APPROXI MATI 0 N PROBLEMS IN ANALYSIS AND PROBABILITY
NORTH-HOLLAND MATHEMATICS STUDIES 159 (Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.
NORTH-HOLLAND -AMSTERDAM
' NEW YORK OXFORD TOKYO
APPROXIMATION PROBLEMS IN ANALYSIS AND PROBABILITY
M.P. HEBLE Department of Mathematics University of Toronto Toronto, Canada
1989 NORTH-HOLLAND - AMSTERDAM ' NEW YORK
OXFORD TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the U S A . and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas NewYork, N.Y. 10010, U.S.A. Library o f Congress Cataloging-in-Publication Data
H e b l e . M. P. Approximation problems i n analysis and probability / M.P. Heble. p. c m . -- ( N o r t h - H o l l a n d m a t h e m a t i c s s t u d i e s ; 159) Includes bibliographical references. I S B N 0-444-88021-6 1 . A p p r o x i m a t i o n t h e o r y . 2. M a t h e m a t i c a l a n a l y s i s . 3. P r o b a b i l i t i e s . I. T i t l e . 11. S e r i e s . O A 2 2 1 .H375 1989 511'.4--dc20 89-16147 CIP
.ISBN: 0 444 88021 6 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989
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i
No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in the Netherlands
To My mother Girijabai My uncle Rama Rao and Sushila, Ajay and Sucheta
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Vii
Table of Contents
ix
Introduction Chapter I. Weierstrass-Stone theorem and generalisations - a brief survey
8 1. Weierstrass-Stone theorem 2. 3. 4. 5.
Closure of a module - the weighted approximation problem Criteria of localisability A differentiable variant of the Stone-Weierstrass theorem Further differentiable variants of the Stone-Weierstrass theorem
Chapter II. Strong approximation in finitedimensional spaces 1. 2.
H. Whitney’s theorem on analytic approximation C” -approximation in a finitedimensional space
Chapter III. Strong approximation in infinitedimensional spaces § 1.
5 2.
§3. §4. 5 5. §6. § 7.
§8. §9. 0 10. fill.
Kurzweil’s theorems on analytic approximation Smoothness properties of norms in LP-spaces C”-partitions of unity in filbert space Theorem of Bonic and Frampton Smale’s Theorem Theorem of Eells and McAlpin Contribution of J. Wells and K. Sundaresan Theorems of Desolneux-Moulis Ck-approximation of Ck by Cw-a theorem of Heble Connection between strong approximation and earlier ideas of Bernstein-Nachbin Strong approximation - other directions
Chapter IV. Approximation problems in probability 1.
5 2. 53. $4. 5.
Bernstein’s proof of Weierstrass theorem Some recent Bernstein-type approximation results A theorem of H. Steinhaus The Wiener process or Brownian motion Jump processes - a theorem of Skorokhod
Appendix 1 : Appendix 2: Appendix 3: Appendix 4: Bibliography Index
Topological vector spaces Differential Calculus in Banach spaces Differentiable Banach manifolds Probability theory
1
2 6 19 31 34 41
41 61 77 77 95 99 101 103 107 111 121 127 153 154 169
170 172 178 183 189 20 1 215 223 229 237 243
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ix
Introduction The classical Weierstrass-Stone theorem and the Bernstein-type weighted approximation theorems were greatly extended by L. Nachbin. Another aspect of approximation theory, now called strong approximation and initiated by H. Whitney, had simultaneously developed, with contributions in finite-dimensional spaces as also in infinitedimensional spaces, by various individuals. At the same time, several approximation results in a probabilistic setting - from the elegant probabilistic proof of Weierstrass’ theorem by S. Bernstein to the later results on convergence of stochastic processes established by A.V. Skorokhod and other later authors - were being added to the literature. The material in this book covers some special aspects of the approximation theory of functions, viz. strong approximation in function spaces, as also certain approximation results concerning stochastic processes. The choice of topics reflects only the author’s taste. Within the narrow range of topics chosen, I have tried to do as thorough justice as I could, to the subject as also to the contribution of various individuals active in their respective areas; any possible omission of names is unintentional. This book is meant to be a monograph, of interest to research workers in the fields of analysis, probability, and stochastic processes. Graduate students, hopefully, will find it useful not merely as a source of information but also as an incentive to spur then on to do further work. The author has noted other monographs recently published, covering related topics. However, the contents of these books show that the overlap between these and my present monograph is negligible (e.g., cf. K. Sundaresan and S. Swami-
Introduction
x
nathan: “Geometry and non-linear analysis in Banach spaces”, Springer Verlag Lecture Notes in Math. No. 1131, 1986; and
J.G.Llavona: “Approximation of continuously dif-
ferentiable functions”, Notas de Matematica No. 130, North Holland 1986).
A quick description of the contents of this book appears to be in order. The material is divided into four chapters. The first chapter gives a quick survey of the classical Weierstrass-Stone theorem, Bernstein’s weighted approximation problem and Nachbin’s extension of the classical Bernstein approximation results. The material in this chapter excluding sections 4 and 5 , is mostly a summary of Professor Leopoldo Nachbin’s monograph: “Elements of approximation theory”. In Chapter I1 we present strong approximation results in a finite-dimensional space R” - first H. Whitney’s theorem on strong approximation by real analytic functions, and then some results on Coo-approximation (strong sense); the latter appear to have been commonly known and there are excellent expositions in several monographs, hence we have been content with only a summary in this book. Chapters I11 presents strong approximation results in finite- or infinitedimensional separable spaces, starting with Kurzweil’s extension of Whitney’s theorem (on analytic approximation), and ending with some recent results established by this author, as also an indication of possibilities in other directions. We also explain a connection between strong approximation results and the earlier Bernstein-Nachbin ideas. In Chapter IV we present some probabilistic approximation results, starting with a quick look at Bernstsein’s well-known proof of Weierstrass’ theorem with some recent developments, and ending with some results by A.V. Skorokhod on approximation of stochastic processes. Here, again, individual choice was the guiding factor. We thought it necessary
xi
Introduction
to leave out the enormous area of weak approximation - an area which has found excellent exposition in several monographs, e.g., M. Rosenblatt: “Markov processes, structure and asymptotic properties” (Springer Verlag 1971), and D. Pollard: “Convergence of stochastic processes”, (Springer Verlag 1984).
As for organisation of the the material, theorems, lemmas, etc., are numbered according to chapter and section; thus Theorem I1 2.1 means Theorem 1 in section 2 of Chapter 11. Equations and formulae are numbered consecutively, but the numbering is separate for each section and each chapter. We have used the common symbols: “3” for “there exists”, “V’ for “for all” or “for any”, “3” for “such that”,
“+” for “implies”,
and “H”for “if and only if”, C denotes the set of complex numbers, R the set of real numbers and R” the n-dimensional real Euclidean space. There are four appendices at the end of the book explaining basic background material without proofs, and with sufficient further references. The writing of this monograph was partially supported by an NSERC operating grant. Thanks are due to Shirley Chan and Pat Broughton for patient and expert typing of the manuscript. I am indebted to Professor Leopoldo Nachbin, first, for encouraging me to write this book for the Notas de Matematica series, and secondly for permission to summarise the material of his monograph: “Elements of approximation theory”.
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CHAPTER I Weierstrass-Stone theorem and generalisations
- a brief survey
In the first three sections of this chapter we shall review known results concerning the classical Weierstrass-Stone theorem (cf. [SO]),Bernsteiris approximation problem, (cf. [41])and further generalisations by L. Nachbin. The material in this chapter, excluding sections 4 and 5, is taken from L. Nachbin’s published lecture notes [39]and for this reason we shall often present only a summary, leaving it to the reader t o refer to his monograph for further details. Any missing proofs will be found in [43].For convenience in presentation, the chapter is divided into five subsections. For the concepts of functional analysis used we refer the reader to Appendix 1. Throughout this chapter we shall assume E to be a completely regular space, i.e., a Hausdorff topological space such that for any a E E , and any closed subset F c E not containing a, there is a continuous real-valued function f on E such that 0 5 f 5 1, f ( u ) = 1 and f ( z ) = 0 for any
2
E F.
We shall denote by C ( E ;C) the commutative algebra with unit of all continuous complex-valued functions on E ; if E = 4 we set C(E ;C) = ( 0 ) . For convenience we shall often write C ( E ) for C ( E ;C). Every compact set K norm
l l f l l ~ ‘kfmax{ If(z)l, z € K } on C ( E ) . Let I?
cE
determines an algebra semi-
be the family of such semi-norms (cf.
Appendix 1). We shall understand that C ( E ) is endowed with the compact-open topology viz. the topology rr determined by the family I? of semi-norms. Then C ( E )becomes a topological algebra, i.e., a topological vector-space which is also an algebra. If E is compact then C ( E ) becomes a Banach algebra.
Chapter I
2 We note that the topology
7,.
is the topology of uniform convergence on compact
sets. This fact will be utilised in certain proofs. In proving results for a C ( E )with E completely regular, it will be often convenient to prove any one such result on the assumption that E is a compact space, the result will then follow for a completely regular space. Sometimes it may be necessary to use the result: every element in C ( K ) ,where
I< c E is compact, has an extension to C ( E ) . The subalgebra of C ( E ) consisting of real-valued continuous functions on E will be denoted by C ( E ;R), and will be endowed with the subspace topology inherited from r r .
$1. Weierstrass-Stone theorem. The first non-trivial theorem that we should note is the following result, due to S. Kakutani and M.H. Stone. A subset L
c C ( E ;R) is called a lattice if f , g
E L
=+
sup(f, g) E L and inf ( f , g ) E L. Theorem 1.1.
(Kakutani-Stone)
Then f belongs to the closure of L
Proof.
Let L
c
C ( E ;R) be a lattice and f E C(E;R)
Vxl,3'2 E E , and VE > 0 , 3 g E L 3
Only the sufficiency part of the assertion meeds some attention. As noted ear-
lier, it is enough to prove the statement on the assumption that E is compact. So suppose E is compact. Let f € C ( E ;R) satisfy the condition (l),and let compactness, it follows that for any given t E E , 39, E L satisfying:
E
> 0. Then, using
Weierstrass-Stone theorem and generalisations
-
a brief survey
3
Then again using compactness, we find that 3g E L satisfying:
f(u)- E < g(u) < f ( u ) Thus 119 - f
+
E
for any u E E
.
l l ~< E , and hence g E z.This completes the proof.
The next step is to note a result concerning the closure of an ideal I
ideal in C ( E )is a nonempty subset I
c
c C ( E ) . An
C ( E )such that for any g E I, f g E I Vf E
C(E). Proposition 1.2.
Let I
c
C ( E ) be a n &:id, and let f E C ( E ) . Suppose N is the
closed subset of E consisting of all x E E 3 g(x) = 0 Vg E I. Then f E
Proof.
We note that N = nfEr f-l({O}),
f %f
= 0 on N .
hence N is closed. Also for each z E E the
delta function 6, is continuous (cf. Appendix). For each z E N , 6, vanishes on I, hence
6, = 0 on f. This proves
“j”.
Next suppose E is compact and suppose f E C ( E )vanishes on N . Let define X = {z E E 0
[If
I If(.)[
5 h 5 1, and h ( z ) = 1Vz
2
E}.
Then X is closed, and X
nN
6. Now let
c C ( E ;C) is said to be
+ f E X , where f is the complex conjugate of f .
(Weierstrass-Stone) Suppose A
c C ( E ) is a subalgebra,
assume to be self-adjoint in the complex case. Let f E C(E).Then f E A H (1)
Vz1,22
EE
h EI3
“+”. The proposition follows.
We turn now t o the Weientrass-Stone theorem. A subset X
Theorem 1.3.
> 0, and
E X (one such h does exist). Now set g = f h E I. Then
- 9113 < E , and hence f E f. This proves
self-adjoint if f E X
=
E
3 f(z1) # f ( ~ ) , 3 E g
A 3 d.1)
(2) Vx E E 3 f ( z ) # 0, 3g E A 3 g(x) # 0. For the proof the following lemmas are needed.
# 9(z2);
which we
4
Chapter I
Lemma 1.4. If k 1 0, and
I Ip(t) - It1 I<
E M
E
> 0, then 3 polynomial p
: R + R, 3 p ( 0 ) =
3 It1 5 k.
Since the statement is trivial for k = 0, we shall suppose k
Proof.
0 and
Lemma is true for k = 0 then it is true for any k
It1 5 1; then I k p ( i ) - It11 < k& for
It1
> 0. Also if the
> 0. For suppose I p ( t ) - It1 I<
5 h. Hence we shall assume k
1
tER .
Then pn(0) = 0, and by induction it follows that
Next for n 2 1 we find (by induction)
and hence for n = 0,1,2, . . . ,
Now let
E
3 0
< E I 1. Then from (2) and (3) it follows that 3 integer no 2 0 3 0
I It1 - pn(t) I It1 I E
if
It( I E ,
E n o I It1 - pn(t) 5 (1 - 5) IE
if
E
5
Jtl5 1,n 2 no
.
Hence 0 I It( - pn(t) I e if It1 5 1, n 2 no. Thus the lemma is proved.
Lemma 1.5.
(Lebesgue) Every closed sufmlgebra A
for
= 1. Now we define
the polynomials p,, for n = 0, 1 , 2 , . . . , on R, by:
PO = O,pn+l(t) = pn(t) t $ [ t 2 - pn(t)’],
E
c C(E;R) is a lattice.
Weierstrass-Stone theorem and generalisatwns - a brief survey
Lemma 1.6. Let A
c R2 be a subn.lgebra and b E
R2. Then b
#A
($
5
at least one of
the following conditions holds:
(1) b
# R x 0 and A c R x 0;
(2) b E 0 x R and A
(3) b # A and A
c 0 x R;
c A.
Proof of Theorem 1.3. Again we shall attend only to the sufficiency part of the proof. Suppose the conditions (1) and (2) of the theorem hold. If q , z 2 E E the mapping g € C ( E ;R)
-+
(g(zl), g(z2)) E R2 is an algebra homomorphism, which we shall
call 0. By (2) if 0(f) !$ R x 0, then @ ( A ) @ 0 x R. Again by (l),if 0(f) $! A, then
@ ( A )@ A. Then by Lemma 1.6, @(f) E @ ( A )i.e., 3g E A c A; A is a lattice by Lemma 1.5, hence f E
(using Theorem l . l ) , i.e., f E A. Thus the theorem is proved.
Next, using the result of the next lemma, Theorem 1.3 follows in the complex case as well.
Lemma 1.7. Let A c C ( E ; C )be a self-adjoint algebra. Then the set R e A of the real parts Ref of all f E A is a subalgebra of C(E;R) and A = R e A
+iReA.
The following is a consequence of Theorem 1.3.
Corollary 1.8. Suppose A c C(E)is a. subalgebra which we assume to be self-adjoint in the complex case. Then A is dense in
C(E)H
(1) A is “separating on E” i.e., V z 3 , ~E E with z1 #
52,
(2) A is “non-vanishing on E”, i.e., Vx E E!, 3g E A 3 g(z)
3g
E A 3 g ( q ) # g(z2);
# 0.
We also note here that the classical Weierstrass theorem follows as a consequence of the Weierstrass-Stone theorem.
Chapter 1
6
$2. Closure of a module, the weighted approximation problem. The closure theorems of the preceding section can be seen to be special instances of closure theorems concerning modules; this is the topic of this section. Given a set A of C-valued functions on a set E , we introduce an equivalence relation on E , denoted EIA, as follows: if x1,x2 E E , the x1
N
x2
modulo E I A if f(x1) =
f (x2)V-f E A. Theorem 2.1.
Suppose A
c C ( E )is a subalgebra containing the unit,
to be self-adjoint in the complex case, W module over A, i.e., AW class X
cE
c W ; and let f
modulo E/A,V compact set
and is assumed
c C ( E ) is a vector subspace which is also a E C(E). Then f E
E
C(E)
I( c X , and VE > 03w E W
V equivalence
3 lw(x) - f(x)l
<
EVX E K . The proof requires the existence of a “continuous partition of unity”, and the following result is just enough for this purpose at the moment; later on we shall note a stronger result on partitions of unity. A continuow partition of unity subordinate to a
.
finite open covering V1 U . . U V, of E is a finite sequence fi
2 0, and is 0 outside K, and
Lemma 2.2.
fi,
.. . ,fn
E
C(E;R) 3 each
EL,fi = 1.
(DieudonnB and Bochner) Suppose E is a normal space. Then 3
continuous partition of unity on E subordinated to any finite open covering of E.
Proof of Theorem 2.1.
Iff E
then clearly f satisfies the conditions stated in
the theorem. Conversely suppose the condition stated in the theorem holds. We shall assume E compact. Let F be the quotient space of E modulo the equivalence relation
E/A,?r the natural projection E + F , and we shall understand that F is endowed with
Weierstrass-Stone theorem and generalisations - a brief survey
I
the quotient topology. For every f E A define g on F by: gn = f. The mapping f defines a mapping A separating on Now let
--t
C(F);let B be the ima.ge of A under this mapping. Then B is
F ,which we note to be a cninpact Hausdodf space.
E
> 0, and y E F .
Then
T I ({ y ) )
c
E is an equivalence class and is com-
pact. By assumption 3w, E W 3 Iw,(x) - f ( ~ ) l<
K, =
{X
+g
E E
I
Iw,(z) - f(z)l
y $ 7r(Ky). It follows that
n
E
for x E ~ - ' ( { y } ) . The set
1 E } c E is compact and n-'({y}) n IC,
=
0, hence
x ( K , ) = 0, and by the finite intersection property
YEF
3 ~ 1 ,... , y n E F 3 7r(Kvl)n . . . n 7r(Kv,,)= each $i
1 0 and
0.
By Lemma 2.2 3$1,.
= 0 on n ( K y i ) ,i = 1,.. . ,n , and
IC; Let M = sup{ lwYi(r)l I x E
#i(X)wvi(.)
- f(x)
$1
I E
. . ,&
E C(F)3
+ .. + $,, = 1. Then di
Vz E E
= $;n,
.
(1)
E,i= 1,. . . ,n,},and choose 6 I &. Then from (4)and ( 5 )
we find
IC Since h;w,, E AW
hi(S)W,,(x) - f(x)l
c W,i = 1,. . . ,n, therefore f
5 2~ E
VX E E
.
r.This completes the proof.
We shall simply state the theorems of Dieudonnk and Choqukt-Deny on closure in tensor products and on the closure of a coilvex sup-lattice, respectively.
Chapter 1
8
Theorem 2.3.
(Dieudonn6) Suppose E and F are completely regular spaces, and
let f E C(E x F ) . Then V compact K
cE
x F , and VE > 0, 3gl,. . .,gm E C ( E ) ,and
Before stating the theorem of Choqukt-Deny, some definitions are in order. A subset
S c C ( E ;R) is called a sup-lattice if f , g E S if f , g E S
3 sup
( f , g) E S , and is called an inf-lattice
+ inf(f, g) E S. If 4, $ are continuous linear functionals on C ( E ;R), we write
If 4 2 0, we say that d is positive. Theorem 2.4.
(ChoquBt-Deny) Suppose S C C ( E ;R) is a sup-lattice and let f E
C ( E ;R). Then f E 3 (in (C(E; R))
@
(1) V positive 4 E C(E;R)' (the dual of C ( E ;R ) cf Appendix 1) and Vu E E
(2) V positive 4 E C ( E ;R)')
d(f) 2
inf(#(g) 1 g E S } .
At this point it is necessary to explain Bernstein's weighted approximation problem and for this purpose we should first explain the concept of a weighted locally convex space of continuous functions. The next step thereafter is to explain L. Nachbin's contribution towards extending the classical Bernstein approximation problem, viz. his work on the weighted approximation problem for modules.
We first turn to the concept of a weighted locally convex space. Let V be a set of upper-semicontinuous positive functions on E. We shall assume that V is directed, i.e., if
9
Weierstrass-Stone theorem and generalisations - a brief survey v1,v2
E V , then 3A > 0 and 3v E V 3
called weights. The vector subspace of
v1
5 Xv and v2 5
C(E ) consisting of
Xu. The elements of
V are
all f 3 v f is bounded on E ,
for each v E V , will be denoted by CVb(E). Then each v E V determines a semi-norm p,(f) = sup{v(z) If(x)l +
Iz
E E } on CVb(E). w e shall understand that CVb(E) is.
endowed with the natural topology i.e., the locally convex topology determined by the The vector subspace of C ( E ) consisting of all f 3 Vv E family of semi-norms {p,,(.)}uE~.
V and VE > 0 the closed subset {z E E
I v(z). If(z)l 2
E}
is compact, will be denoted
by CV,(E). It is clear that C V , ( E ) c C\’b(E), and the natural topology on CV,(E) is understood to be the topology induced by CVb(E).
A few observations are in order at this point. The family of semi-norms { p , , ( - > } , , ~ v in the preceding paragraph is directed because V itself is directed. If V consists of a single function v(-) then we shall denote CVb(E) and CV,(E) by C u b ( E )and C,,,(E), respectively, and if V consists of the constant function 1 then C,,,(E),Cuoo(E) will be denoted by Cb(E),and C,(E) if v x E E 3 v E
v
3 v(z)
>
respectively. CVb(E) and CV,(E) are Hausdorff spaces 0. C h ( E ) is a module over Cb(E),and CV,(E) is a
sub module over Cb(E). Furthermore, i f f E CVb(E),g E
C(E),and 191 5
If1
then
g E CVb(E);a similar remark holds for CV,(E).
We further note the following: (i) if 1’ is the set of characteristic functions of all compact subsets of E , then C ( E ) = C&( E ) = CV,(E) as locally convex spaces; (ii) if
V consists of just the constant function 1, then C&,(E) = Cb(E),and the topology on Cb(E) is defined by the single norm same norm
I l f l l ~ ; (iii) if E
I l f l l ~ ; also in this case CV,(E)
= C,(E), with the
= R”, and V consists of the Ipl for all C-valued polynomials
Chapter 1
10
#
C(R"); in this case if a norm 11x11 is fixed on R",
+ 11x11)"'
for m = 0,1,2,. . ., on R" then it is known that
p on R" then CVb(R") = CV,(R")
and W consists of functions (1
CWb(R") = CW,(R")
= CVb(R") = CV,(R") as locally convex spaces; in this case the
C(E)thus obtained are said to be rapidly decreasing at infinity.
elements of
We note the following result.
Proposition 2.5.
Ca(E)n CV,(E) is dense in CV,(E).
For stating the next theorem of Dieudonnb on dense subsets in tensor products
some terminology should be explained. For each i = 1,.. . ,n let E; be a completely regular space, Q a directed set of upper-semicontinuous positive functions on Ei; let
-
E = El x .. x En,and V = V1 x Vz x ... x V,. The following theorem of DieudonnC holds.
Theorem 2.6. fi
x
... x
(Dieudonnh)
fn, f; E
The set of all finite sums of tensor-products f = fi x
(CQ)=(E;),Z= I , . . . , n , is dense in CV,(E).
We turn now to the Bernstein approximation problem (cf. [43]). Let f be a C-valued function on R", and suppose f is locally bounded, i.e., bounded on every compact subset of R". Then f is said t o be rapidly decreasing at infinity if the following equivalent conditions hold: (1) pf is bounded on R" for any p E 'P(R") (the set of polynomials on W);
(2) pf
-+
0 at infinity for any p E P(R"). The implication (2) + (1) is clear. To see that
(1) + (2) define q on R" by q ( z ) = x:
+ xg + + xi, x =
(l),pqf is bounded for any p E P(R"), q
Bernstein problem (first form).
-+ 00
. . ,x,)
( ~ 1 , .
E R"; assuming
at infinity, hence pf -+ 0 at infinity.
Let w 2 0 be upper-semicontinuous on R" and
rapidly decreasing at infinity, i.e., P(R") C Cw,(R"),
or equivalently, P(Rn)C Cub@").
Weierstrass-Stone theorem and generalbations - a brief survey The weight w is said to be fundamental if P(R") is dense in Cw,(R");
11
and the Bern-
stein problem consists in finding necessary ;i.nd sufficient conditions for a given weight w to be fundamental. We remark here that the Weierstrass theorem means that every characteristic function of a compact subset of R" is a fundamental weight; and this implies that every w 2
0 which is upper-semicontinuous on R" and has compact support, is a fundamental weight.
Bernstein problem (second foTm). at infinity, i.e., P(R"). w
c C,(R")
Let C(R"), and suppose w is rapidly decreasing
or equivalently P(R") w
that the load w is fundamental if P ( R " ) .w is dense in C,(R").
c Cb(R").
We then say
The Bernstein problem
consists in finding necessary and sufficient conditions for a given load w to be fundamental.
For convenience in the sequel, we shall call these problems Bernstein's problem I and
Bernstein's problem 11, respectively. The next proposition follows.
Proposition 2.7.
Let w E C(R"), w 2 0. Then w is a fundamental load if and only if
w is a fundamental weight and w ( x ) > 0 for any x E R".
In order to explain the work of L. Nachbin in this area we have to explain the
"weighted approximation problem" for modules. Let A
c C(E)be a subalgebra containing the unit, and W c CV,(E)
subspace; we shall also assume 14' to he n inodiile over A i.e., AW
approximation problem consists in determining
be a vector
c W. The weighted
in CV,(E) under these circumstances.
In the special case in which A consists only of the constant functions, W is the most
Chapter I
12
general vector subspace of CV,(E). sists of all f
e CV,(E)
3 every.
In this case, all we can say about
+ E CV,(E)*
w is that
con-
vanishing on W must also vanish at f .
The general case of the weighted approximation problem is reduced to the special case just mentioned, by considering the subsets of E on which the functions belonging to A are constant, i.e., by introducing on E the equivalence relation E / A mentioned earlier. The following definition is formulated with this view in mind.
We say that W localisable under A in CV,(E)
Definition 2.8.
Vf E CV,(E), f
E
w (in CV,(E))
#
if the following holds:
Vv E V , VE > 0 and V equivalence class X modulo
EIA, 3w E W 3 .(.)a
lw(x) - f(x)l < E
vx E x .
The strict weighted approximation problem consists in finding necessary and sufficient conditions for W to be localisable under A in CV,(E). We note that if the following conditions are satisfied:
(1) A is separating on E ; (2) W is everywhere different from 0 in E i.e., Vx E E 3 w E W 3 ~ ( x #) 0; then W is localisable under A in CV,(E)
($
W is dense in CV,(E). Hence if the con-
ditions (1) and (2) are satisfied then corresponding to every sufficient condition for localisability to be established below there will be a corollary asserting density of W in
CV,(E). Furthermore the strict weighted approximation problem can be seen to be a generalisation of the Bernstein approximation problem, as follows. Consider the Bernstein problem I; let E = R", V = { w } , A = P(R"), W = P(R");or consider the Bernstein problem
Weierstrass-Stone theorem and generalisations - a brief survey
13
11; let E = R", V = {l},A = P(R"), W = P ( R " ) .w. Then condition (1) in the preced-
ing paragraph is satisfied; the condition (2) is always satisfied in the case of Bernstein's problem I; and as for Bernstein's problem 11, the condition (2) amounts t o saying that w(z)
# 0 for any z E
R", and in this case Proposition 2.7 justifies assuming the condi-
tion (2). Hence if these conditions (1) and ( 2 ) hold, then finding necessary and sufficient conditions for P(R") = Cwm(R") in Bernstein's problem I, or for P(R")w = Cm(R") in Bernstein's problem 11, is equivalent to finding necessary and sufficient conditions for localisabilit y. The next step is to consider how the weighted approximation problem can be reduced t o a finite-dimensional Bernstein problem. We shall denote by fl, the set of all upper-semicontinuous functions w 2 0 on R" which are fundamental weights in the sense of Bernstein. Let G ( A ) be a subset of A which topologically generates A as an algebra over C with unit i.e., 3 the subalgebra over C of A generated by G ( A ) and 1 is dense in A (in the topology of
C(E)); also let G ( W ) be a subset of W
3 G ( W ) generates W as a mod-
ule over A i.e., the submodule over A of W , generated by G ( W ) is dense in W for the topology of CVm(E). The following theorem now holds.
Theorem 2.9.
Suppose C(E)= C(E; R); if we let C(E)= C(E;C) then we shall as-
sume that G ( A ) consists of real-valued functions. Suppose further that Vv E V,Vul,. 3w
. . ,a,
E G ( A ) and Vw E G ( W ) , 3a,+l,.
E f l 3~ v(z) Iw(z)I 5
. . ,U N E
G ( A ) with N 2 n, and
w(al(z),. ..,nn(x),... ,aN(z)), foranyz E
E . Then W is
Chapter I
14
localisable under A in CVw(E). For the proof we shall need the following two lemmas.
L e m m a 2.10.
Let E = IIierEi be a Cartesian product of Hausdorffspaces and
K: a
collection of compact subsets E with an empty intersection. Then 3 finite subset J
I 3 if I I j denotes the natural projection E
4
c
IIiejEi then I I j ( K : ) has an empty inter-
section.
Lemma 2.11.
Let f E CVw(E),v E V , and
E
> 0.
Further suppose V equivalence
class X C E modulo E / A 3w E W 3 v(X)lw(x) - f(z)l
G ( A ) ,h i , . . .W , E G ( W ) and (~1,.. . ,Y(,
v(x).
12
cr;(al(z),
< EVX E X. Then g a l , . . . , a , E
E Cb(R") 3
. . . ,a,(r))wi(z) - f ( r )
IE
.
Vx E E
i=l
Proof.
Let F denote the space of all real-valued functions on G ( A ) ,and we shall as-
sume that F is endowed with the Cartesian product topology. Let
:
T
E
-t
F be the
continuous mapping which to x E E associates n ( z ) E F 3 the value of ~ ( x at) a E G ( A ) is a(.) E R. Let y E n ( E ) , then every a E G ( A ) is constant on n-'(y). topologically generates A, hence every a E A is constant on ~-'(y).
Now G ( A )
Therefore ~-l(y) is
-
contained in an equivalence class modulo E / A ~-l(y) is actually an equivalence class, for 7r is constant on every equivalence class.
Now by the assumption of the lemma, for each y E n ( E ) 3w, E f(z)l
< E Vx
E n-'(y).
W 3 v(r) Iwy(x)-
We shall assume wy is in the vector subspace of W generated by
G(W); for let 6 = SUP{+).
Iwy(.)
- f(x)l
I
5
E .-'(y>}
;
15
Weierstrass-Stone theorem and generalisations - a brief survey
then 6
< 8, for v(z) -
Iwu(z) - f(z)l attains a maximum on n-'(y).
generates W , hence 3 ~ 1 ,... ,a, E A and 3wl,.
Let
G ( W ) topologically
. . ,w, E G(W ) 3
X1, . . . , A, respectively, be the constant values of a l , . .. ,a, on the equivalence class
n-l(y). Then
I .(z).
+..*+Xr~r(~)-.f(z)I Ial(z)wl(z) + + ar(z)wr(z) - wy(z)I
+
Iwy(z) - f(z)l <
lXlwl(z)
.(.)a
.(.)a
* *
We can replace wy by
(E
- 6)
+6 = c
vz E n-l(y)
.
Xiwi, hence we may assume that wy belongs to the vector
subspace of W generated by G ( W ) . Now let
Vy E n(E). Then K , nn-'(y) =
n v e * ( E ) A(&)
= 0, hence ~ ( l
0. Now, using Lemma 2.10, we conclude that
if Q, is the mapping t
-+
( a l ( t ) ,. . . ,a,(t)) from E
-+
R" then
%I,.
.. , a , E G ( A ) 3
fluE*(E) @(Ku) =
0.
By the finite intersection property and compactness, 3y1,.. . ,yp E n ( E ) 3 Q,(KyI)n
.. . n @(K,,) C(R") 3 p1
=
0. Then, R"
being normal, 3 continuous partition of unity
+ + Pp = 1 and Pi = 0 on d(Kui)i = 1, . . . , p .
A little argument now shows that
PI,. .. ,PP E
Chapter 1
16
Finally each wya belongs to the vector subspace of W generated by G ( W ) ,hence 3101,
. .. ,w, E
G ( W ) 3 each wyi is a linear combination of w l , . . . ,w,.
linear conbinations
a1,.
. . ,a ,
of
P I , . . . ,,L$,
Then for suitable
the lemma follows. Note that
ai E
Cb(R")
for i = 1,.. . ,rn because 0 5 pi 5 1 for i = 1, . . . , p .
Proof of Theorem 2.9.
One preliminary remark would be in order here. If
.
a l , . .,a, E G ( A ) ,w E G ( W ) ,a E Cb(R") and 6
> 0, then 3w'
E
t~
E
V,
W3
To justify this we note that by assumption 3un+l,. . . ,U N E G ( A ) where N 2 n and 3w E R N 3
for any z E E. Now a E Cb(Rn) determilies /3 E C b ( R N ) by the formula p ( t 1 , . . .t,,
. . . , t N ) = a(t1,.. .,t,)
for
tl,.
. ., t N
E
H. Also Cb(RN) c CW.&P) since w
at infinity. Now because P ( R N )is dense in C w m ( R N )and 3p E
p
.. . , t N
o
E Cwm(RN),therefore
P(RN) 3
for any t l , . . . ,t,,
--t
E R. Hence by ( 5 ) and (6)
..
17
Weierstrass-Stone theorem and generalisations - a brief survey
Now to complete the proof of Theorem 2.9, let f E CV,(E) 3 Vv E V ,VE > 0 and V equivalence class X
c E modulo E / A , 3w
EW3
a ) . ( .
~w(x) - f(i)l < E Vx E X. Then
by Lemma 2.11, gal,.. . , a , E G ( A ) , 3 ~ 1 ,... ,w, E G ( W ) and gal,.. . ,am E Ca(R") 3
By the preliminary remark in the last paragraph 3w:,. . . ,w k E W 3
for any x E E , and i = 1,.. . ,m. Putting together (7) and (8),and taking 6 =
2,we
find
v(x)* I W(X) - f(z) I< 2~ Hence f E
VX E E .
r.This proves localisability of W under A in CV,(E).
Corollary 2.12.
Suppose C ( E ) = C ( E ; R ) ;or suppose
C(E)= C(E;C) and that
G ( A ) consists of real functions. Suppose hrther that G ( A ) ,G ( W ) are both finite: G ( A ) = {q,. . . , a , , } , G ( W ) = {wl,.. .,to,,,};
and that
Vv E V,Vi= 1,. . . ,m 3w E 0,
3
v(z) I wi(z)15 w(al(z),.. . ,a,(z))V~ E E . Then W is localisable under A in CV,(E). Before stating the next theorem and its corollary, it is necessary to explain some notation. If x = (XI,.. .x,) E R", we shall denote by
151 the
point
. . , lznl). We
(1~11,.
shall also denote by 0: the set of all w E 0" which are decreasing in the sense that if
r,y E R" and IzI 5 IyI then Theorem 2.13.
W(X)
2 w(y);this implies W(X) = ~ ( 1 ~ 1 ) .
