant Julia sets of rational semigroups. PhD thesis, University of Illinois, 1998. 6. R. Stankewitz. Completely invariant Julia sets of polynomial semigroups. Proc. Amer. Math. SOC., 127(10):2889-2898, 1999. 7. R. Stankewitz. Completely invariant sets of normality for rational semigroups. Complex Variables Theory Appl., 40(3):199-210, 2000.
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8. R. Stankewitz. Uniformly perfect sets, rational semigroups, Kleinian groups and IFS’s. Proc. Amer. Math. SOC., 128(9):2569-2575, 2000. 9. R. Stankewitz, T. Sugawa and H. Sumi. Some counterexamples in dynamics of rational semigroups. Preprint. 10. H. Sumi. On hausdorff dimension of julia sets of hyperbolic rational semigroups. Kodai. Math.J., 21(1):10-28, 1998. 11. H. Sumi. Skew product maps related to finitely generated rational semigroups. Nonlinearity, 13:995-1019, 2000. 12. H. Sumi. Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th.& Dynam. Sys., 21:563-603, 2001. 13. H. Sumi. Semi-hyperbolic fibered rational maps and rational semigroups. preprint .