Suppose C(E) =
C(E;R); or suppose C(E)= C(E;C ) and that
A is self-adjoint. Also suppose that Vw E V,Val,.. . , a , E G ( A ) and Vw E G( W)
Chapter I
18
3a,+l,.
.. ,a N E G(A) with N
2 n and 3w E Rd, satisfying
Then W is localisable under A in CV,(E).
Remarks (1) We note that corresponding to Corollary 2.12, there is an analogous corollary of Theorem 2.13.
(3) Using the weighted Dieudonnk's Theor(im 2.6 on density in tensor products, the arguments of Theorems 2.9 or 2.13 are reduced to one-dimensional arguments. We shall denote by
rn the set of all upper semi-continuous y
a fundamental weight in the sense of Bernstein for any k that I?,
c R; however there are examples showing that
a certain k
> 0 then yf E R,
set of all y E ~ ( 5= )
for all f?
> k.
> 0.
I?,
2 0 on Rn 3 yk is
By taking k = 1 we see
# R,.
Also if yk E R, for
Furthermore we shall denote by :?I
I?, which are decreasing i.e., z,y E R", 1x1 5
IyI
+ $5)
the
2 y(y) hence
~(IzI).
Theorem 2.14.
Suppose A is self-adjoint and that Vv E V,Va E G ( A ) and Vw E G ( W )
37 E I'f 3 V(Z) lw(x)I 5 y( Ia(z)I)Vx E E. Then W is localisable under A in CV,(E).
19
Weierstrass-Stone theorem and generalbatwns - a brief survey
53. Criteria of localisability In this section several criteria of localisability due t o Nachbin, will be established; we find that each of these turns out to be a special case of the one immediately following.
Theorem 3.1.
Suppose C(E)=
C(E;R);
or suppose C ( E ) =
C(E;C) and that
A is self-adjoint. In either case, suppose further that Vv E V , Va E G ( A ) and Vw E
G(W)3C > 0 and 3c > 0 satisfying
Then W is localisable under A in CV,(E).
For the proof we need the following two lemmas.
Lemma 3.2.
Suppose V is a directed set of upper-semi continuous positive functions
of E . Suppose Cb(E) C CV,(E) and that A is a subalgebra of Cb(E) which is separating on E , contains the constant function 1 and is self-adjoint in the complex case. Then
A is dense in CVm(E).
Lemma 3.3.
Let y 2 0 upper-semicontinuous on R be 3 3C
y(t)
Then y E
Proof.
5
> 0,3c > 0 satisfying:
Ce'l*l V t E R.
rl. Clearly y(.) is rapidly decreasing at infinity, since trne-'ltl
any rn E R. Let t , x , y E R, z = z
+ i y E C, and define e ,
--f
0 aa t
--f
00,
for
E C(R;C) by e L ( t ) = eiZt for
Chapter I
20
t E R. We then note
provided (yI < c. Hence, e, E Cb(R,C)if JyI < c. Denote by S the open strip:
Now let
9 be a continuous linear functional on Cy,(R; C ) , and define f : S --+ C by:
f(z) = d ( e , ) , z E S.
Our claim now is: f(-)is analytic on S. To show this we write
for z E C, t E R. We shall show that if z E C and Iz1
<
c,
then the above series (1)
converges uniformly for t E R. We use the elementary remark that if m and if c
> 0, then the function tme-ct
Therefore for m
>0
Now Stirling's formula
> 0, is an integer
on t >_ 0 attains its maximum at t =
$; and hence
Weierstrass-Stone theorem and generalisations - a brief survey
21
implies that
m -- e.
lim
(m!)llm
m-w
Hence using Cauchy's criterion for convergence of a series of positive terms, we see that
Thus from (2) and (3) we see that the series in (1) converges uniformly for t E R and z E C with IzI
< c.
Now define u, E C(R;C) by u,(t) CI(,(R;
C ) because tme-'l*l
+
0 as t
+
= (it)" 00.
for t E R, m = 0,1,. . .; then u, E
Because of the uniform convergence of (1)
for t E R we see that
where convergence is understood in the sense of Cy,(R;
C). Now let a E R: then we also
have m
where convergence is understood in the sense of C-y,(R;
C), hence eau, E C,, (R; C).
Thus we find
provided Iz - a1
< c.
Here a E R is arbitrary; thus f is seen to be analytic on S. Further-
more
Now suppose
4 vanishes on P(R;C).
Then f("'(0)
= 0 for m = 0,1,2,.
. . by (4)
(taking a = 0). Since f is analytic on S, it follows that f is identically zero on S, hence
22
Chapter I
Vx
on R, i.e., q5(ez) = 0 all
ez,z E
E R. Denote by A the vector subspace of Cb(R; C) generated by
R. Then q5 = 0 on A .
It is then clear that A is a self-adjoint separating subalgebra of C @ ; C) containing By the last lemma A is dense in Cy,
1 and Cb(R;C) c Cy,(R;C). on Cy,(R;
(R;C), hence q5 = 0
C). Hence, we see that +=O
on
P ( R ; C ) + q5=0
Hence by the Hahn-Banach theorem we see h a t
on
Cy,(R;C).
P(R;C) is dense in Cy,(R; C).
7 E i l l . But now yk satisfies the same kind of assumption as y for integer
yk E il, for k
> 0 hence y
E
Proof of Theorem 3.1.
Hence
k > 0, hence
rl. This completes the proof of the Lemma. We now apply Theorem 2.14, and the last Lemma, taking
7 ( t )= Ce-'l*I for t E R , where we notice that y E I?!.
This proves the theorem.
We shall next turn to the quasi-analytical criterion of localisability. The following theorem will be established.
Theorem 3.4.
We shall suppose A is self-a.djoint, and also that Vv E V,Va E G ( A ) and
V w E G (W )we have
whereM,
= sup{v(z).
I
~ ( z ) ~ w ( Ix In: ) E
E} for rn
= 0,1,2, .... Then W is
localisable under A in CV,(E). Before turning to the proof of Theorem 3.4,we recall the following concepts from the area of infinitely differentiable functions. Suppose M = { M ,
1m
= 0, 1,2,
. . .} is a
sequence of strictly positive numbers. We shall denote by C(M) the set of all indefinitely
23
Weierstrass-Stone theorem and generalisations - a brief survey
differentiable complex-valued functions f, each defined on some open interval I C R (I depending on f), and satisfying the following estimates for its successive derivatives: for every compact subset K
c I, 3C > 0, and
3c
>0 3
Vx E I< and m = 0,1,2,. . .. We say thn.t C(M) is a quasi-analytic class if the following is true: i f f E C(M) and 3a E I such that f ( " ) ( a ) = 0
V m = O , l , 2 ,... , then
f E 0 on I.
Clearly this amounts to the requirement that every
f
E C(M) is determined within
C(M) by the knowledge of its Taylor series at a single point a E I (though the Taylor series o f f at a or at any other point in I are not assumed to be convergent
-
in fact may
fail to be convergent).
If M , = m!(m = 0,1,2,. . .) then by a theorem of Pringsheim, C(M) consists of complex valued functions which are analytic on open intervals of R , and hence can be called the analytic class. In this case C(M) is clearly quasi-analytic. On the other hand it is known that not every class is quasi-analytic. For instance, if
for m = 0, 1,2,. . . , then C(M) is not quasi-analytic, for the function f defined by: f ( x ) = e-i,O
< 2 < 1, and f(x)
= 0 for -1
<
I
5 0, is indefinitely differentiable on the
interval I = (-1,l); furthermore, f E C(M), and f(")(O) not the zero function on I .
= 0 for m = 0 , 1 , 2 , . . . , yet f is
24
Chapter 1
We shall assume Denjoy's criterion of quasi-analyticity. This is contained in Theorem 3.5 below, by which Denjoy solved the problem of Hadamard, to find necessary and sufficient conditions on a given sequence M in order that C(M) should be quasi-analytic. T h e o r e m 3.5.
(Denjoy-Cademan) Let M be given and set = inf{Mt, k = m,m
p,
+ 1,.. .)
for m = 1 , 2 , , . .. Then C(M) is a quasi-analytic class Corollary 3.6. If C:=,
%=
00
c:=,& =
00.
then C(M) is quasi-analytic.
M,"
We note that the Corollary follows from Theorem 3.5. Remark.
If M , = m!,m = 0,1,2,. . . , then the class C(M) is the analytic class, as
noted above. Since (by Stirling's formula) 3X
> 0 3 m! 5 Xmmm(m= 1,2,. . .), it follows
that C(M) is also quasi-analytic. On the other hand, suppose C(M) is quasi-analytic by virtue of the existence of some a
>0
2 z ( m = 1,2,. . .) and by application of
3 M,m
the last Corollary. Since 3X
>0
3 mm
5 A" . m!(m= 1 , 2 , . . .), it follows that C(M)
is then contained in the analytic class. We conclude that the analytic class is the largest quasi-analytic class which is tied up with the divergence of the harmonic series
c:=,A.
We shall need two lemmas before we turn to the proof of Theorem 3.4. L e m m a 3.7. Let a, m = 1,2,. . . If
1 O(m = 1,2,. . .) and suppose 30 > 0
c:=,a,
=
M
then
Cz=,a,,,
=
will be omitted. L e m m a 3.8.
Let y 2 0 upper-semicontinuous 3
00
3
a,+1
5
OCY,
for
for p = 1,2,. . . The formal proof
25
Weierstrass-Stone theorem and generalisations - a brief survey where M,,, = sup{r(t).
ltmllt E R} form = 0,1,2,. . .. Then y is fundamental weight on
R. To be precise, 7 E rl.
Proof of Lemma 3.8. If M,,,
=
0, for some rn, then y ( t ) = 0 for t
#
0. The support
of 7 ( . )is reduced to 0, or empty, hence by an earlier remark (re: the Bernstein Approxi-
mation problem) we see that 7 E rl. Next suppose M,,,
> 0 for rn = 0,1,2,. ...
diverge, it follows that M,,,
< 00
Since the series
Cz=,
is assumed to
M,"
for infinitely many values of rn. On the other hand, if
M,,, < 00 for some m, then M p < 00 for all p = 0,. . . ,m, as we find from the relation:
outside a compact neighbourhood of 0. Thus y is rapidly decreasing at infinity. Using the earlier notation for e, and urn, if x E R, then e, E Cb(R;C) C Cy,(R;C). Let
4 be a continuous
linear functional on Cy,(R; C ) . Define f : R
-t
C as before by
f(z) = + ( e , ) , z E R. We then claim that f is C" on R, and
a E R, m = 0,1,2,. . . (here we note that e, E Cb(R; C) and u,,, E CT"(R; C), hence e.u,
E C7=(R; C)). For suppose
This is true for m = 0. If h E R , h
f'"'(Q
f is m-differentiable and ( 5 ) is true for some m 2 0.
# 0, then
+h)h
f'"'(~3)
= 4(eo
. eh - 1
urn).
By Taylor's formula, if g is twice differentiable on the interval I ( h ) = [0, h] (or [h,O]), then
ha
I d h ) - d o ) - hg'(O)l I2""P{lg"(x)lIx
E I(h)).
Chapter I
26
Apply this to g(x) = e i x t , t E I?being fixed; then we obtain
Hence,
Therefore, eh -
1
e,.T.urn
-+ e , . ~ , + ~
C ) . Then from (6) and (7) it follows that
as h
-+
f(”)
is differentiable and (5) holds with m replaced by m
0, the convergence being in Cy,(R;
(7)
+ 1. This proves our claim (5).
From (1) it follows that
where 11q511 is the seminorm of q5 on the semi-normed space Cy,(R; with
C ) . Hence f E C(M)
M = { M r n , m= 0,1,2,. . .}. By Denjoy’s Corollary above, C(M) is quasi-analytic. Now suppose
4 = 0 on P(R;C ) . Then f(”’(0)
0. Hence by quasi-analyticity, f
E
= 0, for m = 0,1,2,. . . by ( 5 ) for a =
0 on R. The rest of the proof that y E
proceeds
along the same lines as in Lemma 3.3 above. Next, y k satisfies the same assumption as y for any k
> 0, i.e. if we define
for m = 0,1,2,. . . , then an inductive argument combined with the hypothesis of the Lemma and an application of the last Lemma shows that
27
Weierstrass-Stone theorem and generalisations - a brief survey
From the earlier part of the proof we then conclude that yk E
yE
fl1
for any k
> 0, i.e.
rl. This proves the Lemma. We now turn to the proof of Theorem 3.4.
Proof of Theorem 3.4.
Define y on R by
where we understand that y(0) = 0 if some ,A!
= 0, and y(0) = Mo otherwise. Then
y 2 0 and y is upper-semi-continuous, being an infimum of a family of continuous functions. By the definition of 7,
for rn = 0,1,2,.
. . , hence 7
E
rl, by the last Lemma.
Then clearly y E I?:.
By the
definition of M , v ( x ) . la(z)mzu(z)l
5 M,,
rn = 0,1,2,. . .
,
hence v ( x ) Iw(x)I 5 y(la(z)l) Vx E E . Now we apply Theorem 2.14. This proves Theorem 3.4.
Chapter I
28
Distinguishing between the bounded analytic and quasi-analytic cases of the weighted approximation problem By “the bounded case of the weighted approximation problem” is meant the one in which every a E G ( A ) is bounded on the support of every vw,forv E V,w E G(W). We note that each of the following assumptions leads to an instance of the bounded case: 1. A c Ca(E); 2. every a E G ( A ) is bounded on the support of every v E V;
3. every a E G ( A ) is bounded on the support of every w E G ( W ) ;
4. each ow,for v E V , and w E G( W ) ,has compact support; 5. each v E V has compact support; 6. each w E G ( W ) has compact support. The next few propositions in this section are meant to distinguish between the different cases of localisability.
Proposition 3.9.
The bounded case arises -++ in Theorem 3.4, y can be taken to have
compact support; or equivalently, if y can be taken to be a constant times the characteristic function of a compact subset of R .
Proposition 3.10. because 3c
>0 3
The bounded case arises
+2
c
++ in Theorem 3.4 the series is divergent
form = 1 , 2 , . . .
M,m
Proof.
Suppose 3c
>0
3
4 1 c.
Then Iv(z)w(z)Ik
. Ia(z)I 5 f
for m = 1 , 2 , . . . ,
M,“
and z E E . If v(z)w(z)
# 0, let
m
-+
00,
and we obtain Ia(z)I 5
$.
Hence a(.)
is
bounded on the set { z E E
I
v W . Conversely suppose
5 k, k > 0, on the support of vw. Suppose vw is bounded
la1
v(z)w(z)
#
0}, hence on its closure, i.e. the support of
29
Weierstrass-Stone theorem and generahations - a brief survey by
C > 0 on E. We note that w
1 .(I) 15 C1l"'k f o r m
CV,(E) c CVa(E) and v E V . Then I W(I)W(I> (I/"'
= 1 , 2 , . . . and
1,2,. . . , then we shall have
Theorem 3.11.
E
I
E
E . If we choose c 3 0 < c <
c-'/m
> c for m = 1,2,. . . , . This proves this
Mkm -
for m =
proposition.
With the preceding definitions and terminology, localisability always
holds in the bounded case of the weighted approximation problem, provided we deal with real valued functions, or complex-valued functions and A is assumed to be selfadjoint.
Proof of Theorem 3.11.
Use the first of last two propositions and Theorem 2.14.
By the analytic case or the quasi-analytic case of the weighted approximation problem is meant the one in which the sufficient condition of Theorem 3.1, or of Theorem 3.4, holds. This terminology is justified by the fact that in the proof of Theorem 3.1 or of Theorem 3.4 (respectively) use was made of analyticity (or of quasi-analyticity). The next proposition distinguishes the occurrence of the analytic case.
Proposition 3.12.
The analytic case arises if and only if in Theorem 3.4 the series is
divergent because 3c > 0 3
Proof.
Suppose 3c
>0
cmM, 5 C , i.e. c"'v(I).
2 3
MLm
> - c for m
= 1,2,
... .
l a ( ~ ) ~ w ( z5) lC for m = 0,1,2,
m!, and summing up, we obtain
hence
2 for m = 1,2, . . . . Let C = sup(1, Mo}. Then
. . . , and
I
E
E. Dividing by
30
Chapter I
Hence the sufficient condition of Theorem 10 holds. Conversely suppose 3C
> 0, and 3c > 0 3
We now note the elementary inequality
which has already been noted above in the course of the proof of Theorem 3.1. Then we find:
Choose c' 3 0
v ( z ) . la(z)mw(z)l
5 c
< c' 5 ceC-'/"' for m
= 1,2,
This proves the proposition.
(--)m m
for any
zE E
. . . and then -& M,
2
.
& for m = 1 , 2 , . . . .
31
Weierstrass-Stone theorem and generalisations - a brief survey $4. A differentiable variant of t h e Stone-Weierstrass t h e o r e m
In this section and the next we shall give an account of differentiable analogues of the Stone-Weierstrass theorem for certain algebras of r-times continuously differentiable functions. We shall explain a theorem due to L. Nachbin (cf. [44]), and in the next section mention some generalisations by Aaron and Prolla. Suppose M is a differentiable manifold of order r 2 1, and dimension n 2 1. Let
A be the algebra of Cr-functions i.e., r-times continuously differentiable functions on M endowed with the topology of uniform convergence of C ' functions up to order m on compact subsets of M . Nachbin established the following theorem ([44])p. 1550).
A necessary and sufficient condition for the algebra A ( B ) generated by
T h e o r e m 4.1.
a subset B c A to be dense in A is that the following conditions are satisfied:
(1) for each E E M 3 f E B 3 f ( E )
# 0;
(2) for each pair of points (,q E M , E (3) for each
Proof.
# 71, 3f E B
3
f([) # f (q);
< E M and for each tangent vector 0 # 0 at E, 3f E B 3 3 # 0. c M be compact, and W an
Only the sufficiency needs justification. Let K
open connected subset of M containing I< and such that
is compact. For each point
in M 3 a function which is not identically 0 in a neighbourhood of this point; hence 3 finite number of functions
Now let 3f1 f2
E B 3
E B3
E
f1,.
E M , and let
# #
. . ,fn
01
#
E B 3 { fl(z), . . . ,fn}
#
(0,. . . ,0} for z E
0 a tangent vector to A4 at
0. If n 2 2, then 3 tangent vector O2
0. Then if n 2 3, 3 tangcnt vector 0 3
Thus we obtain tangent vectors O1,.. . ,On at
#
v.
6. Then by hypothesis
0 at
# 0 at 6
E
3
3
= 0. Then let
$=
a
6 a.nd functions f1,. . . ,fn E B
3
= 0, etc.
$# 0
32
Chapter I
(i 5 i 5 n) and
= 0 (1 5 i
< j 5 n). with values in R",
Consider the linear mapping defined on the tangent space at which maps 6 +
- + cn&,
{ %,. . . %}. Each vector in R" is the image of a vector 6 = c161 +
i.e., this mapping is an isomorphism on R".
The implicit function theorem shows that the mapping z is a homeomorphism of order
T
2
. . ., gh E
B and 3 open subsets
w is compact, hence 3 functions
c M (1 5 i 5 b) covering
.
+ {gi(z),. . ,gk(z)} is a homeomorphism of order r of
Now set fa+(;-*),,+,
= gf. If [ , q E M , €
( z , y ) in a neighbourhood of
{ f l ( z > , . . . ,fn(z)}
(cf. DieudonnC: Foundations p. 272) of a neighbour-
hood of [ in M onto an open subset in R". The set gi,
+
([,r]).
#
r],
3 each mapping
onto an open subset of R".
then 3f E B 3 f(z)
#
f ( y ) for all
The space
is compact and disjoint from the diagonal of
w x w and thus 3 functions h l , .. . ,hc E
B 3 for ( z , y ) E Q we have
Write fa+bn+i = hi. Then we consider the mapping 9 : M
+RN
with
N = n+bn+c defhed by the mapping z
+
(fl(z), . . . ,f~(x)}. This mapping 9 is a homeomorphism
) order r# in of order r of W on the submanifold @ ( W of Consider the inverse @-'
RN.
: @ ( W )+ ( W ) . Let f E A. Then
tinuously differentiable of order
T
f@-l
is r-times con-
on 9 ( w ) . By a special case of a theorem of Whitney
Weierstrass-Stone theorem and generalisatwns - a brief survey
(cf. [63]),3 r-times continuously differentiable function on
@(K), hence f(z)
+ on RN
33
3 + ( z ) = f(@-'(z))
= @(fl(z),. . . , f N ( z ) ) on I<. We note that O(w)does not con-
tain the origin RN. Hence we can suppose
+ is 0 at the origin in AN. Now we apply the
classical Weierstrass theorem; the proof is thus completed.
Remark.
The above theorem can be alternatively formulated as follows: any proper
closed subalgebra
B of the algebra A
is contained in a maximal closed subalgebra.
For maximal closed subalgebra is precisely one of the following types of sets: either the set of functions vanishing at a point, or the set of functions taking the same value at
2 distinct points, or the set of functions with a zero derivative along one tangent.
Chapter I
34
$5. Further differentiable variants of the Stone-Weierstrass theorem
The "differentiable" result of Naclibin explained in the last section was generalised, first by Lesmes and Restrepo who dealt, respectively, with C'-functions defined on a Hilbert space (Lesmes [34]) or on certain reflexive Banach spaces (Restrepo [49]);and later by Aron and Prolla (cf. [2]) who dedt with C kmappings between Banach spaces
E and F with E' satisfying the bounded a.pproximation property. Here we shall briefly summarise the results of Aron and Prolla [2]. To explain these, it is first necessary to explain some notation and definitions.
Notation and definitions. N denotes the set of natural numbers, N' = N U {m}, and 0 E N. E and F are real Banach spaces, their normed duals being denoted by E' and F', respectively. For
each n E N, P(,E; F ) is the space of cont,inuous n-homogeneous polynomials from E to
-
F , each polynomial being a composition of the form A o A,, where A is an element of the space .C("E;F ) of continuous n-linear mappings of E x
.. . x E
-+ F and A, is the
n
diagonal operator A, : E
--t
E x . . . x E. For n = 0, C("E;.F)and P ( " E ;F ) are both n
identified with F . P("E; F ) is a Banach space with the norm
The subspace P f ( " E ;F) of 'P("E;F) is generated by the collection of functions of the form 4,
@I
y ( n E N, 4 E E',y E F ) where
4,
@I y
( x ) = 4"(1)y for each
I
E
E. The completion of P f ( " E ;F ) with respect to the norm of P(,E; F ) is denoted by
Pc("E;F ) c P(,E; F ) . P ( E ;F ) is defined to be =
c,"==, 'P(,E; F ) .
Let U be a nonempty open subset of E. The space Cm(U;F ) , m E N', is the space
Weierstrass-Stone theorem and generalisatwns - a brief survey of all mappings f : u +
35
F 3 V j E N*,j 5 rn and z E U , the jth successive Frkchet
derivative D j f ( z ) E P ( j E ;F ) exists and is a coiitinuous function of z E U . If the range space F is not specifically identified, it is nnderstood that F = R'; so C"(U) denotes
C"(U, R1), etc. A subspace A
c
C"(U; F ) is a polynomial algebra if Vg E A , and V P E P f ( E ;F )
the composition P o g E A . The work of Aron-Prolla shows that the Stone-Weierstrass theorem holds for polynomial algebras in C"(U; F ) .
A Nachbin polynomial algebra is a polynoinial algebra A
c
C"(U; F ) (rn 2 1)
satisfying: (a) V x E U,39 E A 3 g ( z ) # 0; (b) VXlY E u , x
# Y,3g E A
(c) V x E U and Vw E E , v
3 g(x) # g(!/);
# 0 , 3g E V 3 D g ( z ) # 0.
When F = R1, a polynomial algebra is an algebra; in this case (a) and (b) are the usual conditions in the real version of the Stone-Weierstrass theorem. Nachbin (cf. $4) showed that in a finite-dimensional E , an algebra A the compact open topology
@
c
Cm(U) is dense in C"(U) w.r.t.
all three conditions (a)-(.) hold.
We note that P f ( E ;F ) , P ( E ;F ) , C"(U; F ) , Cm(U;F ) are examples of Nachbin algebras. Suppose E has an m times continuously differentiable norm; in this case [f E
C"(E; F ) I f has bounded support] is also a Nachbin algebra. A certain property called the Bounded approximation property is related to one condition which will be often assumed in this section. A Banach space E is said to have the approsimation property if V compact Ii
c E , V E > 0 , 3 operator T (depending on E
Chapter I
36
and K ) of finite rank 3 IITz - x(1 <
E
Vx E K ; and E is said to have the bounded ap-
proxamation property if it is further possible to choose the approximating T of finite rank 3 VE > 0 and V compact K
c E , llTll 5 X where X is independent of E and I(.
The following condition will be often assumed in this section: 3C 2 1 3 V compact
K c E , V compact L c E' and VE > 0, 3.rr E L ( E ;F ) of finite rank satisfying:
It is shown in [25] that "E' has the bounded approximation property" is equivalent to the property (*).
Weakly uniformly continuous mappings. We find that it is necessary to consider polynomials and functions which are weakly uniformly continuous when restricted to any bounded set.
Definition.
Let E , F be real Banach spaces. A mapping f : E
+
F is weakly uni-
f o r m l y continuously on bounded subsets of E if V bounded B c E , and VE
>0
3d1,.. . , 4 b E E' and 36 > 0 3
Definition.
PIU("E; F) is the subspace of P("'3; F ) consisting of those
m homoge-
neous continuous polynomials which are weakly uniformly continuous on bounded subsets of E (equivalently, on the unit ball in E ) .
37
Weierstrass-Stone theorem and generalisations - a brief survey
Deflnition. f :E
C,"(E; F ) is the space of m-times continuously differentiable mappings
F satisfying:
( a ) D j f ( z ) E Yw("E; F ) , for z E E , j 5 m.
(b) DJf : E
4
Y,JmE; F ) is weakly uniformly continuous on bounded subsets of
(i) If E is reflexive then f E CE(E;F ) iflfor each j 5 m, D; : E
4
Pw("E; F ) is
weakly continuous on bounded subsets of E (cf. Restrepo [49] for the case m = 1, E reflexive). (ii) C,"(E;F ) contains all functions of the form g o T where T is a continuous linear operator of finite rank and g E Cm( T ( E ) ;F ) (iii) C,"(E; F ) contains no non-zero function with bounded support except when F = 0 or dim E
Deflnition.
< 00.
f'
is the locally convex topology of uniform convergence of order m on
bounded subsets of E , endowed upon C,"(E;F ) , and is defined by all semi-norms of the form
where B is an arbitrary bounded subset of E. Each such semi-norm is well-defined. The topology 7s" on C,"(E; F ) is defined in an obvious manner. Deflnition.
A function f E Cm(E;F ) is said to t o be uniformly differentiable of order
38
Chapter I
m if V bounded B
cE
and VE > 0 36
> 0 3 if x E B , and y
E
E with llyll < 6 then
( N o t e : Restrepo [49]investigated uniform differentiability of order 1.) The first main result proved in [2] is the following theorem on uniform approximation of
C" mappings up to order m on bounded subsets of E. (cf Aron and Prolla 121p. 207) Suppose E , F are real Banach spaces
Theorem 5.1.
with E' having the bounded approximation property with constant a polynomial algebra A
c C,"(E; F ) is rr-dense
@
C;let m > 0. Then
the following conditions hold:
(a) A is a Nachbin polynomial algebra; (b) V continuous linear map
T
: E + E of finite rank, with
composite g o T belongs to the $'-closure
1 1 ~ 1 15
C, and Vg E A , the
of A.
Approximation up to order m in the compact-open topology. Definition.
Let U
CT(E;F ) (for m
c E be an open set, where E , F are real Banach spaces. Then
E N') is the space of functions
f E Crn(U; F ) 3 for each I
E
U and
v j 2 m, D j f ( z ) E Pw("E; F ) . C T ( E ;F ) is endowed with the locally convex topology of uniform convergence on compact sets of order m, defined by the family of semi-norms of the form
where j
5 m, and K is an arbitrary compact subset of U .
Remark.
39
Weierstrass-Stone theorem and generalisations - a brief survey
(i) When m = 0 or 1, C,"(U; F ) = C"(U; F ) , and for m
> 1, the two spaces are
generally different, though in certain cases, e.g. when E = co, F = R1,C,"(U;F ) =
C"(U; F ) Vm E N'. (ii) Cz(E;F ) is always a proper subset of C T ( E ;F ) if E is infinite dimensional. The second main result of [2] is the following theorem.
Theorem 5.2.
(cf 121 p. 210) Suppose E l F are real Banach spaces, E' having the
bounded approximation property with constant C. Let m E N', and U nonempty open set. A polynomial algebra A
c
E be a
c C,"(U; F ) is rF-dense in C,"(U; F ) e
the following conditions are satisfied: (a) A is a Nachbin polynomial algebra; (b) V continuous linear operator r : E open V
cU
C,"(U;F ) .
3 .(V)
+
E of finite rank and 11r11 5 C, Vg E A and V
c U , the composite g o ( T I v )
belongs to the closure of A
IV
in
This Page Intentionally Left Blank
CHAPTER I1 Strong approximation in finite dimensional spaces
The concept of strong approximation appears to have originated with H. Whitney (cf. [SS]), though he does not use the words "strong approximation" in his paper. In this chapter we shall present some results on strong approximation in a finite dimensional space R". The fist of these (Theorem 1.8 below) is Whitney's theorem on strong approximation by real analytic functions. We have presented the original proof (Lemma 6 in [66]), for we feel that this proof might suggest further possibilities (see also the Appendix by Stein in [l]).The second result in $2 of this Chapter, is a weaker result than Whitney's but is still interesting because it uses different techniques. This result on strong approximation by C" functions appears to be rather commonly known (cf. Munkres [42], Hirsch [21]); however, we have a.ttempted to be guided by the presentation
in [XI.
$1. Whitney's theorem on analytic approximation We shall first explain some notation. A point in R" shall be denoted either by a single variable, e.g., z,or by an ordered n.-tuple of real numbers e.g., ($1, we shall write z =
(21,
. . . , I,,) for a point in R". A
.. . , z"), and
multi-indez is a n ordered n-tuple
a = (all .. ., an)of non negative integers ai. With each multi-index a is associated the
differential operator
where Di =
&;so D pf(x) means
PI+...+..L
...
f(z1,
. . ., x,,).
The order IaJof D p is
Chapter II
42
defined by: la1 = a1 cy
+ .-
8
+a,; if la1 = 0, then Do f means f . Clearly
f p means (a1 f PI, . . . , a, f Pn),and cy 5 p means ai 5 pi, i
Ia+PI
= 1,
= IaI+
IpI,
... , n. We shall
write
where k =
(k1,
. . . , kn),
between two points z =
. .. , In)
1 =
(11,
(51,
. . . , z,)
d ( z , y ) = (x;!,Czi- yi)l)
112
=
and y = ( y l ,
. . ., y,)
in R" will be denoted by
- yII. However, a little further on we shall allow z
((2
and y to be complex: 2
are multi-indices with 1 5 k. The distance
= (z!
Y = (Y:
+ iz;, . . . , z:,
iz:),
+ iy;, . . . , y:, + i Y 3 ,
in which case d(z,y)' shall mean x;=l{(z> - y>)
+ i(y7 - y;)}',
where i =
in C.
d(z, E ) shall denote the distance from the point z to the set E , i.e.,
while d(A,B) shall denote the distance between the sets A and B, i.e.,
We shall have occasion to consider functions indexed by multi-indices, e.g. fo(z)= fo ,...,o(z),
OT
fa(.)
= fa,,...,a,,(z). We shall suppose A to be a closed set in R", bounded
or unbounded. Suppose f ( z ) is defined in A, and let m 2 0 be an integer. We shall say: f(2)
= fo(z) is of class
functions
fk(Z)
C" in A
in terms of the finclion~fk(Z) (with JLJ5 m ) if the
are defined in A for all k with lkl
5 m and satisfy, with z , d E A:
43
Strong approximation in finitedimensional spaces
meaning:
-k
Rk(z';x),
for each fk(z), with Ikl 5 m; here R k ( z ' ; z ) is assumed to satisfy: Vx" E A , VC 36
> 0 3 if 2,s' E A with
112 -
zoll < 6, llz'
>
0,
- zoll < 6 then
Note that if rn = 0, these conditions (1) and (2) mean that f ( z ) is continuous on the set
A , and also that these conditions are satisfied automatically at all isolated points of A, regardless of how the
fk(z)
It is clear that the
are defined there.
fk(Z)
are continuous and hence bounded in a neighbourhood of
each point in A. Thus i f f is of class C" in A in terms of the f k ( 2 ) with lkl
f is of class c"', rn'
< m in terms of the f k ( 5 )
5 rn, then
(with lkl 5 m'). w e shall say that any
arbitrary function f(s) is of class C-' in A , and that f(z) is of class
C" in A in terms
of the f r ( ( z )if the f k ( z ) are defined for all k and f is of class c" in A in terms of the fk(z)(lkl
5 rn) for each integer m 2 0.
Suppose f ( z ) is defined in a region R and is of class C" in terms of the
m). Let z = ( z 1 , . . . , z,),~' = (21, . . . , z
h
+ Azh, . . . ,
Zn);
fk(Z)
(lkl 5
then if Ik( < rn, we find:
Chapter ZZ
44
in R. Hence in this case f ( z ) is of class Cm in the ordinary sense, and the f&(z)are the partial derivatives of f(z). Also Taylor's theorem shows that the converse is true. We shall need a few lemmas, before turning to the proof of the approximation theorem of Whitney in question. L e m m a 1.1.
Let w ( z ) be a continuous function of one variable defined in an interval
I containing zo, let B be a closed set in I, and let
20;
be a fixed number. Suppose VE >
0 36>03
( I ) ifz E B and Iz - 201 < 6 then
-1
- wkI < e;
(2) i f l z - 201 < 6 and z fZ B then ~ ' ( z exists ) and J w ' ( z )- whI Then w ( z ) has a derivative at
20
< E.
and w ' ( z 0 ) = w i .
The proof of this lemma is omitted since it is elementary. We shall make use of the functions denoted below by $&(z'; 2). If z E A, and z' E E ( m < 00) then we define:
Thus $&(z'; z) is the value at z' of the polynomial of degree 5 m - lkl which approximates
fb(z)
to the (m- Ikl)th order at
2.
For fixed z, it is a polynomial in
Taylor's formula in terms of its value and derivatives at
I.
2'
given by
From (1) and (4) we then see
that
The P'' derivative of $b(z'; z) (as a function of
2')
at z', is $&+f(z';z). If we then ex-
45
Strong approximation in finitedimensional spaces
press
$k(Z";
x) in terms of its value and derivatives at z', we obtain $k+dz'
$k(Z'' - Z) =
e!
f
-
(,ti
(5)
This identity shows that +k(z";x') =
ce!
.fk+dz')(zft- ,r)f
f
Our next objective is to construct a suitable C" partition of unity on the set C ( A ) .
This will be done through several steps. We define a function, which shall be denoted by 0(z). Let R be the region defhed by:
1xhl
<
1, ( h = 1,2,
. . . , n), R'
be R minus the origin; and dR the boundary of R.
We define the functions 6, a', 0 as follows: e ( x ) = 2 ( l - x ~ ) ( l - x ~...( ) l-x;)-l,
Z E R'
x E R'
Then we see that e ( x ) has the following properties:
hence 6'(z) + +a as z + 0;
and 6'(z)
-+
-m as
z .+ d R
Chapter II
46
Therefore
O(x)
to infinite order as x
+w
+0
,
and O(z)
Also O(x)is C" for x
# 0.
-t
0
to infinite order as x
If O'(x) =
--$
aR .
& in R' and O'(x) = 0 for x = 0 then O'(z) is
C" for x E R. The next step in the construction of a C" partition of unity on C(A) is to define a suitable subdivision of this set. We first divide the space R" into n-cubes of side 1 (we shall only consider cubes with sides parallel to the co-ordinate ones). Let K O be the set of all these cubes whose distances from A are at least 6n'I2 (if any). In general, having constructed the cubes of K,-1, we divide each cube which is now present but not in UfZ;Ki into 2" cubes of side from A are at least
&; let K , be the set of all these cubes whose distances
$ (if any).
The following facts concerning this subdivision of E - A will be needed. The distance from any cube C of K , to A is
Lemma 1.2.
Proof.
< -(s
2 1).
For it lies in a cube C' of the previous subdivision not belonging to K,-l and
whose distance from A is therefore
<
=
-.
Any cube C of K , is separated from any cube C' of K,+z by at least f o w
Lemma 1.3. cubes of K,+1.
Proof.
1ZJ;; This is true, because the distance d(C,A ) 2 p , the distance from any point
of C' to A
<
g, and the diameter of any cube of IC,+l is &,which means that any
47
Strong approximation in finitedimensional spaces
cube C’ of K,+2 is separated from any cube of K , by definitely more than 3 cubes of
K,+l; this number of intervening cubes of I<,+, has to be a whole number of cubes and hence at least four. Our objective now is to introduce the functions q&,(x), v = 1,2,
. . .. Let y l , y2 . . . be
arranged in a sequence; the set of all the vertices of Uj>oKi
r , = d(y”, A ) = distance from y” to A ; xu a fixed point of A 3 d(x”, y”) = r,;
b, a fixed point of the side of the largest cube of UiioKj with y” as a vertex; I, be the set {x E R,
IXh
- yhyl 5 b,,
v =
1,2, . . . , n} and B, = aI,
.
Then we define r,,(x) =
o
r~(x> =
01
(9, ...,-)
v)
(y, m-v” ...,
x - yp, p
Lemma 1.4.
in E - Y Y ;
in
I, - B,;
# v.
Let y* E E - A, and set 6, = d(y’, A ) (or 6, = d(y+,xO)for a point
xo E A). Suppose y* lies in a cube C E I<,, and suppose I,, with centre y”, has points in cornmon with C . Let d, = d(yY,A ) (or d, = d(yY,2 0 ) ) . Then
d
-1. < d,
2 -
Proof.
< 26. .
Let C’ be a largest cube with y” as a vertex, and suppose y‘ E Kt. Then t 2
s - 1. The diameter of C’ as
g. Therefore the distance from yy to any point of I, is at
Chapter I1
48
most
$ 5 $.
The diameter of C is
follows that d(y’,y’)
g,hence d(yY,yL) 5 %$.
Since 6, 2
9,it
5 $*.
Let z E A 3 d ( y * , z ) = 6,. Then d, 5 d ( y Y , z )5 S,+d(y’,y’)
5 6,++6
+ < 26,.
- 36
+ d(y’, y”); hence 6, 5 d(y”, z ) + is,, i.e., % 5 d(y’, This is true for any z E A, hence % 5 d,. Thus the Lemma is proved. Also, d(y+, z ) 5 d(y’, z )
Next, each function T,,(z)is and 0 to infinite order as x
-+
y”
z).
> 0 in I, - B,, - yy, and only in this set. It tends to 00 and x
-+
B,, respectively. Each point x E E - A is in
I: for some v, hence a,,(z) > 0 for some v, i.e., C ? T X> (X 0 in ) E - A. The function &(x)
# 0 in I,,- B,,
and only on this set. Further
C 4,,(~)= 1
if
Z E
E-A.
Y
It is clear that d,,(x) is C” at points x y’, X
#
v . The function ?~\(x) is C” in
ilarly c$,,= i G $ p -
#
y”.
Let Ux be a small neighbourhood of
U’, hence so is q5,, =
in
u’. Sim-
is C” in a small neighbourhood U,,of y”. Thus &(x) is C”
on E - A. We shall next derive convenient estimates for the derivatives of the functions d,,(z). Let C, C’ be two closed cubes of Ui>oI(i. The cubes C, C‘ are said to be of the same type if the sets in J’ can be brought into coincidence with the sets in J by a translation and by stretching of the axes. There are at most a finite number, say d, of possible types of cubes, and for some number
c,
3 at most c sets I,, with points in a given cube C.
Let C be a fixed cube of KO, and k 2 0 a fixed multi-index. Each &(x) is C”, and &(x)
# 0 only for a finite number of v, hence lDbq5,,(x)l
< Nk(C) V x E C and V v
= 1,2,.
..
49
Strong approximation in finitedimensional spaces
for some positive number
Nk(c).
Now let C E K,, and let C E KO of the same type as C'. If I q , .. . , I,y are the sets
Ix with points in C', let Ix,, . . ., Ix, be the corresponding sets with points in C which can be carried into the former by translation of the axes and stretching by a factor Each function
$.
4 x q corresponding to Ix, thereby is mapped into the function
corresponding to 1 x 4 . On differentiating lkl times w.r.t. s, we find
and therefore, as &(s) = 0 in c' for v
# A:, . . . , A:,
For a fixed k, there are at most d distinct values of
Nk(c);
let
N k
be the largest of
these. The following lemma has thus been proved
Lemma 1.5. For any multi-index k, 3 a number N k 3 if C is any cube of K, then
smooth extension of f(z).
The next result establishes a C"-
Lemma 1.6. Let A be a closed set in R" and let f(s)= fo(z) be of class C" (m finite or infide) in A in terms of the
fk(.)
(lkl 5 m). Then 3 function g(s) which is c" in
R" - A, and has the properties: (1) g ( z ) =
f(.)
(2)
= f k ( z ) in A, lkl
Dkg(z)
in A ,
5 m.
50
Chapter II
Proof.
Case I. First suppose m finite. Let
where the &(z) and $(x, c y ) = t,bo(x; 5") have been defined above ( v = 1,2,
. . .).
The
&(x) and $(x; 5") are Cooin R" - A , hence so is g(z). The function g(x) = f(x) is C" at all inner points of A. We only have to show that Dkg(Z) exists, equals
fk(Z),
and is
continuous, at all boundary points of A , for 1k1 5 m. Let xo be a fixed boundary point of -4, and let
9 where N = max{Nk, lkl
[
E
30
< E < 1. Let r] > 0 3
< min i 9 2 c { ( m+ 2)!}"(108fi)"N E
1
5 m } . Let M > 0 3
and let
'< min(l'
6(m
so small that lRk(x;
.")I
5 Ilt - z 11 rn-(kl 7 . '
0
Now let y* E R" - A 3 y* E B a l r ( t 0 ) .We now assert:
Contention. IDrg(y*) - jk(xoI<
To prove this contention, suppose d(y*,A) = y.11 = $. Consider the function
E
for
I~CIIm .
%, (so 6,
< S), and let x*
E A 3 11x*
-
51
Strong approximation in finitedimensional spaces
(which represents $ k ( z * ,
5')
in the notation adopted earlier). Each
above sum contains at most ( m
+ 1)" terms.
lh
is
5 rn, hence the
On removing the term with
el
en = 0 to the other side, we find in each remaining term a factor (zh - zf)'h Each
=
.. .
with
=
lh
> 0.
Izt - z i l < 6 < 1; hence
Also &(z*,
< 9 < f , hence (using:
3')
Similarly, we find:
It)k(y*,z')
-fk(z*)I
Now suppose y* E C E K,; let
xxq
is at a distance
characteristic property of
&(2*,
= $k(z*,
2')
+ Rk(z*,
5')
< i.Hence
Ixl,. . . , Ixt be those sets Ix with points in C. Each
corresponding point yxq is at a distance sponding point
fk(Z*)
from z ' (cf. Lemma 1.4),hence each corre-
<
< 6 from zo.The same is true of z*;
hence using the
z"), we find
Now set Cu,k(z)
then as llzy - 21 .1
= $ ' k ( z ; z') - $ k ( z ;
< 6, and
1zh -
z i l < 6.
I*) (v = X I ,
.
. I
At)
;
for z E C, it follows that I(z - z")'I
hence (using the earlier identities for the function $ k ) :
<
52
Chapter II
Since
$,,(z)
= 1 in R" - A , we see that
Y
Hence as t I c, and
(ti)I m!, we find
To complete the proof we now use Lemma 1.2. Use the inequality established above: I&g(y')
- fk(zo)I
< E(lkl 5 m ) , with k
= 0; this shows that g ( x ) is continuous through-
out R". Next let k = (k1,.. . ,kn) with (kI _< rn, and k' = (kl,... ,kh Suppose D h g ( z ) is continuous in R"; we shall show that tinuous in R". Let x o = ($;, W(Z)
. .. , $:)
= w ( q ) = Dkg(z;, . . . , z h ,
with sp= zg (for p
+ 1 , . . .,6").
Dklg(Z)
exists and is con-
be any boundary point, and write zo =
. ., , z:),
wk = f p ( i " ) . Let
# h). Then letting Axh = $1, - z,:
A' be the set points of A
= (xi,. . .,
d'
$:,
-th
h ,
... , zi),
and
and ( D k l g ( Y * )- f k l ( x o ) l
<&
(for
lkl
show that the conditions of Lemma 1.2 are fulfilled; hence equals f k I ( $ ' ) .
5 m) = D k l g ( z o )exists and
Again the last inequality shows that Dk'g($) is continuous at
g ( z ) is C" in R".
LO.
Hence
53
Strong approximation in finitedimensional spaces Case 11. Next we shall consider the case rn = 00.
For any given rn, let $,,k(x';
x)( lkl
I rn) be the function defined earlier:
Choose the axes so that the point 0 = (0, whose distances from 0 are
5 2P,p
A n S,}, and N(P) = max{Nk 6,
. . . , 0) E A.
= 1,2,
I lkl I ) . p
Let S, be the set of points of R"
. . .. Let M p = max{lfk(z)l
I
(kl I p , E~
For each position integer p let 6, 3
< 1/{22p+1c[(p+ 2 ) ! ] " ( 3 6 m P N ( P ) PM+ l h
6,.
6,-1
<2 .
The required extension goo(x) of f(x) is determined as follows. For any given v , deter-
6 r , < 6, (recall: r ,
mine 7, 3 6,+1
= d(y",A)). Let 7, = 0 if r ,
>
Then we
define
w e shall establish an inequality for Dkgoo(s) similar to one for Dkg(S) (case I), for any k. Let g("')(z) 1,2,
. . .).
be the extension of class C" obtained in the proof of case I (rn =
Let zo be a boundary pont of A , and let
zo.E S, and 3
$ < E.
Then choose 6
< 6,
E
> 0.
Then choose p
3 for any y* E R"
2 lkl + 2
3
- A with IJy*- zoll < 6, we
have IDkg(lk""(y*)- fk(zo)l
Let q 3
&,+I
I 6, < 6,,
<E .
where 6, = d ( f ; A ) ; then q 2 p . Define C, I<,,IA~, . . , , IA, as
with preceding arguments (Lemma 1.5). Now for u = any
Ah,
SyY+lI r,
< 26 < 26, <
Chapter II
54
&-I,
hence 7”
+1>p -
I, thus 7” > p - 2 1 ILI. We now write
Using the definitions of g ( l k l ) ( z ) (from case I) and of gw(z) we find:
Note that DjEv(z) = qbyvi,(z;z”) - qbl,+j(x; z”). Replacing Ic by j in the expression for
5 rn - ( j l occur. Now replace rn by 7”
we see that only those terms with
&(z‘;z),
and Ikl successively; then on subtracting we obtain
Also
T,
>
4, hence
T,
>
bq+2,
number of terms in the sum is Ifj+t(T”)I
<
hence 3; 5 q (q
+ 1 (v
= XI,
. .., A t ) .
Therefore the
+ 2)”, and in each term Ijl + It( 5 q + 1. Hence
I MP+i in each term. Further
(zh
- zX( < 26. < 2 4 , and It( 2 1Ic( - ( j (+ 1 in
each term. Therefore
Hence
The distance
I>
, therefore
55
Strong approximation in finitedimensional spaces
in particular at y*. Thus we find
for any point y' E R" - A within distance 6 of zo. Now again apply Lemma 1; and it follows that D k g ( " ) ( S ) exists and is continuous throughout R". This is true for every
Ic = (k1,. . . , kn). This proves the Lemma. We shall now turn to the following extension of the familiar Weierstrass approxima-
tion theorem.
Lemma 1.7. Suppose g ( z ) is of class Cm in R" (rn R". Then for each t > 0
Proof.
< co), and S be a compact
3G analytic in R" and satisfying:
Let Ra = Bk(0) ( b 2 0 ) , and consider the n-fold integral
where T is 3 @(m= 1. Then 0 5 @ ( b ) 5 1 Vb. Now replace y by tcy and b by
Kb,
obtain
Let v ( z ) of class
Drv(z)
set in
C"
be E 1 in
S1 E 0 outside some neighbourhood of S and 3
0 in S Vk (cf. Lemma 1.6). Put g'(z) = v ( z ) g ( z ) ,and let
and we
Chapter II
56
where n will be suitably chosen presently. Then G(s) is analytic in R". The function
111
- yll' is a function of z - y only and differentiating under the integral sign gives
where
Dp),Dp)denote differentiation with respect to s and
y, respectively. Then inte-
gration by parts llcl times yields
If we recall that @(m)= 1, we see that
Now let M
> 0 so large that
The functions Dkg'(s) are uniformly continuous on R", hence 3 6
Then let n
>0 3
> 0 so large that 1- q n 6 )
Let U = B&(s), and denote by
J1
and
J2
4M
the regions formed by replacing the domain of
integration on the right side of I D k G ( s ) - D k g ' ( s ) I we then obtain:
E
< -.
above, by U , and R"-U, respectively;
Strong approximation in finitedimensional spaces
hence
This completes the proof of Lemma 1.7. The preceding Lemma can be generalised to yield the next theorem which is the main theorem of Whitney that we are aiming at in this Chapter.
Theorem 1.8. Let R be an open set in R", and R1, R2, . . . bounded open sets 3
u
nzl
R,
= R, and 3
Rp c Pp+l for each p .
finite or finite), and suppose 61 2
~2
Suppose g is defined and of class C" in R (m
2 . . . are given positive number. Then 3 analytic
function G(z) in R, satisfying
where (YP =
Note that if R1,
. .. , R,
{
m p
if nz is finite i f m = 00, p = 1 , 2 , ...
are empty, then this statement means
= QLUQPUQ;.
In Lemma 1.6 we replace the closed set A by this set, and replace f(s)(of Lemma 1.6)
by a function which is function,
C"
in R", 3
3
1 in Qp, and
0 in
QL U Q:.
Then for each p , let up(.) be a
Chapter II
58
and D k u P ( z ) = 0 in QI,UQpU Q: (1k1 > 0). (If Rp+l = 0 we put up(.) but Rp-l = 0, then up(.) = 0 in Q:) and = 1 in
?zp+l.)
Now let 2,
Now define, successively, the analytic functions Gl(z), Gz(z),
= 0; if Rp+l # 0
2 1 be a number 3
. . . , by:
(If p = 1, the factor in brackets is g(y).) The constant tcp is so chosen that if we set
in Rp+i(IkI 5 ap+l;cf. Lemma 7; the constant
K~
will be further restricted a little later
on). From the definition of u p ( z )we see from the last two relations that
We then differentiate H p ( z ) ,replacing p by p - 1 in the preceding relations and we see that (cf. proof of Lemma 1.6, m
< 00):
The function up(.) and its derivatives are 0 in Rp-l, hence the preceding holds also in
Rp-l. Hence
2
I ~ k ~ ~ ( z ) l < in
R~ ( I ~I I ap>.
Strong approximation in finitedimensional spaces
59
We shall show that G(s) yields the required approximation to g(s). We note that
Dk{Gl(s)
+ + Gp(z)} converges uniformly in any compact subset of R (lkl 5 m);
hence G( x) is defined in R and
<
& ( P
-1+ - +1. . . ) 22
= -EP
2
23
vx
RP+l
( I k I I QPtl)'
Hence from a previous estimate we obtain:
This proves part of the Theorem. Now we want to show that G(z) is analytic in R. We extend the definition of each
Gp(x) t o complex values of z = (xi
+ zxyl . . . , x: + is^), using the definition of Gp(x).
Consider the analytic function of x:
The domain of integration in the integral defining Gp(x) is real, hence y l = 0, hence n
Re(rt,y) =
C { z i - 1~1)' - x i 2 } .
h=l
Chapter II
60
Now let z o E R and
U be the complex open ball B p ( x O where ) p > 0 is so small that
the real points in the complex open ball 133p(xo)lie in some
&. Now if p
> q, x
E
U and y
E
R - Rp-l, then
cxi2 <
R,;we take q
so that 3p2
p2 and c ( x k -
>
~ 6 2)4p2, ~
hence
Re (r&) > 3p2 Furthermore Hp(y) = 0 in R, and in R" - R,+z for p (recall: Hp(y) is determined before
K,),
> q.
Hence if Mk = max IHp(y)l
and if Vp = volume of Rp(p= 1,2,
~ G p ( z ' + i x f f<) ~T K ;
J
Rp+.-Rp-
. . .), then
MLe-3k:P' dY I
< TK,"e-":IzbMfp vP+2 if z E U and p
>
q. Now choose tcP successively for p = 1,2,
the right side in the preceding inequality is
. . . so that
the term on
< &, then the series defining G ( x ) converges
uniformly in a complex neighbourhood of any point of R. Hence G ( x ) is analytic in R. This completes the proof of the theorem.
61
Strong approximation in finitedimensional spaces 52. C” approximation i n a Aiiite diineiisioiial space
We shall now turn to the proof of a slightly weaker theorem than Whitney’s on strong approximation in a finite dimensional space. The methods used, however, are sufficiently different from those used by Whitney to merit special attention. It is now necessary to define the concept of approximation in a strong (or fine) topology. The compact open topology (Chapter I) is suitable for investigating closeness of maps over a compact set E, whereas if E is not compact then the compact open topology does not yield sufficient control over the behaviour of a map, and a strong (or fine) topology is more useful. To be more precise, the word “fine” is prefixed by words such as rrckv
T“” or
, as will be made clear presently.
We shall define the concept of Cj-fine topology on C k ( U ,Y ) where
U is a nonempty
open set in a Banach space X, Y is a Banach space, k is a given nonnegative integer and j is an integer 3 0 5 j
5 k. When speaking of differentiability of a map defined in a Ba-
-
nach space we shall always understa.nd Frhchet differentiability unless specifically stated otherwise. As is customary Ej denotes the j-fold Cartesian product E x
. . . x E. Cj-
i smooth means j-times differentiable and the j t h derivative, as an element in the space Lj(E, F ) of continuous j-linear operators E j ous in
-t
F , depending on x E U , is continu-
U hence is symmetric and belongs to the space L j ( E ,F ) , of continuous j linear
symmetric operators Ej
--t
F . Coo-smooth means Cj-smooth for every integer j 1 0.
Djlc, denotes the j t h successive derivative of 1c, if it exists. C k ( U , Y )is the space of C k smooth mappings U
Y . The Cj-fine topology on C k ( U , Y ) ,0 5 j 5 k is defined to be
Chapter II
62
the topology for which sets of the form:
~ ( hq), = [g E
c ~ ( Yu>,I v integers i E [ ~ , jIlo'g(x> l, - ~ih(x)ll
< q(x) v x E u] where h(.) E C k ( U , Y )and q(.) is an arbitrary positive continuous function on U , form a base. The Ck-fine topology on C k ( U ,Y ) is also denoted by C:(V, Y ) . We should point out that although the preceding definition is stated for Banach spaces we are emphasizing results in finite dimensional spaces in this chapter. So we shall now onward in this section deal with functions defined in an open set U in R" with values in R". Before proceeding to the theorem in question and its proof we shall need some preliminary lemmas. Let
and d(x) =
p(llz11'). The function b(.) is C" on R',P(.)
(b,m) while
4(.)
function
is C" on R",d(.)
1 on BJ;;(O)and
1 on (-m,a),P(.)
d(-)
d(.) also enables us to construct a Coo-function 0 : R"
0 on
0 outside B d ( 0 ) . This -t
R' 3 0 = 0 outside a
compact set and its Lebesgue integral in R"' equals 1. By the support of a continuous real-valued function f , denoted by Supp f, is meant the set f-l(R1 - {0}), so the complement of this set, C(Supp f ) is the largest open set on which f = 0. Let U =
{ U a } ,be ~ ~an open cover of U . By a Cj-partition of unity
subordinate to U we mean a family of Cj-maps A, : U -+ [O,l],aE A , 3
63
Strong approximation in finitedimensional spaces
(i) supp A,
c U,,
a E A;
(ii) {supp A a } , E ~ is a locally finite family; and (iii)
C
Aa(z) = 1, z E U .
uEA
The local finiteness property of {Supp X u } a E ~ ensures that each point of U has a neighbourhood on which all except a finite number of A, are 0, and the sum is locally a finite sum. We note that condition (iii) ensures that
where E" denotes the interior of E. We also note the following simple observation: If
V
= {V a } a E ~ is an open cover of U which refines U = { Up}pE~ , and if
V
has a
subordinate Coo-partition of unity, then so does U.
Lemma 2.1.
Every open cover of U has a subordinate CJ-partition of unity, for any
given j 3 0 5 j 5
00.
This result is well-known; for a sketch of the proof see 53 in Chapter 111. To approximate C'-maps by Cm-maps in the strong topology we need to approximate locally f on the
Ui,where { U i } ; E h is an open covering, by Coomaps whose deriva-
tives up to order r uniformly approximate those of
f. In a finite dimensional space this
is achieved by using the technique of convolutions, which we shall explain next. Let 6 : R" + R be a function with compact support. There is a smallest u 2 0 3 supp 6 is contained in the closed ball B,(O) C Rm. We call u the support radius of 6. Now suppose
U c Rm is an open set and f
:
U
+ R" a map. If 6 : R" -+ R has
compact support then we define the convohrtaon o f f by 6' to be the function denoted by
Chapter ZI
64
8 * f : U,
R", defined by
e*f(.)
=
J
@(Y)f(.
- YWY,
xE
uu
9
B..(O)
I Bu(z) c U } .
where U, = {x E U
Here we understand integration to be in the
Lebesgue sense, dy denoting Lebesgue mensure in Rm. We note that the integrand is 0 on the boundary of B,(O);we define the integralid to be 0 outside B,(O) and thus extend it to all of Rm. Then @
* f(.)
=
JRm
@(Y)f(.
- YWY,
5
E
UU .
Now for a fixed x E U, we make the change of variable: z = x - y. Then
e*f(.)
=
J
Be(=)
q x- Z)f(t)dZ
=
@(x - z)f(z)dz,
x E U,
,
where the integrand is defined to be 0 outside B,(x).
A function @ : R" port and
-+
R is called a convolution kernel if @ 2 0, @ has compact sup-
s,.,, @ = 1. Earlier we noted that such kernels which are further C",
do exist.
Since @ * f ( x )can be considered to be a weighted average of values of f near z, it is plausible to expect that @ * f might be an approsirnation to f in a neighbourhood of x, and the approximation might be smooth. We shall use the notation
i f f : U + R" is C', U
c Rm
is open,
K c U is any subset, and IlD'f(z)ll is the norm
of the kth derivative (as a continuous I:-linear symmetric operator) o f f at z. IlD"f(z)ll
We now need the following result.
Strong approximation in finitedimensional spaces
Lemma 2.2.
Let 8 : R"
--*
R have support radius u
65
> 0. Let U
set and f : U + R" be a continuous function. Then 8 * f : U,
+
C R" be an open
R" has the following
properties: (a) If 81,upp(0)0is C k ,1 5
kI 00, then
so is 8 * f , and for each finite
k
D k ( 8* f)(x) = Dk8 * f (z) . (b) I f f is C k Then
Dk(8*f ) = 8 * D k ( f ) . (c) Suppose f E
C', 0 5 r I 00. Let I< c U be compact. Given E > 03u > 0 3
K c U, and 8 is a C'-convolution kernel with support radius u , then 8 * f E C '
Proof.
The part (b) follows by differentiating under the integral sign. The part (a)
follows by a change of variable, viz. z = x - y. (c) It suffices to let r = 0. Since dist (K,R" - U ) > 0, we can choose u
> 0 sufficiently small so that I( c U,. Also let
sufficiently small so that, if z E K and Now use the fact that
,s,
112 -
y1I
u
5 u then If ( z ) - f(y)l < E .
8 = 1. On integrating over R", we obtain:
This proves the Lemma. Thus we see that a proximated by
C"
C ' map from an open subset U of R" to R" can be C ' ap-
maps in neighbourhoods of compact subsets of
Kurzweil (see(301) for the proof of the next lemma.
U. We shall follow
Chapter II
66
Suppose X is a metric space and G
Lemma 2.3.
c
X an open set. Suppose G =
m
u
B; where we have written Bi for B t i ( z i )= the open ball in the centre zi and radius
i=l
&, i = 1,2, . . .. Then 3 locally finite open covering Remark.
The
{K}rlof G 3 K c Bi, i = 1,2, . . ..
in this lemma will turn out to have further properties: viz. each V;
will be “scalloped” to use the terminology of Lang [32] p. 35. Proof. 1 > el
Choose a sequence of positive numbers
< E~ > ... with lim
having defined
Vj-1,
u j= W
Clearly G =
~i
t-+ w
w
=0
.
Then define
{&i)zl 3 3ci V1 =
< 6;, i
= 1,2, . .
. , and
B I , VZ= Bz fl CB61-c1(21); and
define
Vj, and Vj c B j , j = 1,2, . . . ,
Let y E G, and let k - inf[rnly E B m ] . There is an integer e
>k
3y E B~~-3~~(zk).
Then
Hence
This means that 3 neighbourhood viz. B6,,-3rr(zk)of the point y which intersects only a finite number of the sets Vj, j = 1 , 2 , . . .. This shows that the covering
{Vj}glis locally
finite. This proves the lemma.
Lemina 2.4.
Suppose G
c R”
{ C i ) z l of G by compact set C;.
is an open set. Then 3 countable locally finite covering
67
Strong approximation in finitedimensional spaces
The proof is a slight modification of the proof of the last lemma. For each
Proof.
G3 open ball Bza(z)c G. Then 3 countable subcollection {Bai(zi)}& 3 G
u
=
i= 1
where we set Bi = Bai(zi). Then G =
U
0 0 -
Bi, and each
5
E
W
Bi,
is compact.
i=l
Let lim
i+w
~i
{&i}g1 be a sequence of positive nunihers 3 3&i < 6i, 1 > ~1 > EZ > ... , and
= 0. Define
generally having defined Cj-1, define
u
m
Then G =
Cj, Cj c Bj and the Cj are compact, for j = 1,2,
. . ..
j=1
Let y E G, and let k = inf [mly E
B&,- 3 c r
(2k)
Barn].There is an integer L? > k
3 y E
Then
hence
This means that 3 neighbourhood via.
B 6 h - 3 c r ( z k )of
the point y which intersects only
a finite number of the C,, j = 1 , 2 , . . . , i.e., the covering { C j } z l of G is locally finite.
This proves the lemma.
Theorem 2.5.
Let
U c R"
be an open set. Then Cw(U,R") is dense in C'(U, R") for
0 5 r < 00, in the Cr-fine topology.
Chapter I1
68
Proof.
Let f E C(U, R"). Let I( =
{I
be a locally finite family of compact sets
covering U , let e = { ~ i } i e , be , a family of positive numbers and define the sets N ( f , K, e) to be the set of functions g E C'(U, R") 3
We want t o show that with a fixed family K, if we let continuous function on U , and
E;
= min
E(.)
{Xi}i3,,
be a C"-partition
> 0 be an arbitrary positive
on I<, where I<; E K , i E A, then
C"(U,R") n N(f,K,e) Let
E(.)
# 0.
of unity on U 3 supp(Xi) is compact and contains
Ki . Given any set of positive numbers { c ~ ; } i e A 3C" maps g; : U;
-+
R" (defined on
suitable open sets Ui 3 ICi, i E A ) satisfying:
Then we define g : U
-+
R by: g ( s ) =
xi X;(z)g;(z).Then clearly g E C".
We need to
estimate llDkg(s)- D k f ( z ) l l for , 0 5 k 5 v. We note the following generalised Leibnitz formula: if X : U are
Ckfunctions and $(e)
= X ( z ) h ( s ) , then
where the products are tensor products. Hence
-+
R, and h : U -+ R"
69
Strong approximation in finitedimensional spaces
for a suitable positive constant Ak-independent of and for fixed i E A, A i = { j E A
I K i u I’
L,
# 0).
2,A,
and h. Let A = maxg
This is a finite set, with cardinality,
mi, say. Put
Now we choose the numbers find: 119 - f l l r , ~ <
ai
so that miApip;
<~
i With .
this choice of a;, i
A, we
< E ; . This proves Theorem 2.5.
We shall now turn to another aspect of strong approximation - strong approximation by C" maps with some further special properties, viz. transversality. Such results are more natural in the context of differentiable manifolds Thus some of the results are formulated for manifolds. One additional technique has now to be used, viz. the MorseSard theorem. Some definition and preliminary results are necessary. An n-cube C c R" of edge A
> 0 is a product
of closed intervals each of length A: say I j =
[ u j ,a j
+ A]
c R. The (n)-meusure of C
is p ( C ) = p n ( C ) = A". We say that a set X c R" has measure zero if VE
> 0, X
can be covered by a family of n-cubes the sum of whose measures is less that
E.
Thus a
countable union of sets of measure 0 is of measure 0. Furthermore if each point of X has a neighbourhood in X of measure 0, then X has measure 0 (Lindelof).
Chapter 11
70
Let U
Lemma 2.6.
c R" be an open set and f : U
-+
R" a C1-map. Then if X has
measure 0, so does f ( X ) .
Proof.
Each point of X is contained in an open ball B
Hence if C
c U 3 for some constant K. > 0
c B is a cube of edge X then f(C) is contained in a cube C'
3 such that
p(C') < L"p(C) where L = &.
u
00
Now X =
j= 1
graph. For each
E
X j where each X j is a subset of a ball B as in the preceding para-
> 0, X j c
uE CEwhere each Ck is a cube and '&,~ ( C E< )
E.
Hence
f ( X j ) has measure 0 hence, f ( X ) has measure 0. This proves the Lemma. Now let M be an n-dimensional C--manifold (cf. Appendix 3). A subset X said to be of measure 0 if for each chart
(4, U),the set $(U n X ) c R"
cM
is
has measure 0.
It follows from the preceding lemma that this will be true if 3 atlas of charts with this property. We note that a cube has positive measure, and hence a set of measure 0 in R" cannot contain a cube, and therefore must have empty interior. Thus a closed subset of measure 0 in
R", or on a manifold A4 must be nowhere dense. Suppose X 00
measure 0 and is a-compact i.e., X =
u
cM
is of
K n where each I(, is compact then each K , is
n=l
nowhere dense hence X is nowhere dense. The complement of X is residual i.e., contains the union of a countable family of dense open sets, and is dense (by the Baire category theorem).
Lemma 2.7.
Let M , N be manifolds with dim M
then f ( M ) is nowhere dense.
< dimN.
Iff :M
-+
N is a C'-map
71
Strong approximation in finitedimensional spaces
Proof.
This follows from the fact that F ( M ) has measure 0.
We define a point x E M to be a critical point for a C1-map f : M map 2'. f : M , -+
f
N f ( , ) is not surjective. Denote by
if the linear
cf the set of critical points of f ;
(c is the set of critical values of f , and N - f (c j )
+N
j )
is the set of regular values of f .
We shall now turn to the Morse Sard theorem and we shall only sketch the proof in the case where f is C"; this however, is not the best result.
Theorem 2.8.
(Morse-Sard)
tively, and f : M
+N
Suppose A4, N are manifolds of dimensions m, n respec-
a C'-map. If r
> max(0, m - n) then
f(c,)has measure zero in
N . The set of regular values off is residual and therefore dense. Proof. (only for the C" case) We shall consider a local result viz. in the case
<
where f : W + R" is a C" map, and W is open in Rm. If m
n then f(W) has
measure zero. So we shall suppose m 2 n. Write f(x) = (fi(x),
. . . , fn(r)).
to be the sum of three subsets
p E
cf such that Afi(p)
i = 1, , . , , m.
C'
We note that the critical set
C', c2, C3,defined as follows.
cf can be realised
c' is the set of points
= 0 for all differential operators A of order _< f and all
is the set of points p E
cf such that Afi(p) # 0 for some i and some
c3is the set of points p E xrsuch that $ ( p ) # 0 for some i , j . Then cf = x' U c2U c3. Each of f(cl), f ( x 2 ) and f(c3) has
differential operator of order 2 2.
measure 0. The Morse-Sard theorem is used to show that transversal mappings form a dense and open subset of the set C'(M, N ) where M , N are manifolds with C' fine topology. Let f : M + N be a C'-mapping, A
cN
a submanifold. If K
c M , then we write
Chapter I1
72
f Q K A t o mean that f is transverse to A along K , that is t o say, if x E K , and f ( x ) = y E A, then the tangent space
N , is spanned by A , and the image ( T , f ) ( M , ) . If K =
M , we write f OA (cf. Appendix 3 ) . Before proceeding to our main theorem on transversal mappings we should observe a few facts which we shall state as Lemmas.
Lemma 2.9.
Let f : M -+ N be a C' map, r 2 1, and y E f ( M ) a regular value.
Then f-' (y ) is a C'-su bmanifold of M .
Proof.
It is enough t o consider the case where M is an open set in Rm and N is an
open set in R", and then the proof follows by using the Inverse Function theorem. Now suppose f : M
A , then A ,
+
N is transverse to a submanifold A
+ T'f(x) = N,.
Lemma 2.10.
Let f : M
c N , i.e., if f ( x ) = y
E
Then the following result holds. -t
N be a C'-map, r 1 1, and A
cN
a C'-submanifold.
Iff
is transverse to A, then f - ' ( A ) is a subma.nifold of M , and the co-dimension of f - l ( A )
in M is the same as the co-dimension of A in N .
Proof. Let q = codimension of A in N , and p 3 p the preceding lemma, it is enough to consider
U
neighbourhood of (0,O) E RP x RQ. Then f : M
+q = n
x V +
c
RP
= dimension of N . As in
x RQ, with
U xV
U x V is transverse to U x 0 if and
only if 0 is a regular value for the composite mapping: 9 : M f U x V w V . -+
Now g - l ( O ) = f-'(U
-+
x 0 ) , and the lemma follows by the preceding lemma.
The next lemma will be stated without proof.
an open
13
Strong approximation in finitedimensional spaces
Lemma 2.11. (a) Let yo E N , then the set
is open and dense in C g ( M ,N ) , 1 5 r
5 00.
(b) Suppose fo E C'(M, N ) , yo E N is a regular value for fo, n/ is a neighbourhood of
fo in Cg(M,N ) , and W a neighbourhood of f-'(yo) g = fo on
in M . Then 3g E N 3 g
0 {yo} and
M - W.
We now define
and
rfi'(M, N ; A ) = $>(M, N ; A ) . We shall turn to the following theorem:
Theorem 2.12.
Suppose M , N are manifolds A
cN
a submanifold, and 1 5 r
5
m.
Then (a) or(M,N ; A ) is residual (bence dense) in C'(M, N ) for the strong topology;
(b) Suppose A is closed in N ; if L
cM
is closed then @ L ( MN , ; A ) is dense and
open in Cg(M,N ) . For the proof we need a "globalisation" lemma. Before stating this lemma we should first explain some preliminary ideas. Let M , N be By a
C' m a p p i n g class we mean
a function
x
C ' manifolds, 0 5
I-
defined as follows. The domain of
5
00.
x is
Chapter I1
14 the set of triples (L,U,V) with U and L
c
c
U is closed. The mapping
X L ( U , V )c C'(U, V ) ,and
x
M,V
c
N , L c U where U , V are open
x associates with each such triple, a set of maps
further satisfies the following property:
Given triples ( L ,U , V ) ,a map f E C'(U, V ) belongs to the class X L ( U V) , if 3 triples (Li,Uj, K) and 3 maps fi E x ~ ~ ( UV;)i , 3 L
CU Li
and f =
fj
in a neigh-
i
bowhood of L; Vi. We note that the class mL(U, V ,V
n A ) is a specific example of such a class.
A mapping class is called rich if 3 open covers U , V of M , N with the property that if U c M , V
c
N are elements of U , V respectively and L
c U is compact, then
X L ( UV , ) is dense and open in Ct;l(U,V ) . The "weak" topology C b ( M , N ) needs to be defined here. Let f E C ' ( M , N ) ,
(4, U ) ,(4, V ) charts on M , N , K c U fine a neighbourhood
a compact set 3 f ( K ) c V , and let 0
p(f; (4,U ) ,($, V ) ;E )
< E.
De-
to be the set of C' maps g : M + N 3
g ( K ) c V and 3 Vz E K and V integers k E [0, r ] ,
Ildk($f4-'(4
- Dk($g$-'(4Il
<E .
The topology Ct;l(M,N ) is defined to be the one for which sets of the type
"(f;
($7
V ) ;(4,V ) ;E )
form a subbase.
L e m m a 2.13. r
(Globalisation)
Let
x
be a rich C' mapping class on ( M , N ) , 0 5
5 00. Then for every closed set L c M , (a) X L ( M ,N ) is dense and open in Cg(A4,N ) ,
( b ) if L & compact, X L ( M N , ) is dense and open in Ct;l(M,N ) .
Strong approximation in finitedimensional spaces
L e m m a 2.14. Suppose K is a compact set in a manifold U,RP space and V is an open set in
R".
Then
is dense and open in Ct;.(U,V), 1 5 r 5
00.
For the details we refer the reader to Hirsch [21] pp. 74-77.
I5
c R" is a linear sub-
This Page Intentionally Left Blank
CHAPTER I11 Strong approximation in infinite-dimensional spaces We shall now turn to some results on strong approximation in infinite dimensional Banach spaces, starting with the theorems of Kurzweil. The concept of CJ-fine approximation in Ck(U,Y)(with 0
5 j 5 b ) has already been defined (cf. Chapter I1
52). Kurzweil's theorems are proved for some of the familiar LP spaces. Since one of his proofs uses complex-analytic methods, we shall first explain the concept of complex extension of a real Banach space.
51. Kurzweil's theorems on analytic approximation Let X be a real Banach space. Then X shall mean the complex space of ordered pairs z = (I,y) = I
+ iy , I,y E X , i = ain C , with the usual operations of addition
and scalar multiplication and norm
Suppose p ( z ) is a real-valued polynomial defined on X . Then 3 polynomial F ( z ) defined on
2 by the condition: 5 ( z ) = p ( z ) if
z =I
+ iy, y = 0.
The following is the
first theorem of Kurzweil.
Theorem 1.1.
([28]) Suppose X is a real reparable Banach space 3 3 real polyno-
mial p*(z) on X satisfying
Chapter IIl
78
c
Let U
X be an open set and f(.) : U
4
Y be a continuous mapping, Y being a
real Banach space. Then 3 analytic mapping g(.) on X , satisfying: ((g(z) - f(z)ll
<1
vx E u . Proof.
Let p * ( z ) be a real polynomial with the properties (l), and suppose p'(.) is of
degree m
> 0.
Then
Po(.> where
pj(.)
= Pl(Z) t m(.>
t
' *
t pm(.)
is a homogeneous polynomial of degree j , 1 5 j
5 m . Now define
Then p ( z ) is nonnegative, and = 0 only if z = 0. Let 7) = inf{llp(z)II I11511 = 1). Then 7)
>o. For y E X ,and r > 0 define
Each set K,(y) is open and bounded. Also for each
1 1 ~ 1 1< r ' .
Now for each z E U 3r,
T'
>0 3 >0
3 z E K,(O)
=+
> 0 3 K,..(z) c U and further 3
Then { K r . } . ~is~ an open covering of U ,and because of separability, 3 countable subcovering { K,.,
oo
(zn)),=,
of
u.
Now going back to the proof of Lemma 2.3 in Chapter I1 we see that 3 open covering
of U which is locally finite and 3 for each j = 1,2,
. . . Dj c
Kr.i(zj).
79
Strong approximation in infinitedimensional spaces
For convenience we shall again sketch the construction of these open sets D j . Let
{ c , } 00 ,=1
be a sequence of positive numbers 3
3~i
1>el>~2>...;
and
asn+w.
c,+O
D3 = CrO1--.a(x2)n Cro2--rg(x2)n KrOa(z3); ...
.
Arguing as in Lemma 2.3 of Chapter I1 we conclude that these sets form a locally finite open covering of U and Dj
c Kroj(x j )
for each j = 1,2,
... .
We shall need another countable locally finite open covering { D ~ } ~ =ofl U 3 Dj
D ; c U foreachj=1,2, 0; =
c
.... Define
Krm1+2al(z1),
D; = Cro1-3c~(~1) n Cr.2-~eg(z~)n X r o a + ~ c g ( ~ 3... ); . Then D ,
c
Di
cU
for each j = 1,2, . . . . The set
in the complement of the set D!, j = t?, C
Kroh-3cr(~k),
C > k is contained
+ 1,t? + 2, . . . and hence { D ; } p l
is a locally
finite open covering of U
As before we shall denote n-dimensional Euclidean space by R" . Define the sets
80
Chapter III
and so on, where Vz, V3,
etc., where b l , bz,
. . . are numbers satisfying:
.. . are positive numbers depending on the constants al, a2, . . .. The
positive numbers al, a2,
. . . , and tl, t z , . .. will be chosen later on.
Next we define the functions q51(z), & ( z ) ,
and so on. Then the functions &(z), q52(z),
. .. ,
with z E X , by
. . . are analytic in
(because
composition of the mapping
-
which is analytic in X and of an analytic functioii of j complex variables).
$j(z)
is a
81
Strong approximation in infinitedimensional spaces
Now let a,, n = 1,2,
. . ., be positive numbers 3
OD
the series n=l
an(l
+
1 1 2-
~~11)~"'
converges Vx E X , rn being the degree of the polynomial p * ( z ) ; for instance the choice:
will serve the purpose. Then the numbers t,, n = 1,2, so that the following three conditions are satisfied (
the set
. . . , are chosen sufficiently large
IT,I being the Lebesgue measure of
T,,c R"):
To show that a choice of such numbers t , is possible, we note that if x E D, then
If x 52),
E
D, then the open ball in R" with centre at the point ( p ( x - q ) , p ( x -
. . ., p(x - x,))
and radius en, is contained in T,. Also, if
2
# DZ,then at least
one of the inequalities
p(.
-xn)
p(z
- xj) >
<
rz,
+2~n;
r z j -3cn,
j = 1,2,
..., n - 1
is false. Then by the definition of T,, it follows that for every z @ 0: the open ball
. . . p ( x -xn)) and radius with centre at (p(x -q),
E,
does not intersect T,. The choice
of the numbers t , satisfying the above 3 conditions now follows.
Chapter III
82
The next step is to show that
and H*(.) are analytic. The uniform limit of a
$(a)
sequence of analytic mappings in a complex Banach space is analytic. Hence it is enough to show the following (Hille and Phillips Theorem 3.18.1. (201 p. 113, 1957): for each zo
E U , 36 > 0 and 3 integer no > 0 3 V z E
(IIf(zn)II + 1)
2 with
llzll
<6
and Vn
> no, we have
I ~ ( X O+ .)I < $i.
Let zo E U ,and let j o be the first integer m 3 number a , and 3 integer n'
20
E Kpmm(zm). Then
3 positive
>03
We shall find a lower bound for
j= 1
The polynomial $(z) is of degree 2m ; hence
where
lZjl
5 M - (1 + 115 -
. 11~11,
llzll
< 1, M
being a positive constant (see
Hille and Phillips [20] Theorem 26.2.4 p. 764, 1957). Then
Strong approximation in infinitedimensional spaces
83
Thus
where
(71,
. . . , 7,)
E
Tn and a is positive. The series
00
C
aj(l+Ilzo-zj11)4m converges;
j= 1
hence 3 positive numbers
hence 3 integer n o
holds Vn
p, 6 3
the inequality
> 0 3 the inequality
> no, and for
llzll
< 6.
Now define
Then H ( . ) is analytic. Furthermore, let
2
E U . Then we see that
84 Let
Chapter III
Il (respectively 12)denote the set of indices j
3 x E
D; (respectively
z @
Dj').
Then
+f +
and we obtain I l f ( x ) - H(x)ll _<
< 1. This completes
the proof of Theorem 1.
The next theorem follows form Theorem 1.1.
Theorem 1.2.
([28]) Let
X be a Banach space satisfying the hypotheses
1.1, U C X an open set, f : X
and
E(-)
-t
of Theorem
Y a continuous mapping, Y being a Banach space,
a continuous positive function on U . Then 3 function g ( x ) which is analytic in
U and satisfies: I l f ( x ) - g(x)(( < ~ ( x ) V xE U . Proof. v(x)I
Theorem 1.1 implies that 3 analytic function q ( - ) in U , satisfying:
< 1 Vx
E
U . Then q(x) >
I& + 1 -
6. Also by Theorem 1.1, 3 function IT*(.)analytic
in U and satisfying:
Ilv(.>f(x) - H'(.)ll < 1 Let g(z) = & H * ( x ) ;
1.2.
then Ilf(z) - g(x)11
vx E
< & < E(Z)
u.
Vx E U . This proves Theorem
85
Strong approximation in infinitedimensional spaces
Remark.
The hypotheses of Theorems 1 and 2 hold in the case of L P ( p ) or
lP
where
p is an even positive integer, in this case the p t h power of the norm is a polynomial with
the required properties, hence Theorems 1.1 and 1.2 are true for these spaces. Likewise the theorems are valid for the Cartesian products of such spaces. The following corollary of Theorem 1.2 is also a special case of Theorem 1.8 of Whitney (cf. Chapter 11, Section I).
Corollary 1.3. Suppose U and
E(.)
R" is an open set, f : U + R' a continuous function
c
a continuous positive function on U . Then 39 : U + R' which is analytic in G
and satisfies:
If(.)
- g(x)l < ~ ( x Vx ) E U.
#
Kurzweils's further results show that in the spaces d or L P ( p ) , with p 2 1, p
an even integer, as well as in the space C[O, 11, uniform approximation t o a continuous mapping by an analytic mapping is not possible. We shall now turn to these theorems. However some preliminary definitions and explanations regarding notation are in order. We shall denote by R+ the half-line of numbers {x 2 0). Suppose f ( - ) is a mapping defined on an open set U
c
Banach space Y . Let k
a fixed integer. We shall denote by P ( z ,h ) a mapping:
(2,h )
E UxX
+Y
>0 k
X , X being a real Banach space, with values in a
such that for fixed
2
E
U ,P ( x , h ) is a polynomial of degree at
most
k in h and that P(z,O) = 0. Definition.
The mapping f is said t o be k-times regularly differentiable if 3 mapping
P ( z , h ) : U x X -+ Y satisfying:
a(z,17) being a nonnegative functional defined on an open subset of C[O, 11 x R+ and
Chapter III
86
and satisfying: Vxo E U ,VE > 0 36
>0
3
llz - z011 < 6 and
05
r]
< 6 + a(z,r])is
defined and 0 5 a ( z , q ) < E .
If L = 1, then P ( z , h ) is usually called the diflerential of f(.) and denoted by 6f(z, h ) . We note that a regularly differentiable mapping possesses a Frkchet differential and that an analytic mapping is k times regularly differentiable for any k = 1,2, . . . . Then the following theorem holds.
Theorem 1.4. Kurzweil [28]) Consider the open ball BR(O) c C[O,l]. Let f
:
BR(O) 4 Y be (once) regularly differentiable, with values in a weakly complete Banach space. Suppose
E, r
are two given positive numbers, with r
+
E
5 R . Then 32
E C[O,11
satisfying: r L
Note:
llzll < 7- + E ,
llf(.)
- f(0)ll
<E .
Before proceeding with the proof of this theorem, we note that the theorem
shows that the function
11211,
z E
C[O,11, is not the uniform limit of a sequence of reg-
ularly differentiable functions in the open unit ball: Bl(0)
).(fI
regularly differentiable f : Bl(0) 4 R' 3
- 11z))1<
c
C[O,11. For suppose 3
f . Such a conclusion would
contradict the conclusion of Theorem 1.4. For Theorem 1.4 implies that 3z E Bl(0) 3
$ 5
11z11
< 1, If(.)
).(fI
If(0)l < f , and
- f(0)l
< f ; but
- 1 1 ~ 1 1 1<
$,
).(fI
the inequality
-
1 1 ~ 1 1 1< f
implies that
and this leads to a contradiction. Hence the function
1 1 ~ 1 1on Bl(0) c C[O, 11 cannot be uniformly approximated
by a regularly differentiable
function.
Proof of Theorem 1.4. numbers
r]
Let z E BR(O)c C[O,11, and let V ( z ) be the set of positive
with the property: 3 open subset H ( z ;r ] ) c C[O,11 x R+ containing all
87
Strong approximation in finitedimensional spaces
points (z,[) with 0 5 [ CY(Z',[')
< 77, and
3 if
(I',[') E
H ( z , q ) , then a ( z ' , [ ' ) is defined and
< t.
Define:
Then y(z) has the properties proved in the next lemma.
Lemma 1.5.
y(.) is positive and lower semi- continuous on BR(O)C C[O,11.
Proof of Lemma.
We shall show that
p(.) is 1.s.c.
Let
10
E C[O, 11, and [ E
( O , p ( z o ) ) . The set H ( z o , [ ' ) is open in C[O, 11 x R + , and contains all points
with 0
< C < [' . The interval
[0,[] is a compact set hence 3 open set U
taining zo and 3 all the points (z,<), z € U,O
1 1 ~ 1 1< R
if z E U . Then p ( z ) 2 [VI E U ,and
(10,
C)
c C[O,11 con-
< ( < [, are contained in H ( z , [ ' ) and p(.) is 1.s.c. at
20.
If follows that y(.) is
also 1.s.c. This proves the lemma. Before turning to the next lemma, we should state some definitions. Let q(.) be a homogeneous polynomial of degree 1 defined on C[O, 11 with values in a weakly complete Banach space (cf. Hille and Phillips [20] Theorem 26.2.4 p. 765). Let c'
> 0,
and denote by T(e',q ) the set of numbers t E [0,1] with the property: U is an open interval containing the point t , the 3 z(.) E C[O,11 satisfying: 11z11 < 1,lMz)Il
'
Lemma 1.6.
The set T(e',q) is finite.
I ( . )
= 0 if
7
6
U,
E'.
Proof of Lemma.
Suppose T(c ' ,q) is infinite. Then 3 numbers t , E T ( c ' , q ) , 3 open
Chapter III
88
intervals U, c [0,1], and 3 functions z,(.)
E C[O,l] for n = 1,2,
. . . , and satisfying
llznll 51 .
and
We now invoke the following theorem of Orlicz:
Lemma 1.7. (Orlicz 1421 Theorem 3, p. 247).
Suppose
is a sequence of
elements in a weakly complete Banach space; and suppose 3 positive number K 3
+ ii, + . .. + ?ti,,11 5 K
Ilii,
gers, k = 1,2,
. . .; then
for every finite sequence 1 5
the series
il
<
i2
< ... <
ik
of inte-
?inconverges unconditionally.
To apply this theorem of Orlicz, let i, = q(z,,), and K = llqll. Then by the preceding lemma (Orlicz) 11ijll
i, converges; this however contradicts the assumption that
= 11q(zjll > E ' , j = 1,2,
. . ..
Thus the Lemma is proved.
We shall now continue with the proof of Theorem 1.4. We shall define a special sequence {z,}
E C[O, 11, with z1 = 0 , with the following properties. If z,
has been de-
fined with the properties:
then we shall choose z,+1
satisfying:
r(4> 1
+")
5
llf(zn+l)ll I
llzn+1
- znll,
11~n+111-
Il~nll,
EIIZnll,
and thereafter we shall show that such a sequence must be finite. We shall suppose 1.
E
<
89
Strong approximation in finitedimensional spaces
Let t' E [0,1] 3 t' $! T (;EY(zn),P(zn,z)), and 3 lzn(t')l > llznll - $Y(zn). Here
P ( s , , z ) is the differential of the mapping f. Then 3 open interval U containing t' 3 llP(zn,y)ll 5 f&r(zn)provided y satisfies: Iy(t)l 5 1 Vt E [0,1] and y ( t ) = 0 if t $! U . Now consider a fixed function y(.) E C[O, 11, and suppose
Let xn+l =
2,
+ y.
Then the above 3 required conditions on the sequence
(2,)
are
satisfied because 3
11zn+111
2 Izn+l(t')l = Izn(t')l+ q d z n ) 2 llznll+
1 ~ ~ ( z n ) ,
and then by the properties assumed regarding the mapping f ,
There must be an element zn 3 r 5 llznll
n = 1,2, . . . . Then the series
r
+ E ; for otherwise there would
satisfying the above 3 conditions and also llznll
be an infinite sequence of points z, T,
<
<
00
C
y(z,) must converge, hence
(2,)
mush be a Cauchy
n=l
sequence, z,
--t
z say
llzll 5 r < R .
However the conclusions:
r(zn)-+ 0
and Y(C)
> 0,
contradict the fact that y(.) is lower semicontinuous; this contradiction completes the proof of Theorem 1.4. We shall now turn to the spaces P , L P ( p ) , for p 2 1, p
#
an even integer. The
methods applied in Theorem 1.4 can be applied again, with slight modifications. Let p 2 1, p
#
an even integer, be a fixed number, and let p be the least integer 2
Chapter III
90
p . The space X is now b , or LP(p). The main result for these spaces is Theorem 1.12
(Kurzweil [28]) below. This result will he proved as a consequence of the next theorem.
Theorem 1.8. (Kurzweil[30])
Suppose R,r, E be positive numbers with R 2 r
+E
and suppose the functional f is defined on the open ball BR(O) C P, p 2 1,# 2,4,6,
...,
and has the property: 3 functional w ( x , h ) defined for x E BR(O)c P , and
h E P , such that for fixed h, w ( x , h) is a polynomial of degree at most p and satisfies
Here a ( x ,q ) is supposed to be defined on an open subset of BR(O)x R+, and is assumed to have the property: Vzo E BR(O),and VE'
6
j
a ( z ,q ) is defined and satisfies: 0
> 0 36 > 0 3
1 1 2
- x011 < 6 and 0 5 q <
I N ( Z , ~ <) E' .
Then 3 c E d satisfying:
For the proof of Theorem 1.8 the following lemmas are needed.
Lemma 1.9. Let v(z) be a real homogeneous polynomial in P ,of degree v < p . Let
Y =(
. . ., yn,O,O, . . .).
~ 1 ~ ~ 2 ,
(here the numbers
Then
ai1i2...ivdo
not depend on n and are defined uniquely). Then as j
aj,j...,j
+
0.
+
91
Strong approximation in finitedimensional spaces
Proof.
This lemma is proved by induction. It holds for v = 1. Suppose it holds for
polynomials of degree v - 1, hi
hj
‘v
Let
= {1,0,0,. ..},
=(.
, j..., r l ,0,1,0, ...} ,
j=2,3,..
Then the differentials
are homogeneous polynomials of degree v - 1 in y (cf. Hille and Phillips [20] Theorem
26.2.9, p. 765). Hence the assertions of the lemma (except the last one) clearly hold; and if the last assertion were not true, then 3 sequence j,,j,, ek = {ak,l,ak,2, ak,jl
ab,j
then V(er)
-t
co as k -+
=
ak,ja
= 0 00.
= if
. . .} IC = 1 , 2 , . . . ,
... = a k , j k
=
. . . with the property:
let
where
1 kl/P’
j # j ~ , j , ,..., j b ;
This contradiction establishes the lemma.
This lemma in turn implies the next one.
Lemma 1.10. Let p- be the greatest integer less than or equal to p , and let
E,
Q be
two positive numbers. Suppose q(.) is a real-valued polynomial on 4, of degree at most
p . Then 32 E ?t satisfying -
(3)
x
has only a finite number of coordinates different from 0
Chapter III
92
Proof of Lemma.
If p is not an integer, the proof is immediate from the last asser-
tion (for v ) of the earlier lemma. Suppose p is an integer, then p must be odd. Then let q ( z ) = q1(x)
+ q2(5) where q1(.)
is of degree at most p - 1,q 2 ( . ) is homogeneous of
degree p . So q Z ( x ) is a continuous odd functional. Thus the lemma is proved.
Proof of Theorem 1.8.
This consists in an almost verbatim repetition of the proof of
Theorem 1.4.
As is easily verified, every polynomial satisfies the conditions of Theorem 1.8. Hence we obtain the next corollary.
Corollary 1.11.
Let
c1r1,r2
valued polynomial in eP(p
be the positive numbers, r1
> 1,# 2 , 4 , 6 , . . .). Then 32 E @' < 1141< 1'2
(1)
r1
(2)
M1 . -
(3)
< rz.
Suppose q(.) is a real-
satisfying
3
< &?
the point x has only a finite number of coordinates different from 0.
Proof of Corollary.
Theorem 1.8 implies that 3y E
ep
satisfying the assertions (1)
and (2) The polynomial q ( . ) is a continuous functional. Hence 3 point x E F satisfying all the assertions (1)-(3). We shall now turn to the crucial theorem.
Theorem 1.12.
([30]). Let e, R,r be positive numbers with R 2 r
functional f(.). is defined in the open ball BR(O) c X = t
p
( p 2 1,p
is p times regularly differentiable. Then 3 z E X satisfying
r
I 1 1 ~ 1 1< r + c , If(.)
-
I ~11x11.
+E.
Suppose the
# 2 , 4 , 6 , . . .), and
93
Strong approximation in infinitedimensional spaces
The proof of Theorem 1.12 for X = d consists again in
Proof of Theorem 1.12.
an almost verbatim repetition of the proof of Theorem 1.8, except that Corollary 1.11 is now used instead of Lemma 1.10. Thus the theorem follows in the case X = b. Furthermore, LJ’(p)(for the given values of p ) , contains a subspace isometric to b . Hence Theorem 1.12 follows for X = L*(p).
Remark.
For a different proof of Theorem 1.12, see [62]. Going back to Kurzweil’s
Theorem 1.1, the condition
(A) 3 real-valued polynomial q * ( z ) on X satisfying
can be shown to be also necessary for uniform analytic approximation to a continuous mapping on X , if X is uniformly convex. In fact the following theorem has been proved.
Theorem 1.13.
Suppose X is a uniformly convex Banach space. Then the
([3l])
following three conditions are equivalent:
(A) 3 real-valued polynomial q*(x) in X satisfying:
(C): if U
c
X is an open set, f(.)
Banach space, and if
E
:
U
-+
Y is a continuous mapping where Y is a
is a positive number, then 3 mapping g : U
is analytic and satisfies:
Ilu(.)
- f(.)ll
<E
vx E
u.
--t
Y such that
g(.)
94
Chapter III
For reasons of brevity, however, we shall refer the reader to Kurzweil’s original paper ([31]), and end here our account of his work.
95
Strong approximation in infinite-dimensional spaces $2. Smoothness properties of n o r m s in LP-spaces
We shall briefly recall the smoothness properties of the norms in the classical LPspaces. Questions of strong approximation by smooth functions in these spaces are related to the smoothness properties of the norms. For basic information about this topic we refer the reader to the monograph [27]Vol. I of Kothe (Chapter V $26, pp. 342ff.). Further properties are derived at length in Bonic and Frampton [5], and also [58].Some notation should first be explained. Definition.
A Banach space X is UFk-srnooth if the norm in X is k-times uniformly
X 1 1 ~ 1 1= 1). X
Frkchet differentiable over the unit sphere Sx = { z E
if the norm in X is k-times Frethet differentiable at I
+
I
#
is BFk-smooth
0 and if further the map
Tt,T,k being the k t h successive Frkchet derivative of the norm 11z11 on X - {0},
is bounded on the unit sphere S x , i.e., s u p { ~ ~ Z 'E~ SX} ~ ~ z < 00 for 1 5 i 5 k. Notation.
We shall denote by R(X,p) an annular region of the type { z E
11z)(< p } for
0 < X < p.
T h e o r e m 2.1.
(cf. [58]) A Banach space X is UFk-smooth
($
the norm
XX <
11 11
in X
is uniformly k-times continuously differentiable in arbitrary annular regions R(X,p ) . A Banach space which is UFk-smooth is also UF'-smooth for 1
5 i 5 k , as also B F k -
smooth. Let (52, C, p ) be an arbitrary measure spa.ce with p a positive measure. p is assumed to be non-trivial, and we shall consider 1 > 2 . When
<
00;
so LP(p) is of dimension
52 is the set of positive integers a.nd each point is an atom of mass 1, we
write P instead of LP. For a positive number p , denote by E ( p ) the integral part of p .
Chapter III
96
The main properties of L P ( p ) -norms are proved in the next theorem. Theorem 2.1. (1)
(See 151, 1581) The norm in L P ( p ) is
uniformly ( p - 1) times differentiable over every region R(X,p ) , if p is an odd integer;
(2)
uniformly k -times continuously differentiable for every positive integer k , over every region R(A, p ) , if p is an even integer; and
(3) uniformly E ( p )-times continuously differentiable over every region R( A, p ) if p is not an integer. We shall follow [5] for the proof.
Proof.
We shall write a(.)
=
11zIIP
integer strictly less than p , and define p ( p - 1).. . ( p - k
=
Ix(w)lPp(dw). Let n denote the greatest
4 E C"(R', R') by: 4(t) = I t l P . Then D k d ( t )=
+ l)ltlp-k(sgn t ) k ,where sgn(t) = +1 if t > 0, -1
if t
< 0 , and
= 0 if
t = 0. An application of Taylor's formula gives us:
The specific form of the function D"d on t 2 0 shows that
For any x(.) E LP, and Vk = 0,1,
. . ., n , let
& ( z ) denote the continuous k-linear
functional on Lp, defined by A k ( x ) ( h l , . . . , hk) = X
D k d ( z ( w ) ) . h l ( w ) h z ( w ) .. .
hk(w)p(dw), h l , . , . , hk E Lp. The continuity of & ( x ) follows by an application of
Strong approximation in infinitedimensional spaces
97
Holder's inequality. Then
Thus (cf. Lusternik and Sobolev [36]$42) we obtain that Dkcr(z) = A k ( z ) ,k =
0) 1) . . .
)
71.
Suppose p is an even integer. Then
and
which shows that Dncy is linear in x. Hence D"+'(Y is constant and Dn+ja = 0 for j 2 2 , i.e.,
(Y
is C".
Chapter IZI
98
and therefore
which shows that a is C". The uniformity part of the assertion of the theorem is also clear from the arguments, and the proof is complete.
99
Strong approximation in infinitedimensional spaces
$3.
C" partitions of unity in Hilbert space The first theorem of Kurzweil (cf. Theorem 1.1 in this chapter) depends on complex
analytic methods; arguments based on partitions of unity are not used there. Partitions of unity provide a very elegant tool, and we shall present in the following sections some
interesting results derived with the help of this tool. J. Eells has shown the existence of C"-partitions
of unity on a separable Hilbert
manifold (for a proof see Lang [32]). Similar C" partitions of unity were constructed by Torunczyk and K. Sundaresan (cf. [58] for an account). We shall sketch below the proof of J. Eells showing the existence of C" partitions of unity in a separable Hilbert space; for complete details the reader should refer to [32]
p. 37.
Proposition 3.1.
A separable Hilbert space admits C" partitions of unity.
Sketch of proof.
Let
3-1. Suppose W
{ B i } g l is a countable covering of U by open balls, where B;
=
U
be any nonempty open set in a separable Hilbert space
B,; ( x i ) ,i = 1,2, . . .. Then proceeding as in Chapter 11, 52, we let
{ E ; } ~be~ a
=
sequence
of positive numbers 3
< r ; , 1 > €1 > EZ >
3 ~ i
Then define: V1 = B1, Vz = Bz
m
then U = (J Vj, Vj
c
1 . .
> 0,
and
E;
+0
as i
+ 00
.
n CBr1-,, , and generally,
Bj for j = 1,2,. . ., and the covering V = {Vj}gl is locally
j=l
finite. We now need the following lemma.
Chapter III
100
Lemma 3.2.
(cf 1321, p. 36) Suppose B , B1,
. . ., B ,
are open balls in H ; let V be
the ''scall~pped~~ set:
v Then ezists C"
Proof.
Let
= BnCB,n
... ncB,.
function O ( . ) : 'FI + [0,1] 3 O ( x ) > 0 and O ( . )
4, $1,
. . . , 4,
= 0 on CV.
be C" functions form 'FI+ [0,1] 3
and for i = 1, ..., m,+,(z)= 1 if
x E Bi
and
0 5 4i(z) < 1 if
x ECB~.
Then define O(z) = +(z)(l - 4 1 ( z ) ) . . (1 - dm(z)). This function O ( . ) is the C" function which we are looking for.
Proof of Theorem 3.1. [0,1] 3 Oi(z) > 0 if z E Then
{Xi}&
For each i = 1,2,
vi and
O,(.)
. . . , let
= 0 on Cvi,
O i be the C" function from l-c to
Nowdefine
is the C" -partition of unity subordinate to V .
Xi
=
&,i = 1,2, ....
101
Strong approximation in infinitedimensional spaces $4. Theorem of Bonk and Frampton
The theorems of this section and the next one serve as good illustrations of the use of partitions of unity. These were originally formulated for Hilbert manifolds. We shall however give the proof only for an open set in a separable Hilbert space. (cf. [5]) Suppose U is a nonempty open set in a separable Hilbert
Theorem 4.1. space
R ,and
f E Co(U,F ) where F is a Banach space. Let
tive function on U . Then 3g(.) E C"(U,F) 3 11g(x) - f(x)ll
E(.)
be a continuous posi-
< ~ ( x Vx )
E U . In other
words C"(U, F ) is dense in Co(U,F ) in the Co-fine topology on Co(U,F ) .
Proof. Let V be an open subset of U . Let
E
>0
and let V = {V,},ei
covering of F by balls of radius e / 2 . Then {Uo},e, where U , = f-l(V,)
be an open is an open
covering of V . We express each U, as a union of open balls { B , , i } i E ~,, and therefrom obtain a countable covering 2, = into a set of diameter 5
E.
{D,}F=l
Now let
{$,}F=l be a
D,where w.1.g. we suppose that no $, Then $,(z) f(Z)II
E
D,
COO-partition of unity subordinate to
0. For each n let x, E V 3 $,,(x,)
> 0.
> 0 =+- Ilf(z,) - f(x)ll < E . Then Cf(xn)$,(x) E C" and llW(xn)$,(z)
arability
U3 open ball Brl(x) c U
3 y E B,.(x)
* 9 < ~ ( y ) .Sep-
* 3 countable covering {Brlm(x,)}~=l of U where for each n = 1,2, . . . ,
E Brmn (2,)
<
j
~ ( y ) .For convenience we shall write
T,
for ~(z,).
By
the arguments of the previous paragraph 3gz, E C"(Br,(zc,), F ) 3 Vy E Br,(z,), llgz,(y)
-
< €VX E v.
Next, for each x E
y
of V by open balls D, 3 f maps each
- f(y)II
<
<
~ ( y ) .Now let
ordinate to the covering {Br,(x,)}.
Define g(.) =
be a C" partition of unity subX,(.)g.,(*).
Then g E C" and
102
satisfies: Ilg(y) - f(y)II
Chapter III
< ~ ( y )Vu E U . This completes
the proof.
Strong approximation in infinite-dimensional spaces
103
$ 5 . Smale’s T h e o r e m
To obtain an infinite dimensiorial version of Sard’s theorem, the concept of “set of zero measure” has to be first replaced by a more satisfactory one which makes sense even in infinite dimensional spaces. One such concept is “set of first category”, and Smale was able to show that Sard’s theorem holds if we substitute the concept of first category for zero measure. In this section we shall present this theorem and a consequence of it. This infinite dimensional Sard’s theorem needs a nonlinear concept of Fredholm operator. First we need some definitions and facts concerning linear operators. A Fredholrn
operator is a continuous linear operator L : X +
Y from a Banach space to another,
with the properties: (a) dim ker L
< 00;
(b) range (L) is closed; (c) coker (L) = Y/range (L) has finite dimension. The index of a Fredholm operator L : X
+
Y is dim ker (L)-dim coker (L); the index
is an integer. The following main result should be noted, the proof of which with more details can be found in [15].
Lemma 5.1.
The set F ( X , Y )of Fredholm operators is open in the space L(X, Y )of
all bounded linear operators, in the norm topology. The index is a continuous function
on F ( X , Y ) . The nonlinear generalisations can be conveniently explained using concepts of differentiable Banach manifolds. We shall always assume that these manifolds are con-
Chapter Iff
104
nected and have a countable base. Let M , V be two such manifolds. A Fredholm map is a C1-mapping f : M
-t
N such that for each x E M , the derivative of(.):
T z ( M ) + T f c z , ( N )is a (linear) Fredholm operator. The index off is defined to be the index of of(.)
for some x. Our assumption is that f is C’ and M is connected; hence
the index does not depend on x. Let f : M
D f (z)
:
T,(M)
-t
-+
N be a C1-map. A point x E M is called a regular point off if Tfc,,(N) is surjective, and is singular if not regular. The images
of singular points under f are called critical values, and their complement the regular values. Hence if y E N is not in f ( M ) then y is automatically a regular value. The finite dimensional Sard’s theorem explained earlier in Chapter I1 will be needed in the sequel. Henceforth in this section “almost all” shall mean “except for a set of the first category”. Smale’s main theorem is the following (cf. [58]).
Theorem 5.2.
Let f : M
-t
N be a Cq-Fredholm map with q > max (index f,O).
Then the regular values o f f are almost all of N .
Proof. We recall that M has a countable base. Also a countable union of sets of the first category is again a set of the first category. Hence it is enough to prove the theorem locally. Hence it is enough to assume that the given f is a map: U
-t
Y where U is an
open set in a Banach space X , and Y is another Banach space. Let xo E U , and A = D f ( q ) : X
+
Y . Now dim Ker A < 00, X can be expressed
as X = XI x Ker A where X , is a Banach space, and
50
= (PO,q o ) ,
PO E
XI, qo E
kerA. Then, V(p,q) sufficiently close to (PO,qo), the first partial derivative D1f(p, q ) :
XI + X maps XI injectively onto a closed subspace of X . Now use the implicit function
105
Strong approximation in infinite-dimensionalspaces theorem: we can choose a product neighbourhood B1 x B2 of
Bz is compact, and if q E Bz then f restricted to B1 x
such that
Xi x Ker A
40) in
(PO,
q is a differentiable
homeomorphism onto its image. We now need the following lemma. A map q5 is called proper if the inverse image of a compact set, under q5, is compact.
Lemma 5.3.
A Fredholm map is locally proper, i.e., i f f : M
x E M , then 3 neighbourhood W of x 3 f
Proof of Lemma. y, where
2;
I
W
is proper.
Choose N ( z ) = Bl x B2 as above and let
= (pi,qi) E
B1
N is Fredholm, and
-+
f(z0)
= y; tend to
x Bz. It is enough to show that the xi have a convergent
subsequence. Now Bz is compact, hence we can assume that qi
-+
q, and since f(pi, q )
-+
y, we can even assume qi = q. Now f restricted to B1 x q is a homeomorphism onto its
image. Hence pi
-+
p . This proves the lemma.
Proof of Theorem 5.2 (cont'd).
Let
20
E
M , and again let
B1
x Bz
c XI
x Ker (A)
as above. The critical points o f f form a closed set. Hence by the preceding lemma it is
enough, given a neighbourhood Let
R
be the projection: Y
U1
-+
of f ( 2 0 ) in Y to find a regular value of f in
Y/Range ( A ) . From the hypotheses of the theorem,
Sard's theorem can be applied to the map q5 : by: 4(q) =
U1.
4 o f ( p 0 , q ) t o give a regular
{PO}
value z of
x Ker (A) I$
-+
Y/Range (A) defined
in xU1. Let y E x - ' ( z )
n U1. Then
such a y is a desired regular value. This completes the proof of the theorem. The next theorem (cf. [58]Theorem 3.1) follows as an interesting consequence of the preceding theorem. First some definitions are in order. Let f : M
-+
N be a C' map, and g: W
-+ N
be a C1 imbedding. We say that
Chapter III
106
f is transversal to g if for each
(2,y)
E
M x W such that f ( z ) = g(y) the two spaces
range (Dg(y)), span the tangent space T ( f ( z ) ( N ) . range (Of(.)),
Theorem 6.4.
Let f : M + N be a CQ-fiedholmmapping, and g : W
+
N a C'-
imbedding of a finite dimensional manifold W , with
q
> max(index f
+ dim W,0) .
Then 3C1-approximation g' of g such that f is transversal to 9'. Furthermore i f f is transversal to gl.4 where A is a closed subset of W , then g' may be chosen so that g' = g on A.
Proof. M as well as W, has a countable base. Hence a standard argument reduces the proof of this theorem to the following lemma.
Lemma 5.5. and for any E
Let M , W as in the preceding theorem. Then 3 neighbourhood
of y
> 0 an sC1 approximation g' of g such that f is transversal to g ' ) v l .
Proof of Lemma. lJ2
U1
We can assume that 3 neighbourhood of g(y) in N , as follows:
c RP, N=Banach space F = RP x
F1,
and g : lJ2
the projection. Let U1 be a neighbourhood of y 3 which equals 1 on
U1,O
+ RP
01 c
x 0 is identity xO, n : F + Fl
int ( U z ) , and
4 a Coofunction
on ext (U2). Then by the main theorem, let z E
F 1
be close to
0 3 n o f has z as a regular value on f -1 ( g ( U 2 ) ) . Now define g' as the translate of g by z on U1, smoothed by
4.
This proves the lemma and hence Theorem 5.4 of Smale.
107
Strong approximation in infinitedimensional spaces
$6. Theorem of Eels and McAlpin In this section we shall give an account of the following theorem of J. Eells and J. McAlpin (cf. [lo]). However we shall formulate the result for Banach spaces rather than for manifolds. Suppose X , Y are smooth connected manifolds modelled on Banach spaces, and
X
--t
T,X
Y is a C'-map. A point z E X is a critical point of 4
mapping
d:
q5 if the differential +.(z)
:
T+(,)Y is not surjective. The set of critical points of 4 is denoted by C,. The
4 is called residual (or of Surd t y p e ) if qh(C6) has
Remark.
no interior point in
Y.
The theorem of Smale (Theorem 5.2 in this Chapter) shows that to estab-
lish an infinite dimensional analogue of the Morse-Sard theorem, strong restrictions are necessary.
Theorem 6.1.
Suppose that 3-1 is a separable Hilbert space, and F is a Banach space.
Then the Cm-smooth residual maps are dense in the Co-fhe topology on CO(3-1,F ) . Some lemmas are needed for the proof of this theorem.
Lemma 6.2.
Let U be an open subset in 3-1. Then for any closed set C
neighbourhood V of C in U , 3 countable collection
{U;}El
c
U and
of open balls in 3-1 such that
(1) those with even (respectively odd) subscripts are in and cover V ( resp., U - C), and
(2) the centres a; of the U;are linearly independent points in 3-1. Proof of Lemma 6.2.
For each z E V let B,,(z) be an open ball contained in V .
{
Then 3 countable collection B';o (2,)
}
00
n=l
00
covering V . similarly let
be a countable covering of U - C by open balls B+(yn)
where each yn E U - C .
n=l
108
Chapter III
Now proceed by induction: for each B+(in) (resp. B+(ym)) U2,
select open balls
labelled with even subscripts (resp. UZm+' labelled with odd subscripts) such that
C
B+(xn)
uzn C Br,,(xn)
a; are linearly independent in
(resp. B+(ym)
c Uzm+i c Br,,(Ym))
and whose centres
31. This completes the proof of Lemma 6.2.
Before turning,to the next lemma, some notation should be explained. For any subset A C U , and any
Lemma 6.3. Then
> 0, let ( A , . )
Let V' = ~
{Vz;+l}Eo is
Also v
T
1 and ,
= {z € U
for k: 1 1, let
d ( x , A ) < .}. Set A' = U - A . v2k+1
= U2k+l
n(u;, $) n . . . n(Uik-,, k).
a locally finite open refinement of the covering {U2i+l}g0.
n (u;,i) n .. . n ( U i k - 2 ,
~ %andJ for ~ k, 1 2, v 2 k e f u 2 k
i)determines a
locally finite refinement of { U 2 ; } Z 0 .
For the proof see Lang [32] p. 32 Let U be an open set in a separable Hilbert space 31. Then for each pair
Lemma 6.4.
of disjoint closed subsets CO,C1
c U 3 PI-residual function q5 : U
+
[O, 1) with +-'((I) =
Co,q5-1(1) = c1. Proof of Lemma 6.4.
Take C = C1 and V = U - Co in Lemma 6.2. Then the
composition of a C"-function f;j : U + R'
(1)
fi;
rij :
R' + R1 and
113
- ajll gives C"-smooth
functions
for j 5 i and (-l)j = (-l)i such that
is strictly positive on Ui and = 0 elsewhere;
(2) for j f;j(z)
we have
fij
= 1 outside
Uj
= Brj(aj), fi,
< 1 in between.
Now use the Hilbert space structure on
a;j(lls -
= 0 on Brj-+(aj);and 0
ajll)(s
-
a j ) for
3-1; we find that
the gradient Vf;j(z) =
C"-real functions aij; we can assume ai;(t) = O only
<
Strong approximation in infinitedimensional spaces
when t = 0 or t 2 r ; . Let f;(z) = and V f ; ( z ) = a;;(llz
c
p;j(z)(z
p ,fij(z)
- u;ll)
nfij(z);
then f;
109
> 0 on Vi,= 0 elsewhere,
- a j ) for suitable real functions p i j . Furthermore P;j(z)
= 0 only when z = a; or
I
$ K. Now set f'(z) =
I<*
a locally finite sum of smooth functions. Then f' larly set f " ( z ) = fine q5 : U
+
> 0 on V,
V4 = (f"Vf' - f'Vf")/(f'
=
= f,(z)+f,,(z). "(') Then q5-'(0)
+ f")2
=
c
kjVfj
c fz;(s),
and = 0 on CO.Simi-
cfz;+l(~). Then f" > 0 on U - C1,and = 0 on
I by 4(1)
=
CO,q5-'(1)
C1.
Finally de-
= C1; further
for suitable Coo-smooth functions
kj.
Next let {Wp}gl be a countable open covering of U - (Co U C,) = 4-'(0, 1) by sets which meet only finitely many Vj. Suppwe x E W, is a critical point of
0= z
e k j ( z ) p j k ( z - ak).
Now
kj(z)
#
0 in W,. Since p j j ( z ) = 0 only when z = a j or
$ vj, we conclude that either z = a j or a non-trivial h e a r combination
(c
-fk)z-c-fkC&
that
c
-fk
4, so that
c
-fk(I-ak)
=
must be = 0. The centres { a & }are linearly independent; hence we find
# 0 , i.e., x
belongs to the linear span of
Let M p be the intersection of W, with this linear span; then M, is a finite-dimensional manifold containing all the critical points of theorem to
4
$1
4
R'. Now apply the Morse-Sard
WP
-+ R1; the set of critical values of this function has p1-measure 0. The IMP
set of critical values of q5 : U
-+
R' is a subset of a countable union of such null sets,
hence itself is a null set of R'. This completes the proof.
Proof of Theorem 6.1. and
E(.)
1c, : U
--t
Let U c 7-i be an open set,
4 :U
--t
F be a continuous map,
a continuous positive function on U . The object is to show that 3 residual C"
F3
I I ~ ( I ) - $(.)I1
< E ( X ) VX E U .
110
Chapter III
First let { U i } z l be a countable cover of U by open balls in U with centres at points { a i } E 1 such that II$(x) - $(ai)II
< c ( x ) Vx
E Uj. Let
{K}El
be a locally finite
scalloped refinement as in Lemma 6.3, and a subordinate partition of unity ing the residual functions in Lemma 6.4. Now set
Vx
bi = $ ( a i ) , i
{
us-
2 I, and define the C"-
E F.
Now use the proof of Lemma 6.3; this proof yields a countable open cover { W,},:, of U , each element of which meets only finitely many
< dim F , then $(W,)
K. If dim span
{bj
I Vj n W, # 8)
lies in a proper linear subspace of F , hence is meager. Otherwise
the set of critical points (resp. critical values) of
$1
-+
F lies in a finite dimensional
WP
manifold (resp. linear subspace). The Morse-Sard theorem ensures that this map has a meager set of critical values in F (more precisely, has Lebesgue mp-measure zero where mp is the dimension of the linear span of the relevant bi associated with W,). Hence the
set of critical values of $ : U
--.)
F is contained in a countable union of meager sets, and
is therefore meager, by Baire's theorem. This completes the proof of Theorem 6.1.
111
Strong approximation in infinitedimensionalspaces
57. Contributions of J. Wells and K . Sundaresan John Wells in 1968 (cf. [63]) showed that the techniques used in preceding results could not be used in certain spaces. Specifically he showed that in the space Q one cannot have a real valued function with bounded support and with a uniformly continuous derivative.
K. Sundaresan (cf. [61]) carried these ideas further, and showed that if a Banach space admits a non-trivial function with bounded support and having a uniformly continuous derivative then the space must be super-reflexive. The result of Wells ([63]) turns out to be a special case of Sundaresan's theorem. However we shall first present the theorem of Wells, for it has a simple proof which does not require the machinery used by Sundaresan, and also because it showed, for the first time, that in the space co, Ca-fine approximation of a C2-function by a
C" function is
not possible.
Theorem 7.1. (cf [63]). Let f E C'(c0, R) with a uniformly continuous derivative
D f (.). Then the support o f f is unbounded. Proof. Suppose the statement is not true. Then 3 f E C'(c0, R) with the properties:
f(0) = l,f(x) = 0 for
llzll 2
IlDf(z + h ) - D f ( z ) l l 5
1, and D f is uniformly continuous. Let N 3 llhll 5
k =+
i. Because of the mean value theorem, we then assert ,
Let E be the subset of co consisting of x such that the 2N - 1 of the first 2N components of x have absolute value
k,the remaining component has absolute value less than or
Chapter III
112
equal to
k,and all components after the first 2 N components are zero. The set E is
connected and even, hence we can choose, inductively, hl, h2, . . . , h N 6 E 3 D f ( h l
. . + hk-1)
hb = 0
and hl
It follows that llhl
+
+ + . + h k has at least 2 N - k components equal to k.
+ + h ~ l =l 1, and
c N
I
k= 1
1
1
-((hkll = 2 2
c N
k=l
1
1
N
2'
-= -
which is a contradiction. This proves the theorem of Wells. We shall next turn to the work of Sundaresan ([Sl]) mentioned above. The main result of his paper requires a good deal of machinery which we shall proceed t o explain. First some definitions and terminology are in order. A Banach space E is said t o be
smooth if for all x # 0, I E E ,
exists Vy E E. If this limit exists at a point x
# 0, Vy E E
then it is known that G. E
E' the dual of E , and also llGzII = 1 (see Kothe [29]). A smooth Banach space E is said to be uniformly smooth if the preceding limit is uniform Vx, y with ((z((= 1 = /(y((. The homogeneity of the norm implies the following lemmas. If E , F are Banach spaces, a function
f
:
E -+ F , having a uniformly continuous derivative on a set A
be said to be U.C.D. on A . A Banach space E is said to be U'-smooth if 3 function on E with bounded support.
c
E will
U.C.D. real
113
Strong approximation in infinitedimensional spaces
Lemma 7.2.
A Banach space E is uniformly smooth iff the norm is U.C.D. on regions
R(X,p) = { x E E
Lemma 7.3.
I X < 11x11 < p where 0 < X < p } .
The norm in a Banach space E is U.C.D. on regions R(X,p ) i$ the norm
is uniformly differentiable on bounded sets away from the origin. We shall also need the following lemmas, which are either results on differential calculus in Banach spaces (see Dieudonnk [9], Lang [32]), or consequences of the preceding definitions.
Lemma 7.4.
Let E be a Banach space, f : E
+
R a U.C.D. function, and D f the
derivative o f f . Then (a) if U is a bounded subset in E l then f
IU
is Lipschitzian: E M
>0
3 Vx,y E U ,
Ilf(x)- f(Y)II 5 MIIX - YII; (b) if the support o f f is bounded then f is Lipschitz everywhere in E , in particular f is uniformly continuous.
Lemma 7.5.
Suppose E is a U'-smooth Banach space, and X
function f on E 3 f(0) = 1, and f ( x ) = 0 for
Lemma 7.6.
11211
> 0,
then 3U.C.D. real
2 A.
I f f and g are two U.C.D. real functions on a Banach space E and the
support o f f (or the support of g) is bounded, then f g is U.C.D. with bounded support.
Lemma 7.7.
If E , F, G are three Banach spaces, f : E
+
F, g : F
+
G are U.C.D.
functions such that the derivatives D f , Dg are bounded on E + L ( E ,F ) , and on F -+
L(F, G ) , respectively, then the composite g o f i U.C.D. For the next lemma we refer ther reader to Nemirovski and Semenov ([45]).
Chapter III
114
Lemma 7.8. Suppose E is a uniformly convex and uniformly smooth Banach space. Then the restrictions of the uniformly continuously differentiable functions on E to any
-
-
closed ball U,(O) is dense in the space of uniformly continuous functions on Up(0) with the uniform topology. We now have to turn t o the concepts of super-reflexive Banach spaces and ultrapowers of normed linear spaces. If E, F are Banach space, then E is said to be finitely
represented in F , in symbols E << F , if for each finite dimensional subspace of E and positive number
T : X"?
E,
3 subspace Y of F , depending on X and
Y with ))T1111T-'l) 5 1
+
E.
E,
3 there is an isomorphism
A Banach space F is said t o be 8uperrefEezive if
E << F implies E is reflexive. For basic results on this topic see James [24], and shall be content with stating one known fact which we shall use.
Fact 7.9. (cf. Enflo [ll]).A Banach space E is super reflexive
E is isomorphic
with a uniformly smooth Banach space. Let S be an infinite set and
r a non-trivial (free) ultrafilter on S. Iff
is a bounded
real-valued function on S let limf(s) = sup [ A l { t E S : f ( t )> A } E r]. If (E, 1)
r
is a normed linear space, and f is a bounded E-valued function on S, then let limllf(s)II. Then r
If1
11) =
I I is a semi-norm on the vector space V of bounded E-valued func-
tions on S, and the quotient space of V modulo the kernel of
I I equipped with
the
quotient norm is known as the ultrapower of E associated with the pair S, I?, and is denoted by E(S, I?).
The following facts are known (cf. [Sl]).
Fact 7.10. If E is a Banach space then E(S, I?) is a Banach space. Fact 7.11. If E, F are Banach spaces then E << F
* E is isometric with a subspace
115
Strong approximation in infinitedimensional spaces
of an ultrapower F(S,I?) of F. Before turning to the next lemma, the following remarks are in order.
Remark 1.
If E is a uniformly smooth Banach space, so that the norm of E is U.C.D.
on regions R(X,p ) then by composing the norm of E with a suitable C'-function R1
-+
R' and using Lemma 7.7, we find that E is U'-smooth. Then apply Lemmas 7.5, 7.7,
and use the fact that the norm of E is U.C.D. on R(X,p), and we verify that if E is uniformly smooth, and P , E are two positive numbers, then 3 U.C.D. function f : E 3 0
5 f 5 1, f
-+
R'
= 1 on B,.(O), and f 3 0 outside Bp+e(0).The next lemma uses these
remarks and the lemma of Nemirovski and Semenov (cf. [45]). The support of a real function is denoted by supp(f).
Lemma 7.12.
(Sundaresan 1611) Suppose G is a nonempty open subset of a super-
reflexive space E . Then 3U.C.D. function f : E + R1, 3 0
Proof of Lemma.
< f < 1 and supp(f) = G.
We shall assume w.1.g that E is uniformly convex and uniformly
smooth. First suppose G is a bounded open set, and let C = E - G. Let g : E -+ be defined by: g(z) = d ( z , C ) = distance of
2
from C. Let
P
>0
3
R'
G c G c Brl2(0).
Consider the restriction of the uniformly continuous function g to B,(O). By the Lemma of Nemirovskii and Semenov, 3U.C.D. functions f, on E sup
If,(z) - f(z)I < ,:
n = 1,2,
-+
R' 3
. . .. Now use the preceding remarks; we find that
*EB,(O)
3U.C.D. function 4 : E
-+
R', 3
= 1 on B,p(O), 4 I 0 outside B,(O); let h,
4
Let a,(.) be the C'-functions on R'
Then g is a U.C.D. function E
-+
-+
=
R' 3 supp(a,) = ( i , m ) ,a,(t) = 0 if t 5
df,. ,:
R', supp(g) = G, and 0 5 g 5 1. This proves the
Chapter IIl
116
lemma under this additional assumption. 00
Now suppose G is an arbitrary open set, then we write G =n=l U G,, where G, =
B,(O) n G,n = 1,2,
. . .. Each G,
is a bounded set, hence by the preceding paragraph, OD
3 U.C.D. function
fn :
E
-t
R' 3 supp(f,) = G,, and 0 5 f, 5 1. Let f =
n= 1
&fn.
Then this f has the required properties. The next two theorems show that U'-smoothness is finitely inherited.
Theorem 7.13.
If E is U'-smooth then every ultrapower E ( S , I ' ) is U'-
(K.S. [61]).
smooth.
Proof of Theorem.
We shall denote the norms in E , E ( S ,r) by
11 11,
spectively. E is assumed U'-smooth, hence 3U.C.D. function f : E -+
and
1 1 (11 re-
R' 3 f $ 0,
with supp(f) c BI(0). Let f E E ( S , r ) and let { ~ ( s ) } , ~ , be a representive of 2 . f is a bounded function, hence lim f(z(s)) exists. Let { ~ ( s ) } , , ~ be another representative of
r
If(.)
2 . f is uniformly continuous by Lemma - f(y)l
<
E.
Since {~(s)},,,,
i E E ( S , F ) ,3 JE I? 3 6
s
6
.
{y(s)},€,,
llz(s) - y(s)ll
Hence if
E
> 0 36 > 0
3
112
- yII
< 6+
represent the same equivalence class
< 6. Hence Vs EJ, If(z(s)) 6
- f(y(s))l
< E.
Hence if f*(f) = limf(z(s)), then f* is a real-valued function on E ( S , r ) . The support
r
o f f is in Bl(0) c E , hence
hence lll2lll 5 1. Hence supp(f*) is contained in the unit ball of E ( S , r ) . Now D f : E + E* is a uniformly continuous mapping with bounded range; hence proceeding as in the preceding paragraph we verify that if
f , 5 E E ( S ,r) and { ~ ( s ) } , ~ ~ ,
I17
Strong approximation in infinitedimensional spaces { ~ ( s ) } , , ~ are representatives of
dent of the representatives
2 , c respectively, then limDf ( x ( s ) )(y(s)) is indepenr
{ ~ ( s ) } {y(s)} ,
of 2 , i respectively. Now defined
&(a)
=
l$n D f ( ~ ( s ) (y(s)), ) 5,c E E ( S ,I?). We then verify that -& is a continuous linear functional on E ( S ,I'), since Df is bounded. Now if
{w},EsE k then f * ( ~+ k) = limf(Z(s) r
+ h(s))
=
f(x(s)> + ~ f ( z ( s )()h ( s ) ) + o z ( a ) ( h ( s ) ) ) 7 where we note that since f is a U.C.D. function, given E
> 0, 36 > 0
3 le,(y)I
5 ~llyll
if llyll 5 6 Vx E E. Then 3 set J E I? 3 Vs E J, ~Oz(a~(h(s))~ I ellh(s)ll. Hence limr JO,(,)(h(s))l
5 E 111k111 if (11k111 5 6 and f' is differentiable at i with
of*(?) = &.
Df is uniformly continuous on E -+ E', hence we verify that the map D f' :
E ( S ,I?) + ( E ( S ,I?))' is uniformly continuous, once again working with suitable members of
as previously. Thus E ( S ,I?) is U1-smooth. This completes the proof.
As corollaries we obtain the next results. Corollary 7.14.
If a Banach space E is U'-smooth and F << E , then F is U'-smooth.
Proof. F << E if and only if F is isometric with a subspace of some ultrapower E ( S ,I') of E . The corollary then follows.
Corollary 7.15.
If a Banach space E is superreflexive, then it is U'-smooth.
Proof. A superreflexive Banach space is isomorphic with a uniformly smooth Banach space. Also U' -smoothness is invariant, under isomorphisms. The corollary follows. The second theorem of Sundaresan in this context is as follows.
Chapter III
118
Theorem 7.16. (cf. [ S l ] ) . Proof. Let 0
<8<
If E is a U1-smooth Banach space, then it is reflexive. 3U.C.D. real-valued function f on E 3 f(0) =
1. By Lemma
I , f ( z ) = 0 if llxil 2 8/4. Since f is U.C.D.) therefore if 0 <
M 3 if h E E , llhll 5
E
<
1, 3 positive integer
then
Suppose E is non-reflexive. Then by a theorem of James (cf. [24]) it follows that 3 set X containing the set of positive integers and a subspace L of the Banach space B ( X ) of bounded real-valued functions on X with the supremum norm, isometric with
3 for n
mitting a sequence
E , ad-
>1
zn(i) = 8
1 5 i 5 n, i E W ,
zn(i) = 0
i > n,
i
E
W
,
and
Let
2,,0
Then
=
112n,kII
1
~2,,20,,,=
-+zn for n 2 1, and
xn,k =
3zn - f 2 n - k if n
2 1,k 2 1.
5 1 for all pairs of integers (n,k) for which x,,& is defined. Now define the
polynomial path P in L by
where M is the positive integer chosen to satisfy (A) in the preceding paragraph. Consider the derivative D f ( 0 ) . By our choice of
Zn , k ,
D ~ ( O ) ( Z ~= M 0, ~if )and only if
119
Strong approximation in infinitedimensional spaces
D f ( O ) ( z o , p ) = 0, and D f ( O ) ( z p , 0
5 0 ++ Df(0)
(20,'~)
< 0. The path P is con>
netted, hence 3 t E P 3 D f ( O ) ( [ )= 0. If
then t(j) =
1 Ze
((j) =
--e
[(j) E
[-;e,28]1
1I j ~ 2 ~ - -1, i o j EW,
if
1 4 1
if
2 M - i 0 + 1 5 j ~ 2 M ,~ E W , if
j =2M -io,
j E W ,
and
Thus if [ ( j o )
< f 0 for some j o
t ( j 0 ) = -:),
then [ ( j ) = - i O V j E W , j o
choose
= E/M, otherwise
t1
E W ,15 j
=
5 2M (which is the case if ( ( j o )
+ 1 Ij 5 2.'
E
(-$,
Now if 2M - io - 1 2 2M-11
-(/Ad. The €1 thus chosen has the properties:
A,Df(O)(G) = 0, and &(j)2 & for at least 2'-1
g ) or
lltlII 5
values of j E W , 1 5 j 5 2.'
Now consider Df(t1). As before 3[' E P 3 Df(&)([') = 0. From the properties
*&, the restriction of to the set Q = { jI 1 I j 5 2M} c W , has range either in the set { &,--&} or {-A,&}
of
noted in the preceding paragraph, since
=
except possibly for one value of j E Q. These observations imply either (i> (ii)
(G + & ) ( j ) 2 (1
s,or
- &)(j) 2
g,for at least zM-'
integers j E Q. Let
62
=
& or = -&
according as (i) or (ii) is the case. Repeating this procedure inductively it follows
Chapter III
I20 for 1 _< k
5 M . From our choice of f,M ,
M
el together with the inequality
k
11 C (ill 2 -$$ it follows that i=l
i=l
a contradiction; thus the proof is completed.
The next theorem provides a characterisation of U1-smooth Banach spaces. Theorem 7.17. (cf. K.S. [Sl]).
A Banach space E is superreflexive if and only if E
is U1-smooth.
Proof. From Corollary 7.15 and the Theorem 7.16 it follows that if E is U1-smoothl and F
<< E , then F is reflexive. Thus E is superreflexive. The converse follows from
Remark The Banach spaces
CO, C ( K ) where
(I is an infinite compact Hausdofl space are not
superreflexive (in fact not even reflexive), the earlier theorem of J. Wells now follows as a consequence of the preceding characterisation.
Corollary 7.18.
If E = CO, or C ( K ) with K as above, then E is not U'-smooth
Strong approximation in infinitedimensional spaces
121
$8. Theorems of Desolneux-Moulis The work of Nicole Desolneux-Modis was a difficult and major step in obtaining information concerning strong approximation in infinite-dimensional Banach spaces, at a time when such information was extremely scanty. It was realised that the tools that
that were known to be effective in finite-dimensional spaces, such as smoothing by convolution, or use of smooth partitions of unity, could not be directly utilised in infinite dimensional spaces., the reasons being rather obvious, viz. partly the lack of a convenient theory of integration comparable to the Lebesgue theory of integration in finite dimensional spaces. The tools devised in [8] provided some of the ideas for the further work which this author was able to carry on ($9). In this section we shall briefly describe some of the theorems proved in [8], with a sketch of some of the arguments. First we shall describe the theorem regarding C'-fine approximation established in [8] for the spaces l p ( p being an integer 2 2) and for
CO.
Earlier in 52 of this chapter we
found that the space l z p ( p 2 1 an integer) has a Coo-norm away from 0, co allowus an equivalent Coo-norm, and the space lzpel ( p 2 1 an integer) has a C2P-norm away from
0. In the case of
co it is understood
in the paper (81 that the original norm is replaced
by the equivalent Coo-norm. The methods used in this paper were the same for the following spaces: l p ( p 2 2 an integer), or the space co (with the equivalent Cm-norm); it is therefore convenient to use the symbol E" to denote any one of these spaces, with the Ca-norm (a= 00 for l z p ,or co, a = 2 p for 12fil).
Theorem 8.1. ((81 Theorem 1). Let Cl he a.n open set in E", F a Banach space. The space C-(n, F ) is dense in C1(R,F ) in the C1-fine topology.
Chapter III
122
For the proof of this theorem, suppose and f : R
-t
Ilg(z) - f(x)ll
&(a)
is a positive continuous function on 52
F a C1-mapping. The object is then to construct a Coo-mapping g 3
< E(Z)
and IlDg(x) - Df(x)ll
< E(Z)
Vx E R. The following lemmas are
used in the proof. (Lemma 1, page 295 in [8]) Suppose B1 is an open ball with centre 0 in
Lemma 8.2.
E", and f : B1 71
>03
-t
F a C'-mapping. Suppose Bo
sup llD'f(x)ll
< 7. Let c > 0.
c B1 is an concentric open
Then 3 constants
X0,Xl
and 3g : B1
ball and
+F 3
g is
zEBo
C" and satisfies: SUP Ilg(x) - f(.)ll
ZEBO
< Xos;
and
SUP zEBo
I l ~ g ( ~ >
The proof of this lemma is carried out in two steps. Let
z : } basis in E", i.e., en = { by ep, 1 5 p
3 z; = 0 if p
# n, x:
= 1. Let
be the canonical
Enbe the subspace generated
m
5 n, and E" =n=l U En.
Step 1.
A function f : Em
1,2, . . . is
C"
Step 2.
A mapping $(x) : E"
and
-t
F is constructed 3 f restricted to Enfor each n =
f is "close" to f . -t
Em is constructed 3
112
- $(x)ll < r for a suitable
number r . Then g(.) is defined by: g(x) = f($(x)).
A second lemma that is needed is the following. Lemma 8.3.
(Lemma 2, p. 301 in [8]).
For each x E R 3 open ball Bp(z)(x) c R
123
Strong approximation in infinitedimensional spaces
(d) sup r ( z ) < 2. Zen The separability of the space implies that there is a countable collection of points
. . .} in R E R = u B w ( a n ) . 00
{a1,a2,
n=l
The second theorem on strong approximation in [8] is the following.
([8] Theorem 2, p. 306). Suppose E is a separable infinite dimen-
Theorem 8.4.
sional Hilbert space, R
c E
an open set, and F a Banach space. Then the class
C"(R, F ) is dense in the class CZk-'(R,F ) endowed with the C kfine topology (denoted by C,",-l(fl, F ) . The proof of Theorem 8.4 proceeds along the lines of proof of Theorem 8.1. One lemma needed in the proof is the following.
Lemma 8.5.
Suppose F : E + F is a CZk-'-mapping. Suppose Bo is an open ball
with centre 0 contained in E and
r]
> 0 a number
3 sup IIDZk-'f(z)11 <
r].
Let
E
>
ZEBO
0 be arbitrary, and set r =
E/V.
function g satisfying: for 0 5 i
Then 3 ( k
+ 1) constants X;(O
5 i 5 k) and 3 C"
5 k - 1, sup 11D'g(z) - D'f(z)ll < X;ET'-'-',
and
zEBo
As in the proof of Lemma 8.2 for the proof of Lemma 8.5 one constructs a function
f: E m
+F 3
f restricted
to E,, for n = 1,2,
. . . , is C"
and further
f is "close"
to f.
The second step in the proof consists in exhibiting a function $ : E + E" 3 (a) $ i s C",
(b) for each L E E
3 positive integer n(z) and 3 neighbourhood Uz of z E E 3
Chapter III
124
(c) 3 constants
C1 3 IlD$(z)ll < C1;
Then g is defined by
for suitable functions L'. The proof of Theorem 8.5 is completed as follows. Let
E(.)
be a positive contin-
uous function on R, f E C2"'(R, F ) . Then the objective is to exhibit a function g E Cm(R,F) 3 for 0 5 a 5
Ic IlD"g(x) - D"f(z)ll <
c(z)Vz E R. For this pur-
pose another crucial lemma which is needed is the following.
Lemma 8.6.
For any point x E R, 3 open ball B P ( = ) ( xC)
a satisfying:
established in [8]. To describe the third interesting theorem (which is a consequence of Theorem 8.1) in [8] we should first state a definition, Let R be an open set in a separable infinite dimensional Hilbert space E , F a separable Hilbert space of finite or infinite-dimension. A mapping f : R
-+
F is said to be of
S a d type if the set of its critical values has no interior points.
Theorem 8.7.
(cf. [8], p. 331)
The cla.ss of C" mappings R
type is dense in C'(R, F) endowed with the C1 fine topology.
-+
F which are of Sard
125
Strong approximation in infinitedimensional spaces
We shall give a brief sketch of the proof. To be specific, the function g(x) exhibited in Theorem 8.1 of this section and satisfying
A, X1 being fixed constants, is itself a funct,ion of Sard type. In the proof of Theorem m
8.1, R was shown to be a countable union: R =
u
B,,/2(un) (where we have written
n=l T,
for the earlier ~ ( a , ) ,n = 1,2, . . .). Let x E R, and n the least integer m 3 x E /2
(urn). On B,n/2(an)the function g(.) coincides with the function gn where
C
gn(x) =
CL~(T [ f)( a p )
+ ~ f ( a p .)(.
- up)
+~p(x)I
7
1I P < n Pp(Z)
= dp(4(1-
dp-l(l))
... (1 - d d 4 )
7
and SP(x) is constructed through the following steps: Let $,,(.)
: E + E"
be the differentiable function 3 IIy - $,,(y)II
< $.
There is an open ball n'(x) with centre x, contained in B , , / ~ ( u , )and an integer m 3 &(R'(x))
mapping
C
Em (Emof finite dimension nz); for any integer p(l 5 p 5 n ) , 3 C"
Sp : Em-t F
3 Sp(.) = 8p($n(.)).
Let R(x) be an open ball centre z 3 R(x)
-
c R(z) c R'(x)
C
Bp,/z(an).
The open balls R(x),x E 0, form an open covering of R , hence R =
u Q ( x i ) , for a 00
i= 1
countable collection 1,2,
{x1,22,
. . .. Then exists g(Z) =
. . .} of points
in R. We shall write Ri as short for R(zj), i =
integer ni 3 in R::
C
Pp(z>[f(ap) + ~ f ( a p.)( 2 - UP>
+ gp(Gni(x))
plni
The remaining part of the proof of Theorem 8.7 consists in showing that the complement of the set of critical values of g on 0: is a countable intersection of open dense sets, F is a Banach, hence a Baire space, and it follows that g is of Sard type.
Chapter III
126
We shall now simply state two more theorems established in [8]. Suppose E" is as in Theorem 8.1 (i.e., E" is one of the spaces l p , p 2 1 an integer,
CO,
with a C"-norm), M a paracompact manifold modelled on E", and N a para-
compact manifold modelled on the Banach space F . For any integer j 2 0 denote by
C j ( M ,N ) the class of Cj-mappings M Theorem 8.8.
---t
N.
(cf 181, p. 325) If M and N are C"-manifolds, then C w ( M , N ) is
dense in C'(M, N ) endowed with the Cl-fhe topology. Now suppose E is a separable infinite dimensional Hilbert space, M a COD-manifold modelled on E , and N a paracompact manifold modelled on F .
Theorem 8.9. dowed with the
(cf 181, p. 328) The class C-(M, N ) is dense in C2'-'(M, N ) en-
C k fine topology.
Strong approximation in infinitedimensionalspaces $9. Ck-Ane approximatioil of C k by C"-:
127
a t h e o r e m of Heble
I. We shall give a proof of the following t,lieorern (cf. [17], [18], [19]) in this section: T h e o r e m 9.1. Let R be a nonempty open set in a separable real Hilbert space 'H, F a real Banach space, f : 52
+F a
Ck-smooth mapping k 2 0 being a given integer, differ-
entiability being always understood in the Fre'chet sense, and &(.) a continuous positive function on R. Then 3g : R
+
F 3 g is C O O - s ~ ~ ~a,nd o o t satisfies h V integers j E [0, k ] ,
llDJg(z)- D j f ( X ) l l j < E ( X ) VX E Re This means: C-(R, F ) is dense in C k ( Q F , ) in the 52, for the definition of the
Remark.
C k fine topology (see Chapter 11,
C k fine topology).
This theorem had been a.lreacly proved ea,rlier in the special case k = 0 (cf.
[ 5 ] , [30]), the proof here (reproduced from [18, [19]) is for any integer k 2 0. This theorem consists of 2 parts, both of which are proved below:
A: 3 dense open subset R' C R, and 3g E Ck(R,F ) 3 g E C-(R', F) and satisfies:
V integers j E [0, k],
l l D j ! / ( ~-)D j f ( ~ ) l l< j ~ ( x )Vx E R .
Before stating part B of Theorem 1, we should explain some notation. Let R' be the particular dense open subset of R and !/ the special mapping in Ck(R,F ) exhibited in Theorem A. We then define for x E 0,
B: 3g E C-(R, F ) 3 V integers j
E [0, k],IIDjg(z) - D j g ( z ) J l j
<~(2).
Chapter IIZ
128
The proofs of parts A and B together will complete the proof of Theorem 9.2. We shall now first prove Theorem A . It is necessary to explain some notation which will be consistently used and also explain some trchiiiques, such as e.g. a Leibnitz formula, and localising - a word which we shall use to
iiiriiii
multiplying by a suitable C" function
q5 : 3-1 + [0,1] with a bounded support.
11. (a) Notation.
Let R, F, f , E be as stated in the theorem. The norm in the
Hilbert space will be denoted by
11 llx, and the norm in the Banach space F
by
11 11.
The subscript j after a norm shall denote that the norm is that of a j-linear mapping,
in the particular Banach space under reference. The closure of a set E is denoted (as is customary) by E . The unit interval [0 5
:c
ways denote the particular C" functioii: H
5 11 in R' is denoted by [0,1]. q5 shall al-+
[O, l] explained in subsection I1 (c) below
of this section, q5Jz - c ) denotes the sainc function after scaling (by l) and a translaP tion of the origin by c. Capital letters C, Ii-?M will denote positive real constants, often depending on a positive integer and one or more subscripts, e.g. C ( n ) ,I<, or Mj,;(n).
(b) A n extended Leibnitz formula.
Suppose q5 : H
+
R' and g : R
-+
F , both
being at least C'-smooth. Suppose j is a n integer 3 0 5 j 5 k. Then a product such as q5g : H
-+
R', is also j-times differeiitiablr for 0 5 j 5
L and we have
where the products on the rightside arc t,riisor products, the coefficients
(i) being the
129
Strong approximation in infinitedimensional spaces
binomial coefficients:
(c) Localising.
(i)= &,
0 5 i 5 j. It follows that
As mentioned above, we shall use this term for multiplication by a
particular Coo-function 4 : H
-+
[0,1] with bounded support. This function g5 is defined
as follows. Let
For our purposes we shall always choose a = I
f , b = 1, and
define 4 by d(x) = P(llxIIL)l
E 3c. We shall have occasion to use the modified functions d P ( z )=
and also $,,(I - c ) = d(;(z
- c)), c being
il
d( :I),
0
< p < 1,
fixed vector in H.
We shall need precise estimates for ~ ~ D J d ( r )=~ 0,1, ~ j , j. . . , k, as also def
llDjW1+’~~*+‘n(z)llj where @ P 1 . . * P n ( z ) = d P l ( . c 00
{P,,},,~
-
11)q5,,~(1
-
12).
. . $,,,(I
- I,,); here
is an infinite sequence of positive iiumhers to be chosen suitably and
00
{I,,},=~
can be chosen to be any sequence of vectors in H .
Convention: We shall choose the x,’s such that the set X =
{11,12,
. . .}
is a count-
able dense set in R. The first lemma gives us estimates for
llDj+,,(z)llj.
Lemma 9.2. (i)
dP satisfies:
V integers j 2 0, ~~DJq5,,(x)~~j 5 M -$for suitable constants M, which are
independent of I, and p
> 0.
Chapter III
130
(ii) The same property holds for D j { l - ‘ b p ( r ) }D, j 4 J z - c) and D j { l - d p ( x - c)) where p
> 0, and c is any fixed point in H , with the same constants as in (i).
Proof of Lemma 9.2.
Each derivative D j p ( z ) , of p(.),( j 2 1 ) , is continuous and
vanishes outside a compact set viz. [
5 x 5 11 in R1 , hence is bounded on R’
.
Next, D(11z11&) = F. where F, is the functional on ‘H defined by F.(h) = 2 ( x , h ) ,
z , h E ‘H, with llF.11~ = 211~11.Further Dz((llxll&)= G where G is the (constant) bilinear functional on ‘H x ‘H defined by G(h1, hz) = 2 ( h l , h 2 ) ,h1,hz E ‘H, with llGll2 = 2 . The further derivatives of
Ilzil& are all zero.
Then D2q5(z) = Djp(llxll&) is the: sum of a. finite number of terms each of which is bounded on ‘H, hence each derivative is bounded on ‘H. Then by the Chain Rule it follows that ((Dq5p(x)lll5
9where ( ( D $ ( z ) l (5 &Il; likewise IlD2dp(x)ll25 3 where
11D2q5(z)1125 M2Vx E ‘H. This proves the lemma. We shall make the convention.
Convention.
The constants
Mj
is chosen to be = 1, and for j = 1 ,
in Lemma 1 will be henceforth chosen as follows. MO
.. ., k ,
is chosen to be =l.u.b
llDjd(z)llj.
Z€?i
Next let @,,(z) = 4(z - z)d(z - 22). . . C$(X - z,,),
and define
Further define Mj,i(l) =
where
(;)Mi,
(I) is the binomial coefficient
0 L: j 5 k , 0 5 i 5 j
,
for integers s,t 3 0 5 t
5 s. Then define
Mj,i(2) = ( : ) [ M i , i ( l ) . M o + A [ i , i - 1 ( 1 ) . M l + . * . + M i , o ( l ) . M ; ]
131
Strong approximation in infinitedimensional spaces
Then by Leibnitz's Rule, for 0 5 j
I 2,O 5 i 5 ,j;
Now suppose (finite) Mj,i(p) for all integers p 3 0 have been defined, satisfying: for integers 0
I p I n - 1 (with 0 5 j I k,O I i I j)
5 q 5 p:
and
Then define
This shows
Now define: C ( n ) = 1
+ max{Mj,i(n) I 0 I j
I k,O 5 z I j }
for
n = 1,2,
. . .. We
also note that this preceding inequality concerning ~ ~ D i @ , , ( still z ) ~remains ~i true if some or all of the
4's
are replaced by (1 - 4)'s.
111. The proof of Theorem A depends upon suitable local C"-approximations in the neighbourhoods of the points
ii,
i = 1,2, . . . , to the given Ck-mapping f . These lo-
cal Coo-approximationsthen have to be put together; however, the customary technique
Chapter III
132
of partitions of unity could not be used directly in our problem. In our next lemma we obtain such a local approximation in a neighbourhood of each point of 0; this approximation is even analytic, and is valid in an arbitrary (separable or non-separable) Hilbert space, or even in a Banach space. Let f E Ck(R,F ) l x E R and
Lemma 9.3.
f E C”(R, F ) and 3
v >
0. Then 3f =
fz
:
R
-+
F 3
in a suitable neighbourhood U of I,f satisfies:
Proof of Lemma 9.3.
Let z E R,II > 0. Then define
f = f.
: R -+
F by
I ! I.
Here, as is customary, “(w)(j)” denotes the j-vector iw, . . . , wj. This expression for
f
is
the usual Taylor polynomial of order b of f around the point x. Changing our notation slightly, write
where
A0
= f(x) E F,Al = E Lt(‘H,F ) . Here
Ak =
-
of(.) E
L(‘H,F),A2 = $ D 2 f ( s ) E Li(‘H,F) ...,
L(‘H,F ) is t,lie space of continuous linear mappings ‘H -+
F , and for any integer j 2 2, Li(’H,F ) is the space of j-linear continuous symmetric mappings ‘H x
. . . x ‘H -+ F . j
Here we use the following notation. Suppose Gm(xl, . . . , 2,) ping V x
..- x V m
-+
is an rn-linear map-
W , where V ,W are vector spaces. For fixed 5 1 , . . . , x j € V
-
Strong approximation in infinitedimensional spaces
(1
5
j
5
m ) , we shall write
G m ( z l , . . . , xj)(j) for the mapping V x
133
x V + W
m- j
defined by
-
Similarly taking z1 = x 2 =
Vx
...
= x j = I, we shall write Gm(x)(J)for the mapping
. . . x V 4 W defined by
m- j
and likewise: Gm(x)(j)(x)m-j
ef j
m- j
A little calculation then shows that (more details will be found in [ I):
provided y
i x. (Here “P”
means “approximately equal to”.) Thus 36 = 6 ( z , v , f ) > 0 3
Then it suffices to let U = U ( 6 ) be the neighbourbood B a ( Z ) of x in
a. This proves
Lemma 9.3.
As a corollary of Lemma 9.3 we obtain:
Corollary 9.4.
The function fz in the preceding Lemma further satisfies (preserving
the notation of the last lemma): 36
> o satisfying v integers j E [o,ICI,sup UEU
~ l ~ j f ~ -( y )
Chapter ZZZ
134
Proof of Corollary 9.4.
We compare the Taylor expansion (cf. [33], p. 110) of f ,
or D f , or D2f, . . . , Dkf , respectively with fz(y), or Dfz(y),
. . . , or Dkfz(y). For any
integer j E [0, k], ~ l ~ j f z ( y-) ~ j f ( y > ljl =
11
[
+
~ j f ( z ) ~ j + ' f ( z )* (y - z)
- [Djf(z)
Now suppose 17
Dkf(z>. (y - z)k+ +(k - j ) ! * *
+ Dj+'f(Z). ( y - z) +
*
j
1
*.
> 0; then let 6 > 0 3su'p IlD'f(z) - Dkf(y)llk < 7. Then for such y UEU
we find that
This proves the corollary. We recall that X = {z1,z2, . . .} is a countable dense set in R. We shall write ~(z,),n = 1 , 2 ,
Convention.
.. .. &(a)
E,,
=
We shall now agree on the is bounded by 1 on R.
For otherwise we can replace
&(a)
by min
{E(.),
1).
Lemma 9.5. (a) For each z E n,3 open ball B,(z) SUP u,u'EB.(z)
c R satisfying
I 4 Y ) - 4Y')I
4Y) .
(b) For each n = 1 , 2 , .. . , and given constant Kn > 1 3 open ball Bp,(zn) c R 3 (a) holds in Bp,(zn) as also the folfowing: 3jn E Cm(R,F) satisfying:
135
Strong approximation in infinitedimensional spaces
Proof of Lemma 9.5. (a) Follows because
is positive and continuous.
E(.)
(b) For each n = 1 , 2 , . . . , 3 open Brn(zn)3 (a) holds in Bvn(zn).Also by Cor. to Lemma 2, 3 open ball Bra(zn) and 3 function
jnE C"(R, F ) satisfying:
Now let pn = min(rn, rk). Then
This proves Lemma 9.5.
From now on, we shall make the following
Convention. for n = 1,2,
The constants I<, in Lemma 3(b) will be chosen to be = 2(k
+ l)C(n),
. . .. Further the pn's shall be 3 both (a) and (b) of Lemma 3 are satisfied in
Bp,(zn) and further 3 pn 1 pn+l for n = 1,2, We shall also recall that the function
Bpn/fi(zn) and dp,,
. , .. We shall let po = 1.
$p,(~
E COD:H
- 3),
0 outside Bp,(=,,),for each n = 1,2, m
m
n=l
n=l
We now define R' =
+
[0,1], C$p,
= 1 on
....
u Bpml&(zn),and 52" = u Bp,(zn). The following observa-
tion will be needed in the sequel; we shall omit the simple justification.
Lemma 9.6.
If Bp,(z,)
fl
Bp,(zn) #
8, then E, < 4en.
In the next two lemmas we shall establish convenient bounds for expressions of the
Chapter 111
136
Lemma 9.7.
5
V integers i, j 3 0 5 i 5 j, 1.u.b. IIC)Di{$pl(~- 2 1 ) . . .dpn(x - xn)}lli 5
y.
+en
The inequality is still true if some or all of the 4’s are replaced by (1 -
4) ’s. Proof. We shall use induction on n. For n = 1 the statement is true, since (for any p = 1,2,
. . .) we know that
for any integer i 2 0, 1.u.b. 11Diq5pp(x- zp)lli 2
en
5
2,and
Mj+(l) = (;)Mi. Suppose the lemma has been established for some integer TI 2 1. Write W”’+‘m(x)
.
= ~$,,~(z-x1). . 4pm(z-xm)for any m. Then
@‘1-+‘*+1(x)
= ~ p l . . . P n ( x).4pn+1(x--5n+1).
Using Leibnitz’s Rule (cf. II(b)), and the property that pn 2 pn+l, we find
+...+-.Mio(n) Pi PL+l Mi
I
This proves the lemma.
Corollary 9.8. A similar result holds fbr IIDi{dpp+l(x- x p + l ) .. . 4pp+,(x - ~
~ + ~ ) } ( l i ,
viz.
A similar estimate is valid if some or all of 4’s are replaced by (1 - 4)’s.
The inequality remains valid if some or all of the 9’s except
dPnare replaced
by (1- 4) ’s.
Strong approximation in infinitedimensional spaces
Proof of Lemma 9.9.
137
Using the preceding notation, and Leibnitz's Rule, we find, for
On the other hand, for z E R - BPn(zn), the 1.h.s. is simply zero, hence the inequality is trivially true. This proves the lemma.
Corollary 9.10.
If z E R, then for any integer j
E [0,k],
The inequality remains valid if some or all of the 4's except
4) 's.
4pp+m, are replaced
by (1 -
Chapter III
138
Lemma 9.11.
For each 2 E
a', the sequence {gn(z)} is constant after a certain stage.
Proof. If 1 5 m 5 n, then for z E B p m l f i ( z m ) ,
Lemma 9.12.
On each B p , / ~ ( z n gn ) , is C".
Proof. We see that each gn can be written as gn = [I - d p l - 4 p 2 ( 1 - 4 p 1 ) +4p2(1-4p1)f2
4p.-i)]f
.#p,(l-
+...+4p,(1-4pn-1)...(1
= 4pif1
-4pdfn
= (1 - 4pl)(l- 4 p a ) - . - ( l- $,,,)f+ (finite number of It follows that on
C" terms).
Bpnlfi(xn), gn is C". Thus the Lemma is proved.
Now define the sequence { h F ) , O 5 p 5 n, by:
1 5 p S n - 1 , hp"-l =+pp(fp-f)+(l-4pp)h;.
11:
= 0, h:-l
= 4,
. (fn - f);and for
Thenbyiterationwefind,forl
Strong approximation in infinitedimensional spaces
Lemma 9.13.
For any integer j E [0,E l ,
sup
llDjgn(Z) - D i f ( z ) l l j <
*EBpn,&an)
Proof of Lemma 9.13.
139
4.
We shall show that for any integer j E [0, k], and any integer
P E [1,nI,
On the other hand if Bp,(x,)
Bp,/fi(xn) and
n Bpmlfi(xn) = 0 for some m 5
3 x @ Bpm(zm) for m
n - 1, then for x E
5 n - 1, the corresponding term in the preceding
finite sum is to be replaced by 0, and hence the estimate is again valid. Hence for x E
hence
IIDJgn(z)- Djf(z)llj <
sup
4. This completes the proof of Lemma 9.
zEBpn,fi(Zn)
Now we define g = lim
n->m
gn on R'. For each x E R' set n, = inf
B p , / f i ( x m ) l . Then 3 neighbourhood V ( x ) of z 3 V ( x ) c Epnm/fi(xnm), and 3 the integer n. is constant in V ( x )or changes to a smaller value in which case we use the result: if x € B P m l f i ( x mthen ) for any integer n conclude that
ic C" on fl' and for each x E R':
> m , g n ( z ) = gm(x). Therefore we
Chapter IIZ
140
In the next lemma we shall show that lim gn(z) exists not only on R' but exists, in fact n+w
uniformly, for all z E R.
Lemma 9.14.
For each integer j E [0, Ic], the sequence {Djgn(z)} is uniformly conver-
gent Vz E R. Hence the function g defined by: g(z) =n-w lim gn(z) at each point z E R is Ck-smooth in R and has the approximation property:
V integers j E [0, Ic], IlDjg(z)- DJf(z)[lj< ~ ( z ) Proof of Lemma 9.14.
Let N , N ' be intcgers 3 N'
> N > 0.
Vx E R
.
Then for any integer
j E [0, Ic], and z E R, by Lemma 9.9:
Hence for such j the sequence {Djgn(z)} is uniformly convergent Vz E R. For j = 0 this means { g , , ( . ) } is uniformly convergent in R. Hence ij(z) = lim gn(z) exists not merely n+w
at points in R' but at all points in R. Furthermore by Theorem 12 in [33], p. 117, we are able to assert that for each integer j E [l,Ic], lim Djg, must be = Djij, and further ,,--too
since {Dig,,} is uniformly convergent and each Djg, is continuous in 52, therefore Djg is continuous in R. Thus ij is Ck-smooth in R. Next for such integers j,and each point
,:R
E X,
IlDjg(zrn) - Djf(zrn)IIj = IIDJgn.,,,(~rn) - DJf(zrn)llj = JJD'gm(zm) - Djf(zrn)Ilj
141
Strong approximation in infinitedimensional spaces
because n.,
Let y E
5 m; and hence
(an')n R if (an')n Q # 0. Then
3 subsequence {xn,}
3, ,z
+y
and because
Djg(.),D j f ( . ) , ~ ( .are ) all continuous on R, we find:
Hence g has the property:
This completes the proof of Lemma 9.14 and hence Theorem A is proved.
V. We shall now turn to the proof of Tlicwrcmim 13 stated at the beginning of this section. We shall need the following two lriiinias. Although these two lemmas are stated for the particular functions g(.) and a(.),they are clearly true more generally. Lemma 9.15.
The function y(.) defined for x E R by:
is continuous on R.
Proof of Lemma 9.15. converging to
5.
Let
3'
E R, and let {x"}:!~ be a sequence of points in R
We first show that 7 = linisup y(x,)
will follow that y = liminf y(xn) 2 y(z). n-w
n-
W
5
y(z); by similar arguments it
Chapter 111
142
Taking a subsequence of {zn}if necessary we shall suppose lim 7(zn) = v. Supn+m
pose v
> 7(z). Then a simple argument
shows that for n sufficiently large, B,(,,)(zn)
would properly contain B,(,)(z). However this would clearly violate the supremum property of 7(z). We therefore conclude that v must be 5 7(z). Hence 7 must be I7(s)*
Similarly we conclude that 7 2 7(z), because otherwise the supremum property of 7(zn)would be violated for many
71%.
This completes the proof of the lemma.
The proof of the next lemma is very siiiiilar to that of Lemma 9.15, and hence will be omitted.
Lemma 9.16. The function X ( i ) defined for .T E R by
[
X(z) = sup 0
<6<1
I7
(1')
c R, v
integers
j E [O,IC],
is continuous on R. We now have to define two functions p ( z ) and N ( z ) . For z E R, let
where ~ ( z = )
I+ max ( ~ ~ ~ j i j ( x ) l .l j ) OSj
The preceding lemmas and the proof of Theorem A show that these functions are positive and continuous on St. We shall now introduce suitable C" mappings
tia(
.) :
+
H which map suit-
able subsets of R into R'. The composite mappings g ( u a ( . ) ) will enable us to define a
Strong approximation in infinitedimensional spaces
143
suitable C" mapping g(.) : f2 -+ F . Each of the mappings u a ( . ) will be defined with the help of an element of the dense countal>leset X ; further it will map a neighbourhood Bp(z)of some point llua(z') - ~'(171<
Let
I
E R into R' nnd will have the properties: for I' E B,(z),
I
q, IID(uP(d)-
z')1I1 <
q, and D ~ U , ( G = ' ) 0 for j = 2,3, . .. .
E 0. Let 6' 3
(a) O < 6 ' <
1
(b) &(I) C R, and
* 1/44- P(.)l
<
( c ) 2' E? (2) 6
q;
and define
3
(ii) t =
(i) v = -p(z); 16 where m
1 m
> 1 is the unique integer 3 1
1
110g(&) log2
-m< - 2
Irn-1'
the logarithms being natural logarithms. Let ,z, from G, which lies in B,,+(r).
To each I' EB 6'
.
of X which we seek. Next let
=
be the first element of X , different
(2) we
assign ,z ,
a the required element
1
3t+'ll~-zn,llx
and 1 r' = (1 - -). 2t
= (1 -
Then let 6" 5 min 6'; f(1- $)(1 - &)
[
I' E B p ( r ) the point
1 1 $)2'+'. ((2- 2,,((31 .
. (15 - zn,II~],such that if we define, for each
z' by:
2'
= z:, = (1-
1 1 + , + g' ,
Chapter III
144
zll
then
E
B ~ ( z : ) .Now if the positive number r'' is defined by:
b
r' r'' - 16
then for every such z', B , . ~ ~ ( z C ' ) B,I (2:).
'
This choice of 6" also ensures that
T
B&l(Z)
fl
B,I(z:) = 8. Now let xnp, be the first element of X which lies in B+(Z:)- B ,-~ ( z k ) For . each r x' E
B 6 " ( 5 ) , determine
the point z =
Z,I
E
Br(xnp)such that the two triangles, one
with vertices x', z:, ,znpl,and the other with vertices z',znp,z , are similar. (The relevant geometrical property in a two-dimensional plane can be deduced as a proposition in analysis, see [19].If dimension of 'H equals 1, our arguments need only slight modification.) Because of the one-one bi-continuous relation between the pairs ( d ,za,) with x' E
B p ( 5 ) ont the one hand, and the points z
= z1 .
determined by the above process on the
other hand, we find that the set 2 of all such points z is an open set.
[
NOW let 7' = min ;{ llz; that Bg,(znp,)c 0'. Let ball Bc(z,)
6'
- xnp,
11%
-
f ;dist (xnpr,do'); i { r ' - 11~1- znP,11311, SO
- (1 - &). Because 2 is an open set, therefore 3 open 3 C' -
c 2 , with 5 min[C'; f{))znp- z . l l ~ ] ; dist
(z.,dZ)]. Let
inp be
the first
element of X , different from z,, lying in BC,~(Z,) n 2. Finally let 7' = (1- $)C, and 26 = min[$v; 6 " ] , where 7 = and u ( z ) = (1 - $)z,,
+ &x.
Now define, for each
2'
This mapping u ( . ) which maps the neighbourhood
$ { v l - IIu(x) -znp,llx),
E &,$(z):
&6(.)
in SZ into a neighbour-
hood of u ( z ) in SZ', is, however, defined everywhere (znqbeing fixed), and Vx': IIu(x') -
Strong approximation in infinitedimensional spaces
Dju(x')= 0
for j 2 2
145
.
It will be convenient to change our notation slightly. We find from the preceding that for each z E 0 3 point qr E X and 3C" function u(.):3-1 -+ 3-1 determined by the above process with the help of q r , satisfying: for some t
IIDju(x) = 0 for j = 2 , 3 , . . .
Furthermore 36
> 0, and Vx:
.
> 0 3 Vx E & 6 ( z ) :
('). ( I D ( u ( x )- x)11, = 1 - - < p -
2t
~ j u ( x=) O
The separability of
2
and
'
for j 2 2 .
3-1 implies:
(i) 3 countable collection {zl,z 2 , . . .} of points in 0; (ii) for each 2-3 a point qr, E X , a number 6,
> 0, with fl =
u Bgnlfi(zn),and a C" 00
n=l
function u,(-):'FI + 3-1 defined with the help of qr, with properties similar t o those
Chapter III
146
For each n = 1,2, . . . , the composite function g(un(.)) is
C” in B 2 6 , ( Z , ) because g(.)
has been shown to be C” on Cl’ and the C“ function u,(.) : 3.1 -t 3.1 maps B2&,(&) into
a‘. Further we note that Djg(.) as also Djg(u,,(.)), for j B26,(z,)
= O,l,
.. . , k , is bounded on
for n = 1,2, . . ..
Now for each n = 1,2, . . . , let q5,(.)
3.1 -t [0,1] 3 4,
=
1 on B6,,6(z,) a n d &
4&,(* - 2,) be the Coo-function mapping
= 0 outside &&,(&)
(cf. subsection II(c)
above).
VI.
We now want estimates for I l ~ J { g ( u , ( z ) )- ~( z ) ) l l jj, = 0, I,
. . . , Ic, z E &6,(Z,),
n = 1,2,. . .. For this computation we shall use the following known formula which is verified by induction (cf. [12] p. 222): for
2 E B26,(zn),
and j
1 1,
(ii) a = ( a l , a z , . . ., aj), the ai being non-negative integers satisfying: a1 + 2az + . . . + jaj = j,
(iii)
C adef = a1 + a2 + . . + aj;
(iv) for each integer j (1 5 j 5 k ) , a further satisfies (in addition to (ii)):
with pi = 1,2, . . . , j , i.e. for each j (with 1 5 j 5 k) we consider possible solutions
Strong approximation in infinitedimensional spaces
147
of each of j pairs of simultaneous equations:
(v) A, are certain positive constants (the numerical values of which do not matter in our calculations); (vi) “a
Sj” simply means: “the j-tuplet
cy
=
(a1,
. . . , a,)
satisfies the preceding
conditions (ii)-(iv)” In the preceding formula ( 3 ) Ca is given the successive values j , j - 1, . . . , 1. The first term in the summation on the right side in ( 3 ) then corresponds to the unique solution:
a1
= j, a2 = 0 = . . . = a, of the siniiilta.rieousequations a1+a2+”’+aj
= j,
(4) a1
+2a2
+ . . a +
ja, = j .
The last term in the summation in ( 3 ) is Dij(un) o (DJu,.,(z>),corresponding to the unique solution
a1
= 0 = . . . = a,-1, a , = 1 of the simultaneous equations a1 +a2
+ ... + a ,
= 1,
a1+2a2+*..+ja, = j .
1
(5)
The remaining terms in the summation for D J g ( u , ( z ) ) correspond to solutions in non negative integers ai of the pairs of simultaneous equations
with i = 2 , 3 ,
. .., j - 1 in turn.
In each of these cases, any solution a (for a given i with
2 5 i 5 j - 1) will contain a non-zero ai with
ai
> 1. The corresponding contribution to
the summation representing Dj?j(un(z)) is zero (if j
> 1).
Chapter ZZZ
148
Then for a given j (with 1 I j I k):
This estimate clearly also holds for j = 0.
VII. We now want to put together the composites g(un(.)), n = 1,2, . . . , in a convenient manner, to obtain a suitable C" mapping. We proceed as follows. We define a C" function h(.) : R
+
and generally for any integer n 2 1, b(1,...,
[0,1] in the following manner. Let b(l) = b l , n)
= min(b1,
. . . , bn).
Let x E R, and define
for m = 1,2, . . , ,
Here the constants C ( j )are the ones defined in subsection II(c). For each m = 1, 2, . . . ,
hm(.) E Cw(R,R1). Further the sequence {hm(z)}z=l is constant from m = n onward, where for x E R, n = n, = inf[rn
I x E B6,,~(zm)].
Hence h(x) = lim h,(z) exists m-w
for each x E R, and the function h(.) E C"'(R, R').
We want t o note further properties of the function h(.). As noted, if x E R, and
Strong approximation in infinitedimensional spaces
where ai =
.(";I. ..i' iiiIfThen a ,
2, c+2
This means that
& E C-(n,
149
> a,+l for any rn = 1 , 2 , . . .. Hence
R').
With the notation of thc prcrrtling pnragraphs, for j = 0 , 1 , . . . , k, and making use of Lemma 9.9 (subsection II(c)), we find
sk+ 1 2{ 1 5+ 1 F + .+.&} . + k+2 1 {; + . .. $} + . . . + lc+2 1 {:+..+}, there being no more than ( k and j = O , 1 ,
..., k,
+ 1) terms in the sum in the right side. Thus, for z E R,
Chapter III
150
In general, induction yields: if m is any integer 2 1, then
This suggests that such an estimate might be valid for rn = 0. We verify that this actually is the case, as follows. For j = 0,1,
. . . , I;,
and, on the other hand, the sum in the left sidr in (8) is
We now need the following lemma.
Lemma 9.17.
Let
W1
= {w = ( w j , wj-1,
integer E [0, k], with norm llwll =
.. ., W O )
I
UI;E
Lt(H,R')}, where j is a fixed
j
IIwilli. r=O
Define T linear:
W1 + Lf(H, F
) by
Strong approximation in infinitedimensional spaces
151
where c; E Lf(H, F ) are fixed, the products being tensor products. Then llTll = max{l)ci(l;, 0 5
5j).
Proof of Lemma 9.17.
Hence llTll Imax{llc;ll;, 0
5 z 5 j}.
(ii) Let Wa = { ( a o , a l , . . . , a j ) } , the a; being elements of Lf(H, F ) , the norm in Wz being defined by: llall = max{)la;((;, 0 I i I j } . Then c =
(CO,
fixed i E [ O l j ] define w; E Lf(H,R1) of unit norm, and w = (0,
. . . , c j ) E Wz. For
..., w;,Ol ... , 0)
E
Wll where the ( j - i)th component is w;,the remaining components being 0's. Then llTwll = llw; llcj-illj-i
. cj-;llj
=
5
~ ~ c j - ; ~ ~ j - llTllllwll ;
= llTll. Hence for each i = 0,1,
.. . , j ,
5 llTll. Hence Omax llcilli I IlTll i.e.7 llcll 5 IITII.
From (i) and (ii) we conclude llTll = IIcII. This proves the lemma. This lemma implies that max IID'(g(z) - g(z))ll; is the best constant C O
> 0 satis-
fying the inequality:
Hence
max, IlD'(S(.)
O<,<J
This is true for any j = 0,1,
- g(4)lli
I 4.) .
. . . , k. This is precisely what we wanted to satisfy.
Finally, putting together all the above conclusions (including that of Theorem A) we
Chapter IZI
152
find that for j = 0,
. . . , k,
llDjg(2) - D J f ( 2 ) l l jI a(2)
This completes the proof of Theorem 9.
1 + o(2) = -{a(2) + 2
E(Z)}
< E(2) .
153
Strong approximation in infinitedimensional spaces
§lo.
Connection between strong approximation and earlier ideas of BernsteinNachbin We can adopt a slightly different point of view concerning the space Ck(R,F ) ,
which will enable us t o view the results of Kurzweil, or of Bonic and Frampton, or this
[19] as falling within the context of earlier ideas of Bernstein and author's results in [MI,
L. Nachbin. In order t o explain this point of view we shall define 1
V = {v(.) = - I E ( . )
41
is positive and continuous on SZ}
,
and denote by Ck&(Sl,F ) the vector subspace of Ck(SZ,F ) consisting of functions f E
Ck(R,F ) 3 max v(z)llDjf(z)ll is bounded on 0 for each v E V . Each v E V determines O<j
a semi-norm by:
and we shall understand that Ckl$,(SZ,F ) is endowed with the locally convex topology determined by the family of these semi-norms {p,,(.)},,,. Now denote by W" the vector subspace of Co(SZ,F ) consisting of analytic mappings
SZ
F and by W" the subspace of Ck(R,F ) consisting of C" mappings R
+ F.
Then
Kureweil's theorem says that W" is dense in COl$,(SZ,F ) . The theorem of Bonic and Frampton says that W" is dense in that Woois dense in
cOvb(n, F ) . Also the results of this author tell us
ckvb(n, F ) for a given integer k 2 0.
154
Chapter III
811. Strong approximation - other directions
The concept of approximation in tlie “Whitney topology” or “strong approxiniation” was seen before to be more suitable in the case of function spaces on non-compact spaces than that of uniform approximation or approximation in the compact open topology, and was not unrelated to tlie Bernstein-Nachbin ideas of weighted approximation. In a somewhat different direction, Whitney’s idea of strong a.pproximation has been modified in tlie realm of what have been called Nash mappings between Nash manifolds. In the case of Nash mappings of non-compact manifolds, a suitable “strong” topology is defined in the same way as the usual topology on the space S of rapidly decreasing Cm functions. We shall give here a brief summary of some recent work of M. Shiota (cf. [55])in this connection. This work of Shiota indicates some further possibilities Some preliminary definitions should first be introduced. Let r = 0 , 1 , 2 ,
. . . , or 00
or w . A sub-
manifold of W” is called a C‘-Nash manzfold if it is semi-algebraic and of class C’; semialgebraic (cf. [l])means it is the finite union of sets defined by finitely many polynomial equalities and inequalities, i.e. a finite union of sets S where z E S if Pl(z) =
...
=
P j ( z )= O ;
and
tlie P’s and Q’s being polynomials in
21,
Q1(z) > 0,
. . . , 2,. A
Q2(z)
> 0 ...,
Qm(z)> 0 ,
CP-map from one C“-Nash manifold
to another is called a CP-Nashm a p if the graph is seini-algebraic. A CP-Nesh vectorfield is similarly defined. N “ ( M ) denotes tlie ring of all CP-Nasli functions on A/r where A/r is a C‘-Nasli manifold. It is convenient and meaningful to restrict
1’
to be
< 03,
since it is
known (see [55]) that a Cm-Nash manifold and a Cm-Nash map are already of class C”.
Strong approximation in infinitedimensional spaces
155
One objective of Shiota's paper is to approximate a C'-Nash map between C"Nash manifolds by a CW-Nashmap. In the compact case an earlier result is attributed to Nash as also to Palais (see [55] for the references). In the non-compact case such a result is obtained in [55].
A stronger topology than either the uniform CP-topology or the compact open topology, is introduced in N ' ( M ) , as follows. Let fine
"fk
-+
0 as k
-+
00"
to mean the following:
for any C'-Nash vectorfields v1,
. . . , up# where T'
fk
v1
5
E N ' ( M ) , k = 1,2, . . . , and de-
. . . vp, f k
-+
0 uniformly as k
-+
00,
r . When M = R" and r = co,
this topology coincides with the usual topology on the space S of rapidly decreasing C" functions (cf. Rudin [52]), i.e., f k
k
-+
00,
for any multi-indices
(Y
-+
and
0 as k
p.
-+
00
+ ."opfk(.)
-+
0 uniformly, as
The space N ' ( M ) with this topology is not a
vector space, because a f f , 0 as a E R converges to 0 unless supp
(f)is compact. The
strong topology defined above is called the C'-topology on N ' ( M ) . The following theorems proved in [55].
Theorem 11.1. Let M I and Ma be CW-Nashmanifolds, and f : M I
-+
M2
a C'-Nash
mapping. Then f can be approximated by a C"-Nash mapping in the C'-topology. Further suppose the restriction off to a given compact Cw-Nashsubmanifold M3 of MI to be of class C". Then f can be approximated, fixing on
M3.
The proof uses a C"-Nash function on R" which is an approximation of 0 outside a small semi-algebraic neighbourhood of X ( X being an algebraic set in R"), and of 1
in another one. This function is required to hold a useful well-known property of a C"partition of unity. Several preliminary lemmas are needed for the proof, and though the
Chapter III
156
complete details of the proof of Theorem 11.1 would be outside the scope of this book, these leinmas themselves are interesting in their own right, and we shall csplain these in some detail. Let f E N‘(R”), and e ( z ) = integer, and
1 ~ =1 z:~ + ... + z.:
and k. Let U c
Leiiiiiia 11.2.
&,
where C is a positive number, k is a positive
Write
e as e c , h
when it is necessary t o emphasize C
W” be an open semi-algebraic neighbourhood
of f - ’ ( O ) .
Put
Define 1
F Then F -+ 0 on Vz and -+
= -2( ( f Z + e ) i + f }
f on Vl in the Cr-topologyas C and b +
00
in such a way
that k2k 5 C . Proof of Lemina 11.2. e ) i -+
We shall suppose r <
00.
T h e problem is to prove ( f z
+
If1 on V1 U V2; hence it suffices to prove the convergence on Vl. The proof pro-
ceeds by induction on r . Case
1‘
= 0.
Let
E
be a Nash function of the same forin as e above. P u t
$ ( t ) = iiif{f(z)
I 1x1
=t
and
N
E VI} .
This function is positive a i d upper seini-continuous, a i d by the theorem of Ta.rskiSeideiiberg, it is a semi-algebraic function on the closed semi-algebraic set
Strong approximation in infinitedimensional spaces
157
By Lojasiewicz's inequality (see Malgrange [40] p.59) and the stereogr~.pliicprojection, 3 C 1 , k l
> 1 3 for any C 2 C1 and k 2 k l , 1
& ( t ) d J ( t1 )
for
t E 14'
where ~ ( tis) defined such that &(IxI) = ~ ( x )i.e., ,
Therefore
Thus the proof for the case r = 0 follows. Now suppose
(f2
+ e ) ) + f in the Cr-'
as above, 3C2, k2 2 1 3 for any C 2 1011
C2,
topology; i.e., to be more precise, for any
k 2 ka with k 2 k 5 C and a multi-index a with
5 r - 1, we have
This inequality has to be proved with all
=
E
01
3 la1 = r . Let a be one such multi-index.
{ N f+2 4 ; - f)] . ( D " ( ( f 2 + e ) t +
f)}
4#a Also 3 constants do, . . . , d,-] depending on r but not on C or on k, 3
Chapter III
158
Now k 5 k z k 5 C j
k
(*-i)(2h-l)-i
5 1. Thus
i=O
This together with the induction hypothesis, implies
on V,, for any c1 as
E,
and sufficiently small e with k z k 5 C, where d =
di. Hence,
~~~~
as in the case r = 0, choosing small 61 we obtain C3, k3 2 1 3 VC 2 C3 and V k 2
li3
with k z k 5 C, we have
Thus the lemma follows.
Definition.
The argument for r = 0 in the preceding lemma is called Argument 1
(Arg. 1 in brief). Now let r' be a nonnegative integer 3 r'
< 00 and r' 5 r. Let r$ be a polynomial on
R 3 d(0) = . . . = $r'(0) = 0. Then
is a CP'-Nash function which is r'-flat at f - ' ( O ) ICY1
i.e., P r $ { ( l f l
+ f)/2} = 0 on f - ' ( O )
for
5 T'.
Leiiinia 11.3. $ ( F ) -+
${(If[
+ f)/2}
k Z k 5 C, F being as defined earlier.
in the C"-topology as C and k
-+
00
so that
Strong approximation in infinitedimensional spaces
Proof of Lemma 11.3.
Let
f1
=
(If1
+ f)/2,
159
and a a multi-index with la1 5
If
TI.
(a1> 0 then
Therefore, for any semi-algebraic neighbourhood U of f - ' ( O ) the convergence Dad(F )
+ Daq5(fl) on Rn - U in the Co topology follows from Lcinma 11.2. Hence it is enough to prove the following: Suppose Let
E
cyl,
.., , cyf > 0
are multi-indices with
1011)
+ la21 + . . . + JacI= r"
5
7'.
be a Nash function of the sarne form as e. Then 3C1, kl 2 1 and 3 an open seini-
algebraic neighbourhood U of f - ' ( O ) 3 VC 2 C1 and Vk 2 bl with k 2 k
I@t(F)D"lF . . . D a c f l l < E I d f ( f l ) D " ' f i . . . D"'f1I < E
on on
< C, we have
U , U - f-'(O) .
Such a neighbourhood U satisfying the second inequality is readily seen to exist; thus consider the first one. By assumption, $(')(F) = FP"--(+'$(F) for some polynomial
6.We have to show:
which is equivalent to:
lf'"-'+'o"'(f2
+f )
.. .
D"'(f2
+f)l
<E
on
U
,
Chapter IIZ
160
where
f2
=
(f2
+ e ) i , because 4
< IF1 <
f2.
Furthermore, by induction on r" (see
below), and Argument 1, this last inequality is reduced to
I ft'
-
Consider t h e case
1''
. ..
f(X)l
<&
011
< 74 x11
,
u.
= 0. Let
u u'
=
(5
E R"
= {(X, t ) E
I
If(X)l
UXR
1
(f"X)
+ t2)1/2
Then U' is a n open semi-algebraic set containing f-'(O).
< E(5)) . By Arguineiit 1, for arbitrarily
sinall e, U' contains the graph of e ( z ) ' I 2 on U ; hence
This proves Case r" = 0. Case
1.''
> 0.
Invoking the argument for Case r" = 0 and the equality
we see that we should prove:
for each i, sonie
C2, k-2
for some constant C
and arbitrarily small e with k Z k 5 C. We note that
> 0. Hence we need to show
161
Strong approximation in infinitedimensional spaces
This is clear if l,f3jl
> 1; if
lpjl
= 1 then this follows from the inequality: IDPiel
5 doe in
the proof of Lemma 11.2. This completes the proof. Next, given C1,k l , C 2 , k 2 put el = let
4 be a polynomial
on
W
eCl,kl,e2
= e c , , k , . Let r' 3
3 d ( 0 ) = 0, d(1) = 1, and 3 if
1.'
l" <
00,
and
1.'
5 r;
2 1, then d' = . . . = d(r')=
0 a t 0 and 1. P u t
Then d ( F 1 ) is a C"-Nash function 3
4
= 0 on the set
{f 2 2) and
= 1 on the set
{f 5 1). We now need the following lemma. Leiiiina 11.4. d(F1) can be approximated by d(F2) in the Cr'-topologyby choosing el and
small.
e2
Proof. Fix e l , temporarily. Set
Then 4 ( F 3 ) is a C"-Nash function. By Lemma 11.3, d(F2) + 4(F3) in the C"-topology as C2 and
k2 -+
03
in such a way that k i k a 5 Cz. Hence we should show that d(F3) +
d(Fl) in the C'"-topology as C 1 and kl The case
T'
+
00
satisfying k t k l 5 C1.
= 0 follows from Lemma 11.3. Suppose I"
> 0. We want an inequality:
for small e l
(1) where
01
is a multi-index with 0 <
(011
5 T' and E is a given function of tlie same form as
Chapter III
162 e. If
INI
> 0 then,
as in proof of Leiiirna 11.3,
outside
where
Yl = {f = I}, Y2 = {f = 2}, and (3 = f +
((f - 1 ) 2 + e,)
By Arguineiit 1, for sufficiently s ~ n a l el, l bourhood of
Y2.
U2
Here Fl and
*
F3
z}
=
el {f = 2 - -}
4
.
is contained in a given semi-algebraic neigh-
of Ifl, Y2respectively, and Clo,klo such that for each a],. . . , at
+ .. + loll 5
la11
Y3
L
Therefore to ensure (1) we only have to find open semi-algebraic neigh-
bourhoods U1 and
o with
Y3
TI
and any
~1
>~
1
and 0 kl
> k l o with k:kl 5 ~
can be replaced by
F30
respectively, because
= (3 - f -
((f - 1)2+ e1)')/2
1
we , have
>
163
Strong approximation in infinitedimensional spaces
and furthermore,
F l o ( s )> 0 w
> O for
F30(~)
z E R" - U1
u Uzaiid small
el
.
We shall denote by (2)o, ( 3 ) 0 , and (4)o the respective replaced quantities.
(30)
is trivial for some small U1
Uz
and
because
p ( t ) = tr'-L+l ( t - l)v'-!+i$(t)
for some polynomial $. Then by Lemma 11.3, F30 + F 1 0 on as C1 and
k1
00
3 k:kl
f1
= (( f - 1)'
+ e)
in the C"-topology
5 C1 and this, together with (3)0, provcs ( 2 ) o on Uz and
Then as in Lemma 11.3, (2)o on
where
W" - U1
U1
(4)o.
is reduced to
'.
This is one of the inequalities in the proof of Lemma 11.3.
Thus the Lemma follows. Next let X
c W"
be an algebraic set, I the ideal of R[xl . . . , x,,] defined by X,
namely consisting of polynomials vanishing on X, and h the square suiii of finite generators of I . Define f = h r ' / e 3 , with
c3,k3
> l and e3
=
eCs,ks.
Define e l , e2, r ' , (6, F1 and
Fz as we did earlier (before Lemina 11.4). Lemma 11.5.
The functions 4 ( F l ) and d(F2) are C"', and C" Nash functions, respec-
tively. Suppose U is a semi-algebraic neighbourhood of X. Tlien for small outside U and = 1 in another. Fix e 3 . Then, for some mation ofd(F1) in the CP'-topology.
el,
and
e3,
e3,
$(F1) = 0
( ~ ( F zis) an a.pp1,ox.i-
Chapter IZI
164
Proof. The first statement follows because of the definition. Next suppose sen that U 2 { h
e3
is to cho-
< 2e3}; this choice is possible by Argument 1. Then the second state-
ment follows. Th e last statement is Lemma 11.4. We shall denote by S i n g X the set of all the singular points of X. T he next leninia follows. Leiiinia 11.6.
Let Y c
X be
a connected component of X - Sing A;' let V be a semi-
algebraic neighbourliood of X - Y in W". Let g be a C'-Nasli function on R n r'-Aat on
Y . Then gd(F1) -i0 on R" - V in the C"-topology as C3,l i g -+
00
in such a way that
k i k s 5 C,. Proof of Leimila 11.6. index with la1 5
T'.
for large C3 and
123
Suppose E E N"(Rb)be of the same form as e , and a a multi-
What requires to be shown now is:
3 ktka
5 Cs. This inequality can be reduced t o simpler inequalities.
Since d(F1) = 0 outside W =
{f 5 2)) it is enough t o consider the inequality on W - V .
Furthermore, since
the earlier inequality can be replaced by
We note that if IyI > 0 then
Strong approximation in infinitedimensional spaces
Then because 4'(2 - f ) ,
. . . , 4(r')(2- f ) are bounded
165
on W , therefore it is enough to
prove:
This inequality in turn can be reduced to (for
1/31+ lal I + . . . +lafl + 17' 1 + :. . + IYk( 5 ? - I ,
and H = h"):
IDpgD"' H on W - V . If
I
e
. . . D"'HDy1e3 . . . Dyhe3/et"
15
E
= 0, then li = -1. In the proof of Lemma 11.2, the inequality:
DYiek+' < ole for some constant a , was established. Hence it is enough to prove: 3
1
... DaLH/HI 5 E
(DpgD"lH
on
liV - V - Y ,
l D P g 1 5 a on W - V where
1/31 5 r'. Here the inequality: H 5 2e3 on W as also the hypothesis: Dpg = 0 on
Y,has been used. Now consider the sets
2 =
{ z E R"
2' =
{ z E R"
I I
IDPgDYIH . . . D Y f H ( z ) l5 IDPg(r)l 5
E(Z)
H(z)} ,
~(5)).
These sets are semi-algebraic and contain Y . Using Argument 1, it is enough to prove that 2 and 2' are neighbourhoods of Y . For 2'this is clear. As regards 2,let
20
E Y
Chapter III
166
and consider a small neighbourhood of
(y,z) = (yl, . . . , ymlzm+1,
. . . , z,.)
ZO.
We can obtain a C" local coordinate system
around 2
h(y, 2) = y: t * . * t ym
20
and
3 (y, z ) = 0 at
Y
ZO
and
= {yl=...=ym=O}
(cf. Lemma 4.11 in Shiota [55]). Then by hypothesis
in a neighbourhood of 0 for some constant d'. Hence
pgD;1H . .. D,"'fqy,
I).
5 r"~.ll~~+~-l~l+~~'-l~ll+""l-l~~l - d")yl2r'+l . <
in a neighbourhood of 0 for some constant d". Hence 2 contains a neighbourhood of
20.
The lemma follows. We shall only state the next lemma. However to state this lemma it is necessary to explain some further notation and definitions. Let h E N"(R"), X = h-l(O), U connected CW-Nashmanifold open in
cX
a
X and g E N W ( U )A. minimal polynomial P ( z , x)
f o r g means a polynomial in n t 1 variables 3 P ( z , z ) l
RxU
f 0, P(g(z),z)
= 0 on U ,
and the degree in z is minimal. Then we say that the pair (9,P) has Property ( A l ) if
P-l(O) n (dP-'/dz)(O) fl R x U = 0, P is of constant degree in z at every point of U and
{P-l(0)U(dP-l/dz)(O)}fIR x U is the disjoint union of the graphs of CW-Nashfunctions on
U. By induction, if the pair of each CW-Nashfunction on
U whose graph is contained
in (dP-'/dz)(O) and some minimal polynomial for it has property (Ak - 1) then we say
(g,P) has property (Ak) f o r k being the degree of P in z.
> 1. Property ( A ) is simply property (Am) for m, this
Strong approximation in infinitedimensional spaces
167
Suppose (9, P ) has Property ( A ) . Then we say (9, P ) is of height 0 if (DP-l/&)(O)
nw x U = 0 and, inductively, (9, P) is of height e if it is
not of height
e-
1 and the pair of
each C"-Nash function on U defined by (8P-l / a z ) ( O ) and soiiie minimal polynoinial for it is of height 5 1 - 1. If (g, P ) has Property ( A l ) then 9 can I x esteiitled uniquely to some semi-algebraic neiglibourhood of will be written as
G p ; the
domain of
U in R" satisfying P ( g ( x ) , x)
?jp
= 0.
T h e exteiisioii
will not be specified.
The next lemma then is as follows. Leiiiiiia 11.7. Let D
c R"
be a closed semi-algebraic set contained in U . Suppose the
pair (y, P ) where g E N " ( U ) a.nd P is a polynomial, has property ( A ) . Then 3 closed semi-algebraic neigbbourliood D of D in R" 3 ijp is clefined on D arid
ip
- can be ap-
ID
proximated in the C"-topology by the restriction to D o f a CW-Nashfunction on R". We shall leave it to the reader to read the further details of the proof in [ 5 5 ] .
This Page Intentionally Left Blank
CHAPTER IV Approxiniation problems in probability
We shall turn now to some approximation results in the realm of probability. Again, as in the earlier three chapters, we have confined ourselves to only a few aspects of this
topic, for several reasons, viz. because this aspect of probabilistic approximations fitted in with the topics dealth with earlier in this monograph, and also because several other aspects have been exhaustively dealt with by other authors. For example, there are beautiful expositions of the topics of convergence of distributions and the classical theorems of P. Levy, Khinchin, Prokhorov, Berry-Essen and Kolmogorov, (cf. Gnedenko and Kolmogorov [14], and M. Rosenblatt [50]). Further the topic of weak convergence has been dealt with thoroughly by various authors such as D. Pollard [4S] and others (cf. Ito and McKean [22], and Ito and Nisio [23]). As is well-known by now, probability provides an interesting and extremely useful tool in analysis as also in applied problems. We shall begin this chapter with an interesting and elementary application of probabilistic concepts to obtain a proof of the familiar Weierstrass’ theorem on uniform approximation of a continuous function on a coinpact interval by a polynomial. To be specific we shall present S. Bernstein’s proof of this theorem (cf. [4]) in which he uses his “Bernstein’s Polynomials” (though this is well-known, cf. [13], [51]).
Chapter I V
170
91. Bernstein’s proof of Weierstrass’ theorem It is enough to consider a real-valued function f(x) defined and continuous on tlie unit interval [0,1]. Let Y be a binomial random variable (cf. Appendix 4) of sample size n, that is to say, corresponding to n coin tosses,
2, being
talcen to be the probability
of one success. We then consider f ( Y / n )as one estimate of f(x). This estimate equals
f ( k / n ) for k = 0,1,
. . . , n, with
p,(x)
2
of degree n in
probability ( ; ) z k ( l - z ) ~ - ’ .The Dernstein polynomiol
is defined by:
this being the “expectation”, or the mean value of the variable f ( Y / n ) .The objective then is to show that pn(z) converges uniformly to f(x) as n -+ Let
E
00.
> 0. Then uniform continuity of f on [0,1] implies that 36(~)> 0
[0,1]with Iz - yI
< & ( E ) , we have
If($)
- f(y)
3
Vx,y
< E . Now we apply Chebyehev’s inequality
(cf. Appendix 4), and obtain
5
u2(;)/q2
u2 being the variance of the binomial variable
max{lf(x)I
This completes Bernstein’s proof.
=
x ( l - x)
-,
nq2
Y . Now set 7 = $6( f), and M =
I 0 5 x 5 l}. Then if > f i ,we find n
E
Approximation problems in probability
171
More recently, Bernstein’s idea of using probabilistic techniques in approximation problems in analysis has been pushed further. We shall present a few of these recent results. However before turning to these recent developments we wish to note that Bernstein’s polynomials have been further used, to obtain accurate estimates of the errors of approximation, and these results have proved useful in semigroup theory (cf. Butzer and Behrens [6]).
Chapter IV
172
$2. Some recent Bernstein-type approximation results In the preceding section we saw that for a function
f
E C[O, 11, the Bernstein poly-
nomials n
where Pn,k(Z) = ( i ) z k ( l - z),-'
converge to f(z)uniformly on
be regarded as the result of an operator B, operating on
[o, 11. Here Bnf(z) can
f E C[O,11, for n
= 1 , 2 , . . ..
In
the same context it is known that
where w(6)= sup{lf(z) - f(y)I
I
Iz - yI
5 6, 0 5 z,y 5 l}, 6 > 0 (Popoviciu, cf. [3S]
p. 20). Further suppose f(2k)(z)exists at z, then (Bernstein, cf. [3S] pp. 22-23)
A number of such operators have been introduced after Bernstein. Many of these are special cases of an operator introduced by Feller [13]. Feller's operator is defined as follows. Let {X,, n 2 1) be a sequence of r.v.'s with distribution function ( d
. f.) F:,z(t)
with expectation E X , = z, and variance a ; ( z ) , z
being a continuous real parameter. For a continuous function f on W' define
Approximation problems in probability
Leiiiiiia 2.1.
If CT;(Z)
-+ 0 as n -+
00,
173
then lim L,f(z) = f ( z ) for every continuous n-ca
bounded function f . If further f is uniformly continuous and u k ( z ) -+ 0 uniformly, w.r.t.
x E W’, then the convergence of Lnf(z) to f ( z ) is uniform. Now suppose the continuous parameter z takes values in an interval I (perhaps infinite); let G(z) be a d.f. on I. We then obtain:
Leiiiiiia 2.2.
Suppose ok(z) 5 g(z),where g(.) is G-integrable and that the conditions
of Lemma 1 are valid. Then
The preceding scheme is now modified as follows. Let {Y,, n
2 1) be a sequence of
i.i.d. (independent, identically distributed) r.v.’s with mean z E I and variance 0”s). n
Let S,
=C yi.
Then the above expression for L,f(z) is modified to the following:
i=l
where F,,Z(t) is the distribution function of S,. The following theorem has been obtained by R.A. Khan (cf. [26] p. 195), thus extending, the above-mentioned result of Popoviciu and Bernstein to Feller’s operator defined in (1).
Theoreill 2.3.
Suppose {Y,,n
2 1) is a sequence of i.i.d.
r.v.’s with mean z E I
c W’,
and variance uz(z).Let A =sup ~’(x). Then for x E I the Feller operator defined in ( 1 ) ZEI
above satisfies:
Chapter I V
Ix-yl
5 6,z,y E R1}. firtherrnore, for x
5 yi and X = X (%) = [$I % xl] where [TI Clearly )%(fI f(x)1 5 w(f;6)(1 + A), and hence
Let Sn =
Proof of Theorem 2.3.
E [a,P] c I ,
-
i= 1
denotes the greatest integer ILn(f;Z)
5 T.
- f(x)l
-
I w(f;G)E(l+ A ) I w ( f ; b ) ( l +
{ + -}w(f;
= 1
EX2)
6) .
Now let 6 = n-’lz, and the proof is completed.
As regard monotonic convergence, the following theorems are proved in [26] p. 199 for the Feller operator defined earlier, when the function f(x) is convex.
Theorem 2.4.
Let {Yn, n 2 1) be a sequence of i.i.d. r.v.’s with mean x E I and vari-
ance uz(x). For a continuous convex and bounded function f on R1 defined the Feller
For the proof, the following lemma is needed. n
Lemma 2.5. Then
Let Y1,Yz
. . . be i.i.d r.v.’s with finite expectation.
Let Sn =
C
i=1
yi.
175
Approximation problems in probability
Proof of Lemma 2.5.
We note that E(Y1 ISn+1) = E(Yz(Sn+l)=
I
E(Y,+l (Sn+l); hence E ( K S,+1)
:+: 1 *.
= E('=-
Sn+l) =
. ..
=
Thus
This proves the Lemma. Proof of Theorem 2.4.
We note that
The function f is assumed to be convex; hence using the conditional version of Jensen's inequality and the preceding Lemma, we find
Thus
Now using the very first lemma in this topic (Lemma 2.1), the proof of the theorem is completed. More such results concerning Bernstein type operators are due, e.g., to Katherine Balazs (cf [3]). Some of the results in [3] are as follows. However, these are not exactly probabilistic in nature! For a function f on the positive half axis, with an,b, positive numbers 3 b,
-+
co,
Chapter IV
176
and a, =
% -+ 0 as n
-+ 00,
define the Bernstein type rational functions
Rn(f ;x) by
This particular positive linear operator has been investigated (see the references in [3]). For a function f defined in
(--00,
co),define the nth Bernstein type rational func-
tion RG(f;z) by:
where n
> 0 is even, and
a,, b, satisfying: a,, b, are
> 0,
b,
-+
00,
a, =
-+
0, as
n -+ co. The theorem established is Theorem 2.6.
(cf Theorem 2 [3], p . 196). Suppose f is continuous in
satisfies: f ( x ) = O(ealZI)for some a
for - A 5 x 5 A ; here n o f f in [ - A - E ; A
> 0. Then
for arbitrary fixed A
(-00,
co) and
> 0 and a > 0,
> 0 is even, w [ - A - ~ , A + ~ I ( ~.); denotes the modulus of continuity
+ a ] , and c1 = cl(ar;A; E ) > 0 is a number independent
of n.
Furthermore, a necessary and sufficient condition for the uniform convergence of
R i ( f ) to f is:
f
C[-00, m] where where C[-00, m] denotes the class of continuous
functions f 3 lim f ( z ) exists (finite) and hence f is uniformly continuous on [-co,co]. 14'~ The precise theorem is:
Approximation problems in probability Theorem 2.7.
lim n+m
177
(cf. Theorem 4, [3], p. 197). Suppose b, = np with 0
sup -ca
).(fI
- ~ : ( f ; s )= l 0,
TI
< p 5 $,
Then
* f E C [ - ~ , O O. ]
> o even
A slightly more abstract situation is considered by R. Wittman (cf. [64]). Suppose E is a locally convex space, and A
cE
a convex set. A function f : A
--t
W 1 is uniforinly
continuous if 3 continuous seminorm p on E 3
The result established in the following: Theorem 2.8.
(cf. Theorem 1, [64], p. 463). Suppose f : A
continuous convex function. Then for every E g on E satisfying: sup Ig(z) - f(x)I I zEA
E.
> 0 3 Lipschitz
-+
W' is a uniformly
continuous convex function
Chapter IV
178
$3. A theorem of Steinhaus Probabilistic ideas have been used in a function theoretic context, specifically in the context of random Taylor series. E. Borel, in 1896, formulated the statement that, with probability one, the circle of convergence of a power series with arbitrary coefficients is its natural boundary, i.e. consists only of singular points. This statement is only about plausibility. The correct formulation was given by H. Steinhaus [59] in 1929. The result can be stated as follows. 00
Theorem 3.1.
rneigmzn, where the
The series
&’s
are mutually independent ran-
0
dom variables uniformly distributed on [0,27r], and limsup r:ln n+cn
< 00,
has almost surely
the circle of convergence as its natural boundary. We shall give here an account of a theorem of H. Steinhaus in this context, formulated for Steinhaus series. This account follows Kahane [26], where also one can find references to further related developments. At this point a brief explanation of the probabilistic ideas involved in this theorem, appears to be in order, and the explanation given in the next paragraph appears to be the bare minimum of “probability theory” needed for this theorem. A more general account will be found in Appendix 4.
Probabilistic ideas; random variables
.
3-1 = [0,1] x [0,1] x ...
of infinite dimensions. An event is a subset E
c 3-1 and its probability p(&) is its (infinite
A random point is a point w
= ( w I , w ~ , . .) in the hypercube
dimensional) product Lebesgue measure, if it exists. If p ( E ) = 1 we say that E occurs al-
m o s t always (a.a.), of that & is almost certain. A r a n d o m variable (r.v.) is a measurable function of w , and is always denoted by a capital letter. With any hypothetical property
Approximation problems in probability
179
of a random variable or of a family of random variables we speak of the event of the oc-
currence of this property; we thus speak of the property occurring almost always. The expected value (or expectation) of a r.v. T is E ( T ) =
s . T(w)dw. For any given sequence
{Tk} of r.v.’s if Tk depends only on the kth component
wk,
we have E(II Tk) = IT E(Tk). k
k
The characteristic f u n c t i o n of a r.v. T is E(eiUT)where u E W’; if E(eiUT)= e-”’I2, T is said to be a (real) normal The r.v.’s Tj(j = 1,2,
T.v..
Two r.v.’s TI and T2 are orthogonal if E(TlT2) = 0.
. . . , ) are said to be
independent if 3 a mapping w
-+
w’ of
‘Ft -+ ‘H, which is measure-preserving 3 under this mapping Tj = function of w i a.a. Two r.v.’s T,T’ are said to follow (or to be subject t o ) the same law if T’(w)= T(w’)a.a.
A complex r.v. T is invariant under
r o t a t i o n if €or each
t E R’, T , and Teit are subject
to the same law. If the r.v.’s Tj,j = 1 , 2 , are real normal and independent, then for each
t
E
W’, TI cost
+ T2 sint is normal, and TI + aT2 is called a complex n o r m a l
r.v. A
complex normal r.v. is invariant under rotation. If T is a normal r.v. (real or complex), then a
+ bT is a Laplacian (or Gaussian) r.v., a , b being constants.
We shall need the following lemmas; for the proofs of the first two cf. Loeve [37], and for the third cf. Zygmund [67].
Kolmogorov’s Lemma.
Given an infinite sequence { T j } of independent r.v.’s, and an
event E which for each n = 1,2, . . . , does not depend upon the realisation (i.e. on the values of) Ti, j = 1,2, . . . , n, then p ( E ) = 0 or 1.
Kliinchin’s Lemma. j = 1,2,
If the Tj are real independent r.v.’s, 3 E(Tj) = 0 , E ( T j ) = 1 for
. . . , then the series
cjTj converges a.a. provided
C
IcjI2
< cm.
j
Zygmuud’s Lemma.
If the r.v.’s T,,j = 1,2, . . . , are subject t o the same law,
Chapter IV
180
then the series
c
cjTj
is not summable under any regular process of summability if
j ICjI2
= 00.
j
Random trigonometric series
Here we are concerned with random functions w into Fw(t)mapping tion on the unit circle. Consider a local property of F, e.g. F ( t ) being
IFI into a func-
> 0; or F ( t )
being continuous in t etc. We say at almost all points t , the property holds almost certainly meaning: the property holds almost with probability 1 at all points t except perhaps for t lying in a set of measure 0 on the unit circle. If this happens we say the property holds almost certainly (surely) almost everywhere, (a.s.a.a.). If a property holds at each t almost certainly, it does not follow that it holds almost certainly for all t . It is often much more difficult to determine the probability that a local property should hold everywhere than it is to determine such a probability at any (fixed) point.
A r.v. is ( s t r i c t l y ) s t a t i o n a r y if the probabilities associated with it are invariant under translations t
--+
t - to, that is to say, if to each to 3 transformation
w -+
w’
(see
earlier in this section) 3 F,t(t) = Fw(t - t o ) .To say that a local property should hold a.s., a.e. is saying that is should hold a s . at a fixed point t o . The convolution of a stationary r.v. with a function which is certain (i.e. not subject to any randomness) is a r.v. which is stationary. It is also quite natural to consider formal trigonometric series
C A,eint, with coef-
ficients A , which are complex r.v.’s. Such a series is said to be s t a t i o n a r y if its convolution with anay trigonometric polynomial is stationary. We shall consider here only S t e i n h a u s series F i n which the coefficients A , are in-
Approximation problems in probability
181
dependent r.v.'s which are invariant under rotations. We shall use the notation F
N
CA,eint, so as to be able to use this symbol to identify F with a function, or a distribution, as the case may be.
Steinhaus' theorem We consider properties P of trigonometric series f on intervals, which are subject to the following conditions:
(1") If f satisfies P on
( a ,b ) ,
any translate ft satisfies P on ( a -t t , b -t t ) .
(2") If f satisfies P on two abutting intervals, then f satisfies P on their union. (3") If f satisfies P on an interval, then so does f
+ p , where p is any trigonometric poly-
nomial.
(4")If F is a Steinhaus series, then the statement: "F satisfies P on ( a , b)" is an event for which the probability exists. The theorem of Steinhaus referred to above is as follows.
Theorem 3.1.
(cf (261). Consider a Steinhaus series F and a property P . Then al-
most surely, F satisfies P everywhere or F satisfies P nowhere.
Proof. Let {I,} be a finite collection of intervals, all of the same length the circle, and
xc,, = p[on I,, F satisfies PI,
n = 1,2,
and
x, = p [ F satisfies P on at least one In]
.. .,
E,
which cover
182
Chapter I V
The condition (4") in the definition of the property P implies that x,,,, and xc both exist. Suppose x,
> 0. Now x,
condition (1") implies that, since F is station-
n
ary, all the x,,, are equal, hence xcn
> 0. Condition (3") and the independence of the
An's implies that x,,, = 1 (by Kolmogorov's Lemma). Then condition (2") implies that
p [ F satisfies P everywhere] = 1. The following corollaries are noteworthy.
Corollary 3.2.
Every Steinhaus-Taylor series has its circle of convergence (if it exists)
as its natural boundary.
Corollary 3.3.
If a Fourier-Steinhaus series represents a s . a distribution F , and if
with positive probability 3 interval (dependent on w ) on which F equals a continuous (resp. analytic) function, then a s . F is a continuous (resp. analytic) function, etc. As a special case of the series dealt with in Steinhaus' theorem, we would like to 00
mention Brownian motion, (or Wiener measure) viz. the series
iZnenit, where n=aa
2, = X, + iY,, X,,, Y, being independent real normal r.v.'s. The series is continuous a.s. and as. satisfies a Holder condition of exponent Holder condition of exponent
> $.
< f, but
a.s. does not satisfy a
It is also known that up to normalisation Brownian
motion is the only process { X t } , 0 5 t 5 27r with a.s. continuous trajectories such that the X t are normal and X t - X, independent over disjoint intervals (cf. L. Schwartz [50]). However, in the next section we prefer to sketch a slightly different derivation of Browman motion using the Haar functions.
Approximation problems in probability
183
54. The Wiener process or Browman motion
First we shall state the definition of the Wiener process. In the following T shall denote the closed interval [0,1] or R+ = [0,03). The reader should refer t o the Appendix
4 for the meaning of the standard probabilistic terminology used here in this section.
Definition.
A process X = ( X t ) defined on a complete probability space
(a,A, P ) is
called a Wiener process with variance parameter u2 if it is a Gaussian process with the properties: 1. X o ( w )= 0 (as.);
2. Vs, t with s 5 t , X t - X , has a Gaussian distribution with zero mean and variance
a y t - s); 3. V t i E T ( i = 1 , 2 , 3 , 4 ) 3 tl 5
t2
5
t3
5 t l , the r.v.’s X t ,
- Xtl, Xt, -
Xt, are
independent,
4. for a.a. w , the trajectories t
+Xt(w)
are continuous.
An equivalent definition is as follows; it happens to be a little more convenient in certain ,circumstances.
Definition.
X = ( X , ) is a Wiener process with variance parameter u2 if X is a con-
tinuous Gaussian process with E X t = 0 Vt and covariance function given by
E ( X t X , ) = u2 min(t, s) . A Wiener process with
u2 =
1 is called a standard Wiener process (or standard Brown-
ian motion). First let T = [0,1]. Several methods of constructing a Wiener process on T are known (cf. Ito and McKean (221). The particular method sketched below is based on the
Chapter IV
184
use of the Haar family of functions on [0,1] (cf. Cisielsky [7], P. Levy [35]). However, since the excellent and complete exposition of this method in ICallianpur's monograph
[27] seems impossible t o improve upon, we shall only sketch the highlights of the proof. First we note that a N ( a , a 2 )r.v.
induces a Borel probability measure P on R ,
by the rule: $ ( A ) = P{E-'(A)}, A a Borel set in
W. Now let R
= R", the countable
product of real lines, B(R") the a-field generated by the Borel-cylinder sets in R, and P the countable product of N(0,l) measure on R. Denote by A the completion of B(R") with respect to P. We then introduce the Haar system of functions {gn,j} on [0,1], where n = 1,2, . , . , and for each n , j = 0,1, . . . , 2"-l - 1. These are given by:
These functions are known to form a compute ortho-normal system in L 2 [ 0 1 ,1 Define Gn,j(t) =
sogn,j(s)ds t
=
(xi, gn,j),
where
xt(.)is the characteristic function
of the interval [ O , t ] ,and (, ) is the inner product in L 2 [ 0 ,11. It is helpful to compute the
exact expressions for the functions G,,,j(t). The maximum value of Gn,j(u) occurs at = (j
+ 4)/2,,-',
and max{G,,j(u)
I 0 5 u 5 I} =
2-(,,+1)/2.
Also ;'f
> 1, the
functions Gn,, have disjoint supports. Now let {yn,j} be mutually independent N(0,l) r.v.'s on (0,A, P ) , and define the series
Approximation problems in probability where Sn = { j
I
185
0 5 j 5 2"-l - 1) if n 2 1; So = (0). The series can be written as
Cz=p=, f n ( t , ~ )where f n ( t , ~ )= CjEs,Y n , j ( w ) G n , j ( t ) .Let Y n ( w ) =max 3ESn
Iyn,j(w)I. Then
for any positive number a,, and n 2 1,
where C =
s,"
~ ~ e - " / ~ dasz ,will be seen after some computation. If we choose
an = 2 ( n l 0 g 2 ) l / ~ where ,
the logarithm is the natural logarithm, we find
Hence if we define
then by the Borel-Cantelli Lemma,
converges uniformly in t for w E
C= ;o
f n ( t , w ) when w E
P(R0)
0 0
= 1. It follows that
(and thus, ass.). Finally, define W t ( w ) to be
Ro, and to be equal to 0 Vt when
w
6 no.
It remains to verify that W = (Wt) is a Wiener process on (R, d,P).If w E R o ,
Wt(w)is a continuous function o f t , and thus condition 4 is satisfied. Further the finite dimensional distributions of W are Gaussian. Next
M
Chapter I V
186
[
where we used: E W t ( w ) -
xr=:=, CjEs,
2
y,j(w)G,j(t)]
-+
0 as m
Also EW, =
-i00.
OVt. Thus the conditions of the second definition of the Wiener process are satisfied.
Next we let T = R+. Define
1
h,j(t) = (2/x)'/2
where j = 0 if n = 0 and j = 0,1,
arctant for 0 5 t
)
. . . , 2"-'
- 1 if n
<m,
1 1. The next step is to verity
{hnj)
is a complete ortho-normal system in L2(W+). We can state this as a lemma. Leiiiiiia 4.1. {h,j} is a complete ortho-normal system in L2(W+).
Sketch of the Proof.
where
bab
We note that
is the Kronecker delta, viz.
bab
= 1 if a = b, 6ab = 0 if a
#
b. Thus the
orthogonal property follows. The completeness of this system is verified as follows. If z ( t ) belongs to the class
Lt(O,m) of functions
then / ; ~ ( t ) ~ d t >_ E
L ~ [ om) , H
satisfying:
t(S," %dt)2,
which means that L2[0,m)c Lh[O,m). Further
Ji w h< m.
0 for n = 0 , j = 0, and j = 0,1,
. . . , 2"-l
NOW
let z(.) E L ~ [ om) , 3 :J
- 1 for n
z(t)hnj(t)dt =
2 1. A little calculation shows that
Approximation problems in probability
The system {gnj} is complete in L1[O, 11 i.e. if
3:
E
L'[O,11 and
I87
si 3 : ( t ) g n j ( t ) d t = 0 Vg,j
then z = 0 a.e. This implies that { h n j } is complete in L ~ [ O , C Ohence ) , in L2[0,co).This proves the Lemma. Now let H n j ( t ) =
soh,j(u)du. t
The next lemma gives a convenient estimate con-
cerning the functions Hnj(t). Zn-l
-1
Leiiiina 4.2. j=0
+ i)for t E [O,a],a <
IHnj(t)l 5 ~ 2 - " / ~ ( a
CO.
Proof. Let 6 =
2n-l
-1
G,j(s) 5 2-("+l)/',
Now since
we find
j= 0
< 6*2-"/2(a
1 + ?) .
This proves the Lemma.
L ynj(W)[~;;2-n/~(a Earlier it was shown th&
P(R0)
= 1, and that for w E
0 0 ,
+ 511 1
9
3no(w) 3 b'n 2 n o ( w )
Chapter IV
188
hence
00
and the series n=O
c
ynj(w)Hnj(t) converges uniformly on each interval
j€.S,,
c c
K a for P -
00
&.a. W . Now we define W t ( w ) to be the sum of
n=O
j€S,,
ynj(w)Hnj(t) if w E
00, and
equal to OV t E R+ if w $ 00. Then W = ( W t ) , t E R+, is a Wiener process on (0,d,P ) . This proves the Lemma.
189
Approximation problems in probability $5. Jump processes
- a theorem of Skorokhod
A. Heuristic motivation. In this section we shall consider convergence to a process having at the worst simple conjump discontinuities. Such a process has been latterly described as CADLAG - 1.e. ' tinuous from the right ("continue h droite") and having a left limit ("limites B gauche") at each point. Our objective is to present some work of A.V. Skorokhod (cf. [48], [56], [57]). For basic information concerning Markov chains and Markov processes cf. Rosen-
blatt [50], [51]. The motivation is as follows. Consider a sequence of Markov chains:
(:"',(in',. . .,
(PI, and a sequence of partitions:
of the interval [0,1]. With each of these Markov chains we associate a random process
(p' for t E [tp',t $ l ] , which is a step function with stochastic discontinuities at the points tp'. If &', - tp' 0 uniformly in probability w.r.t. Ic as n and
(("'(t) =
+
+
00
mtx (tcl - tp') + 0, then the process (("'(t)can be considered to be approximately stochastically continuous. We would like to know specific conditions under which (("'(t) will converge, in some appropriately defined sense, as n
+
00,
to a process
( ( t ) which is
a solution of a stochastic equation (cf. [25]), so that under these conditions, (("'(t) can be considered an approximate solution of this stochastic equation. The processes (("'(t)and ( ( t ) ,as solutions of stochastic equations, with probability one do not have discontinuities of the second kind: the (("'(t)are continuous from the
Chapter IV
190
right, and the [ ( t )can be considered to have the same property. We shall consider some convergence results for such processes.
B. The space D[O,11; Skorokhod’s J-topology. We shall define the basic space of functions and a convenient topology on it. Denote by D[O,11 the space of real-valued functions z ( t ) defined on the interval I = [0,1], and having the following properties.
1. at each t E I , the left-hand limits z ( t - 0) exists, and z ( l - 0)
= z(1),
2 . at each t E [0, I), z ( t ) is right-continuous. Skorokhod (cf. [57]) introduced and investigated several topologies on this space
D[O,11, but we shall use only one of these, denoted in [57] by
J1.
For convenience we
shall abbreviate this symbol to J. Before defining this topology, it is convenient to note some simple properties of the functions z ( t ) E D[O,11. These will be stated as lemmas. If a function z ( t ) E D[O:11 has a discontinuity at say to E I , then the magnitude of the jump at this point is clearly
Iz(t0
- 0) - z(to
+ 0)l; for convenience we shall call this
magnitude the “discontinuity” at t o . Lemma 5.1.
If z ( t ) E D[O,11 then for any E > 0, 3 only a finite number of‘values oft
such that the discontinuity at t is
> E.
Proof. For suppose 3 infinite sequence
{tk}
with t k
+
t o , say, and
Iz(tk
+ 0) - S ( t k -
0)l > E , then at t o , z ( t ) would not have a limit either from the right or from the left. Leiiiina 5.2.
Suppose t l , t z , . . . , tk are all the points in I where z ( t ) E D[O,11 has
discontinuities 2
E,
for some E
> 0. Then 36 > 0 3 if It’ - t”I < S and if both t’,t“ belong
to the same one of the subintervals ( O , t l ) , ( t l ,t z ) , . . . , (tk, l), then Iz(t’)- z(t”)(< E.
Approximation problems in probability
191
Proof. For suppose 3 sequences {t:}, {t:} both converging so some point t o and belonging to the same one of the intervals (0, tl), . . . , ( t h , 1) and 3 lx(tL) - “ ( t i ) [ Then t:, t: must lie on opposite sides of t o (otherwise I z ( t k ) - z(t:)l 2 possible!), hence Iz(to+O)-z(t0 -0)l
>
E.
E
>
E.
would be im-
Then t o must be one of t l , t z , . . . , t k ; but this
would contradict the conclusion above that t;, t: belong to the same one of the intervals
Lemma 5.3.
If z ( t ) E D[O,11, then Vq > 0 36 > 0 3 every t E 1 satisfies one of the
inequalities:
This can be stated a little more conveniently as
Lemma 5.3‘.
If z ( t ) E D[O,11, then
The justification of either statement is in Lemma 2. The next lemma is almost the converse of the assertion of Lemma 5.3.
Lemma 5.4.
If a function z ( t ) E D[O,11 satisfies the statement of Lemma 5.3’ then
3?(t) E D[O,11 which coincides with z ( t ) at all its points of continuity. For suppose z ( t ) satisfies the statement of Lemma 5.3’. Then z ( t ) must have, at each t E I , a limit from the left and a limit from the right. Otherwise 37 points tl
< tz < t3
in I 3 Iz(t1) - z(tz)l > 7 and Iz(tz) - z(t3)I
> 0 and three
> 7 . But this contradicts
Chapter IV
192
(*). Hence z ( t ) must equal either z ( t - 0) or z ( t
+ 0).
Setting Z(1) =t-1lim z ( t ) and
Z ( t ) = lim z ( t ' ) . :'+I :'>I
Skorokhod's space D[O,11 and his J-topology on this space are meant to be a generalisation of the space C[O, 11 of continuous functions on I with the uniform topology. The following definition is meaningful for functions ~ ( tE )C[O, 11.
Definition.
The sequence of functions {zn(t)}r=l
converges uniformly t o z ( t ) at the
point to E I if VE > 036 > 0 3
lim
n-oo
sup lzn(t)- z(t)l I<6
<E .
It-to
Facts 5.5. (1) If { z n t } converges uniformly to z ( t ) at every point of some closed set then {znt} converges uniformly to z ( t ) on this whole set. (2) A necessary condition for convergence of {z,(t)} to z ( t ) in the J-topology (defined further on) is: the sequence { z n ( t ) }converges to z ( t ) uniformly at every point of continuity of z ( t ) . The J-topology (defined below) reduces to uniform convergence for continuous functions. The difference appears when we consider the behaviour of z n ( t ) in the neighbourhood of points of discontinuity of z ( t ) . Uniform convergence would imply the existence of a number N such that for all n 2 N the points of discontinuity of z n ( t ) would coincide with the points of discontinuity of z ( t ) . This would mean that if t were considered to be time, then we would have to assume the existence of an instrument capable of measuring time exactly; physically this is impossible. It seems a little more natural to
193
Approximation problems in probability
suppose that functions which we can obtain by small deformations of the time scale are close to one another. Thus one is naturally led to consider Skorokhod's J-topology. Skorokhod's concept of convergence appropriate for jump-processes as also for continuous processes, is as follows.
A sequence { z n ( t ) }
Definition 5.6.
E
D[O,11 is J-convergent to zo(t) E D[O,11 if 3
of continuous monotonically increasing functions 3 X,(O)
sequence {A,@)}
= 0 , A,(I) =
1 and satisfying:
This type convergence will be denoted by:
An alternate equivalent definition is stated below; however a convenient notation should be first stated. For a given E
> 0 write
2'(t) = z ( t ) -
c
[ 2 ( s ) - z(s - O)]
.st
where the understanding is that the sum is over those discontinuities for which z(s) 2(s - 0 ) > E .
Definition 5.6'.
A sequence of functions {z,(t)} in D[O,11 is said to be J-convergent
t o zo(t) E D[O,11 if (a) for every E
>0
3 z 0 ( t )does not have jumps in absolute value equal to
zk(t) + zO(t) - z:(t) for almost all
& sup Iz:(t) - x $ ( t ) l = 0. (b) clim -+On-+m t E ~
t E I;
E,
~ , ( t )-
Chapter IV
194
Now suppose z ( t ) E D[O,11. Then for every C > 0, write
Ac(z(t)) =
min((z(t') - z ( t ) ( ; Iz(t) - z ( t " ) ( )
sup o< 11'
+
1'< 1 " j l 8"J
-
sup O
( I z ( h )- 2(0)(
+ (.(I) - z(1 - h)l) .
Then each function z ( t ) E D[O,11 satisfies:
Now define p c ( z ( t ) , y ( t ) ) V z ( t ) , y ( t ) E D[O,l], and VC E 0
+
sup O
e
inf
< C < 1, by:
[z(s) - y ( s ) l
.
4€[td+%l
This quantity pc(z,(t), x o ( t ) ) is used in the following convergence theorem.
C . Skorokhod's theorem Theorem 5.7.
(i) IfJ- lim zn(t) = z o ( t ) , then lim
lim
C-to n-m
n-m
(ii) If 3 sequence {Cn} with Cn
-+
p c ( z n ( t ) , z o ( t ) ) = 0.
0 3 lim pc, (zn(t),so(t)) = 0, then zo(t) = J- lim n-m
n+ca
xn(t).
Proof. (i) Suppose ~ ( t=)J- lim z n ( t ) . Then from the second definition of J-convergence it n-+m
follows that z n ( t ) -+ zO(t) at every point of continuity of zo(t). Hence, if C E (0, l ):
195
Approximation problems in probability
Hence Ac(zO(t)) -+ 0 as C + 0. Thus we only have to show that: lim
lim
C-+O n-w
Ac(xn(t)) = 0
.
Suppose the Xn(t) satisfy the conditions of Definition 2. Suppose sup Ixn(t) - tl
<
1
g,then &(.n(t))
5 A + ( ( " o ( ~ )+) 3 SUP t
Ixn(Xn(t)) - "o(t)I
and hence
-
lim &(.n(t))
n-w
5 As((zo(t)) ,
and therefore lim
0-0
lim
n-tao
Ac(z,,(t)) Sirno Ag((xo(t)) = 0
.
Thus part (i) of the theorem follows.
To prove part (ii) of the theorem we need the following two lemmas.
Lemma 5.8. for some h
Let
E
> 0, x(t),y(t)
E D[O,11, and z(t1)- s(tl - 0)l
> 0, p h ( x ( t ) , y ( t ) ) < p where p < t E (t' - h, t ' )
=+=
> E . Suppose further,
t. Then 3' E (tl - i,tl + ):
( y ( t ' - 0 ) - "(tl - 0)l
3
< 3p ,
and
Proof of Lemma 5.8. and 3 Iy(sz) - x(s2)l
3sl E
(tl
- $, t l ) and
3s2 E ( t l ,tl
< p . Now Ah(z(t)) < p , hence
+ $), 3 ly(s1) - x ( s l ) l < p
Chapter IV
196
and
It follows that lz(sl)-z(tl-O)l and Iy(sz) - z(tl)l
< p and 1z(sz)-2(tl)( < p . Hence Iy(sl)-z(tl-O)l
< 2p, and thus
hence (Iy(t) - y(t' - 0)l
5
Iy(s1) - y(sl)l
< 211,
> E - 4p > 3p. Further
p . Similarly I y ( t ) - y ( t ' ) l
< p for t E [t',t'
+ h]. The Lemma
follows. Leiiiiiia 5.9.
Suppose the jumps of z ( t ) and y ( t ) do not exceed E , where E > 0 is
given, and suppose for some h
< f, p h ( z ( t ) , y ( t ) ) < p , then sup t
Proof of Lemma 5.9. We shall first show that
Iz(t) - y(t)l
< 2s + 5 p .
197
Approximation problems in probability
Let tl
< t2
and t' E
(tl,t2)
3 if s
< t' and Iz(t1) - z(t')l >
p then Iz(t1) - z(s)l
5 p.
Then Iz(t1) - z(t')l I Iz(t1)
- z(t'
- 0)l + Iz(t' - 0)
- s(t')l
However rnin(lz(t1) - z(t')l; Iz(t') - z ( t 2 ) ( )< p , hence Iz(t') - z(t2)I Iz(t1) - z(t2)I
Similarly
+
I E p .
< p , and therefore
< E + 2P. It1
- t21 < h
+ Iy(t1) - y(t2)I < E + 2p.
If t E [0,1] then 3 t' 3 It - t'I
<
S
and Iy(t') - z(t')l < p . Therefore
This proves the lemma. We shall now return to the proof of Theorem 5.7 (ii).
Proof of (ii) of Theorein 5.7. E
> 0
Suppose pc,(z,(t),
z o ( t ) ) -+ 0 and C,
3 zo(t) does not have jumps equal in absolute value to
E.
<
t2
<
1
.
.
<
tk
0. Choose
Then 3 p 3 p
and zo(t) does not have jumps whose absolute values fall in the interval Suppose tl
--+
[E
5
<
- GP,E
are all the points at which the jumps of zo(t) exceed
+ 61.~1. E
in
absolute value. Let 6 = min (ti+l -ti), where tk+1 = 1. Let the positive integer n be so Ogrgk
large that Cn
<
and pc, ( z n ( t ) , z o ( t ) ) < p . Then in the intervals ( t l -
Lemma 5.2 3 points t!"' 3 Iz,,(t:") - 0) - so(t, -0)l
< 3 p and
%,t + %), by
Izn(tin)- 0) - z o ( t o ) l < 3 p .
Hence
Iz,,(ti"' - 0) - z,(t{"'I > Iso(ti) -
ti - 0)l - 6 p > E .
By Lemma 5.6 we conclude that zn(t) does not have jumps exceeding 2p in abso,, t!"'), (ti"', ti"' lute value in the intervals (ti"' - C
+ Cn). Hence in each of the intervals
Chapter IV
198
ti -
%, t; + %), 3 only one jump of z,(t)
with absolute value exceeding E . Now zn(t)
# tf"'
for any n.
For otherwise by Lemma 5.1 there must be a point t" 3 It' - t"I
< % and
cannot have a jump exceeding E in absolute value at a point t'
zo(t" - 0)l >
E
- 6 p . But, because of the choice of p , the inequality lzo(t") - zO(t'' -
+ Jxo(t") - zo(t"
0)l >
E
(tj-
%,t j + %) would have two points t'
- 6p
loo(t'') -
- 0)l
>
hence for some j , t" =
E,
Then the interval
tj.
and t y ) at which the jumps of zn(t) exceed e
in absolute value, which is impossible. Hence
c
I,(t) - 5 k ( t ) =
( z n ( t y ) - z,(t:n) - 0))
,
tpst and
By Lemma 5.8, and the condition that pc, (zn(t), z o ( t ) ) z n ( t ) + zo(t) at every point of continuity of
Iz,(tin) zn(t)
~ ( t )Hence . as
- 0) - zo(ti - 0)l -+ 0. Also because C,
- zk(t)
-+
zo(t) - z t ( t ) as t
#
ti.
-+
---t
0, it is clear that
---t 00,
0, therefore tf"'
Therefore z k ( t )
-+
Izn(t!n)) - z o ( t i ) / -+
ti. Hence
z t ( t ) for all points t of
continuity of z f ( t ) (as this is fulfilled for xn(t)). We conclude also that condition (a) of Definition 1 is fulfilled for zn(t). We shall now show that when A c ( z ( t ) )< e, &( z ' ( t ) ) t3
< 2.5. If tl < t 2 < t 3 , with
- tl < C, then
provided z ( t ) has no jump exceeding
E
in absolute value in ( t l l t 3 ) ; and if a jump with
199
Approximation problems in probability
absolute value exceeding E does exist at, say, t' E ( t l ,t z ) , then min ( I z c ( t l )- zc(tz)l, ) z c ( t 2 )- zc(t3)1) I ( z C ( t z )- zC(t3)1
+ min Iz(t' - 0) - @)I,
I(z(t')- z(t3)1
I 2Ac(z(t)) ; and a similar inequality holds if z ( t ) has a jump exceeding E in absolute value in Therefore n-m lim A, Now consider
(t2,tS).
( z k ( t ) ) = 0. inf
f
+ I z i ( t ) - zd(t)I. If there is no point ti, ti"'
< ( 6 k + ~ ) P c(zn(t), , zo(t)) . Using the symmetry of ti"' and ti, we conclude that
in C, C +
%), then
Chapter IV
200
Hence, if Cn <
t, and pc, (xn(t), zo(t)) PC, (zt(t),xX(t))
< p , then I (6k
+ ~ ) P (zn(t), c,
X O ( ~ ) ),
By Lemma 5.9
SUP
t
IzE(t) - z;(t)l 5 2~
+ 5(65 + ~ ) P (xn(t),zo(t)) c,
and condition (b) of Definition 5.6' is fulfilled. Hence
zo(t) = J-
lim zn(t) .
n+w
This completes the proof of Skorokhod's theorem.
APPENDIX 1 Topological vector spaces This appendix aims at presenting the basic ideas and results concerning functional analysis which are needed for this monograph. For proofs we refer the reader to e.g., Rudin [52], or Kothe [29]. The letters R and C shall denote the field of real numbers and the field of complex numbers respectively. We shall use the letter K to denote either R or C, and an element of K is called a scalar. A vector space over K is a set X consisting of elements called vectors, and in which two operations, viz. addition and scalar multiplication are defined, with the usual algebraic properties: (a) for every pair of vectors x,y E X , 3 a vector x
+y E X
3
X contains a unique vector 0 (called the zero vector of X ) 3 x to each x E X31 vector -x 3 x
+ 0 = xVx E X , and
+ (-x) = 0;
(b) for every pair ( a ,x), a E K, z E X, 3 vector ax E X satisfying:
(y(z
Note:
+ y) = ax + ay;
(a
+ P b = ax + Px .
the symbol 0 also denotes the zero element in K.
X will be called real if K = R, or complex if K = C. Suppose A c X , B c X,
Appendix 1
202
x E X , and X E K. Then 2
f A means the set
(5
A
+B
{u
XA
means the set means the set
Thus - A denotes the set ( - a
a
I +b I I
fa a E A } ;
{Xu
a E A,b E B};
a E A}.
E A } . Note: 2A may be
# A + A.
A subspace of X is a set Y c X 3 Y is a vector space with the same operations.
Y c X is a subspace if and only if 0 E Y and aY
+ pY c Y
va,p E K .
A convex set in X is a subset C C X 3 tC
+ (1 - t>cc c vt E [O, 11 .
A set B c X is balanced if a B c B V a E K 3 la1 5 1. A vector space X is ndimensional (or has dimension n ) if 3 basis { u l ,
. . . , u,}
in X , i.e. if and only if each
x E X can be uniquely expressed as
A finite damensional X is one which is n-dimensional for some integer n > 0. Before stating the definition of a topological vector space it appears only appropriate to state briefly some standard vocabulary concerning topological spaces. A topolog-
ical space is a set X in which a special collection 7 of subsets (called open sets) is specified, with the following properties:
203
Topological vector spaces
(i) the entire set X is open (ii) the empty set q5 is open, (iii) the intersection of any two open sets is open, and (iv) the union of an arbitrary collection of open sets is open.
A special family T of sets with these properties is called a topology on X . At times for the sake of precision, the topological space which the family
T
turns X into, is denoted
by ( X ,T ) . In a topological space ( X ,T ) , a set E is closed if its complement is open. The clo-
sure of a set E , denoted by E , is the intersection of all closed sets containing E . The interior of a set E , denoted by E", is the union of all the open sets contained in E. A neighbourhood of a point x E X is any open set containing x. ( X ,r ) is called a Hausdorff space and r is called a Hausdorff topology on X if distinct points of X have disjoint neighbourhoods. A set I covering. A subfamily members of 3 x E V
T',
T'
cX
of
T
is compact if every open covering of K has a finite sub-
is called a base for
or equivalently if for any set U E
c U . A family y of neighbourhoods
T
T,
if every member of
T
is a union of
and for any point x E U , 3 set V E T'
of a point x E X is called a local base at
x if every neighbourhood of x contains a member of y. If a topology
T
is induced by a
metric d, we say that d and r are compatible with each other.
A topological vector space (T.V.S.) is a vector space X with a topology
T
on X 3
(a) every single-point set in X is a closed set; (b) the vector space operations are continuous with respect to
A subset E
cX
T.
is bounded if for each neighbourhood V of 0 in X 3s
>0
3 Vt
> s,
Appendix 1
204
E
c tV. Suppose X is a TVS. To each a E X
Translation and multiplication operators.
we associate the translation operator Ta defined by: Tax = a
+ x, x
E
X ; and to each
scalar X we associate the multiplication operator M A : M A X= Xx,x E X. Then these two operators Ta and M A both homeomorphisms of X onto X . This last statement implies the following: every vector topology on X is translation
invariant i.e. a set E
cX
is open if and only if for each a E X , a
+ E is open.
In a T.V.S. X the term local base means a local base of neighbourhoods at 0. Thus a local base of a T.V.S. X is a collection
B of neighbourhoods of 0 such
that every
neighbourhood of 0 contains a member of B. The open sets of X are then precisely those that are unions of translates of members of
B.
A metric d on a vector space X is translation invariant if d(z vz,y,z E
+ z , y + z ) = d(x,y)
X.
The following definition explains some of the types of T.V.S.'s that we might encounter. X here denotes a T.V.S. with topology
7.
Definition. (a) X is locally convez if 3 local base
B
consisting of convex subsets.
(b) X is locally bounded if 0 has a bounded neighbourhood. ( c ) X is locally compact if 0 has a neighbourhood with compact closure.
(d) X is metrisable if (e) X is an F-space if
T
T
is induced by a metric d. is induced by a complete invariant metric.
( f ) X is a Fre'chet space if X is a locally convex F-space.
Topological vector spaces
205
(g) A n o r m on a vector space X is a non negative valued function denoted by
11211,
hav-
ing the properties: ))z))= 0
112
only if
+ YII Ilbll + IlYll
2
=0
VX,Y
EX
.
A vector space X with a norm on X is called a n o r m e d linear space. If a vector space X is normed then d ( z , y ) =
((3 -
yII defines a distance (or metric) on X. If
X is complete w.r.t. this metric, X is called a B a n a c h space. (h) A T.V.S. X is normable if 3 norm on X 3 the metric induced by the norm on X is compatible with the topology on X .
(k) A T.V.S. X has the Heine-Borel property if every closed and bounded subset of X is compact.
Theorem.
If B is a local base for a T.V.S. X then every member of t3 contains the
closure of some member of B. Hence:
Corollary.
Every T.V.S. is a HausdorfTspace.
Theorem.
In a T.V.S. X
(a) every neighbourhood of 0 contains a balanced neighbourhood of 0. (b) Every convex neighbourhood of 0 contains a convex balanced neighbourhood of 0.
Thus Theorem. (a) Every T.V.S. has a balanced local base.
Appendix 1
206
(b) Every locally convex space has a balanced convex local base. Suppose X and Y are vector spaces over the same field K. A mapping T : X + Y is called linear if
T ( a x + P y ) = aTx+PTy V x , y E X and Va, p E K. For a linear mapping we often write Tx instead of T ( x ) . A linear mapping T : X
Theorem.
+K
is called a linear functional.
Let X and Y be T.V.S.S. If T : X
+
Y is continuous at 0 then T is con-
tinuous, and in fact uniformly continuous, i.e., for each neighbourhood W of 0 in Y,3 neighbourhood V of 0 in X 3
y- x E
V
+Ty -Tx
EW
.
For a linear functional on a T.V.S., the following is true. Theorem.
Suppose F is a linear functional on a T.V.S. X , 3 Tx
# 0 for some x
E X.
Then the following four statements are equivalent: (a) F is continuous.
(b) The null-space N ( F ) is closed. (c) N ( F ) is not dense in X. (d) F is bounded in some neighbourhood of 0. The simplest models of Banach spaces are the standard real of complex n-dimensional Euclidean spaces R" or C" over R1 or 43, respectively, normed by means of the usual Euclidean metric.
Topological vector spaces
For example if z =
($
. . . , zn), z;
(21,
E
207
C is a point (i.e. vector) in C" then
llzll
=
112
(zil')
is a norm on C"; likewise if z = (zl,. . . , zn),zi E R' is a point (or 112
vector) in R", then
( ( ~ 1 1=
is a norm on R".
These are by n o means the only norms that can be introduced on W" or C", respectively.
Theorem.
Suppose X is a complex T.V.S., Y is a subspace of X, and dimY = n
where n is a positive integer. Then (a) every isomorphism of Y onto C" is a homeomorphism;
( b ) Y is closed. Theorem. (a) Every locally compact T.V.S. is finite dimensional. (b) If a T.V.S. X is locally bounded and has the Heine-Borel property then X is finite dimensional. Before turning on to some of the most useful type of T.V.S.s we shall mention the general characteristics of a bounded linear transformation (or linear mapping). A linear mapping T : X
4
Y ,where X, Y are T.V.S.'s, is
bounded
if T maps bounded sets into
bounded.
Theorem.
Suppose X , Y are T.V.S.s and T : X
among the following four properties of X ,
If further X is metrisable then
4
Y is a linear mapping. Then
Appendix I
208
so that for a metrisable T.V.S. X , all four statements are equivalent.
(a)
T is continuous;
(b)
T is bounded;
(c) if z,
-+
0 then {Tz,,n = 1, 2, 3, . . .} is bounded;
(d) if xn
-+
0 then T x n -+ 0.
Among the most useful kind of T.V.S.s occurring in analysis are the locally convex ones, for the topological structure of a locally convex space X can be specified by a special family of non negative (non linear) functions on X called semi-norms.
A s e m i - n o r m on a vector space X is a real-valued function p ( . ) on X with the properties:
(4
P(X
+ Y) I P(X) + dY),
(b) p ( a i ) = IaIp(z), Vx,y E
X
(c) p ( i ) # 0 if x
is a norm.
# 0 then p
and V a E K. If further p satisfies
A family P of semi-norms on X is called separating if to each 5
#
0
3 semi-norm p E
p 3 d X ) # 0. If the vector space X is also an algebra, an algebra semi-norm p ( . ) on X is a seminorm which further satisfies
(4
P(X -Y)I P(.)P(Y)
VX,Y E X ,and
(b) if X further has a unit e then p ( e ) is either 1 or 0.
A subset A c X is called absorbing if each z E X lies in tA for some t > 0. Suppose A c X is absorbing; then the Minkowski functional
~ A ( x )= inf[t
I
PA(.)
of A is defined by:
> 0 x E tA] .
Topological vector spaces
We note p ~ ( x )<
00
209
for every x E X . Also the semi-norms on X will be seen to be
precisely the Minkowski functionals of the balanced convex absorbing sets in
X.
A semi-norm p on a vector space X has the following properties. Suppose p is a semi-norm on a vector space X . Then
Theorem. (a) P(0) = 0.
(b)
IP(X) - P(Y)l
(c) P(.)
L P(" - Y).
2 0.
I
(d) {x p ( x ) = 0 is a subspace of X } . (e) The set
Theorem. (a) P A ( x
I
B = {x p ( x ) < 1) is convex, balanced and absorbing, and further p
= pa,
Suppose A is a convex absorbing set in a vector space X . Then
+ Y> 5 / ' J A ( Z ) + PA(Y).
) t p ~ ( xif) t 2 0. (b) p ~ ( t x =
(c) If A is balanced then p~ is a semi-norm. (d) If B = {x
I
p ~ ( 2< ) l}, C = {x
I
pa(%)
5 l}, then B c A c C, and p~ = p~ = p c .
The next two theorems clarify the relation between families of semi-norms on a T.V.S. and locally convex topological structures on X. A family is said to be separating if for each x
Theorem.
P of semi-norms on X
# 0 3 p E P 3 p(x) # 0.
Suppose B is a convex balanced local base in a T.V.S. X . We associate
with every V E 13 its Minkowski functional. Then {pv
V
E
Z?} is a separating family of
continuous semi-norms on X .
Theorem.
Suppose
P
is a separating family of semi-norms on a vector space X . For
Appendix 1
210
each p E P and each positive integer n define the set vp,n
= {x
I P(.)
1
< };
.
Let B be the collection of all finite intersections of the sets VP+. Then
B
is a convex bal-
anced local base for a topology on X which makes X a locally convex space, in which: (a) every p 6 P is continuous, and (b) a set E
cX
is bounded if and only if every p E P is bounded on E .
Remark. (a) The real use of the separating property of the family P of semi-norms in the last theorem is in showing that this property implies that in the T.V.S. X every singlepoint set is closed. However, we should note that this property of a single-point set being closed is sometimes omitted from the definition of a T.V.S. X . (b) If
B
is a convex balanced local base for the topology
T
of a locally convex space X
then l3 generates a separating family P of continuous semi-norms on X , and turn induces a topology
TI
on X. Clearly then
TI
c T ;as a matter of fact TI
P in = T.
(c) Suppose P is a countable separating family of semi-norms on X , then a complicated theorem explains the construction of a translation invariant metric d on X compatible with the topology on X and such that the open balls defined by d are convex. However a much simpler and direct definition of a compatible translation invariant metric in terms of the countable family P = {pi}zl is as follows: let
then d is a metric, invariant and compatible with the topology on X . However the balls which are defined by d need not be convex.
21 1
Topological vector spaces
One last theorem which should be stated here is as follows.
Theorem.
A T.V.S. X is normable if and only if the origin in X has a convex
bounded neighbourhood. The commonly occurring function spaces provide excellent examples of locally convex spaces.
Example 1. The space C(s1). Suppose s1 is a non-empty open set in a Euclidean space R N . It is known that R is the union of a countable family of compact sets 3 each I<, is non-empty and further each I<,
c
(i.e. in the interior of
I<,+l). In fact the sets K , can be defined as follows. Define
U {aEzn) Bl/n(a)}
w n = ~n(m)
where Bn(oo) = {x E
RN
I
llxll
> n}, and let I<,
= C(W,), for n = 1,2,
. . .. Then we
can verify that the Kn’s form one family of compact sets with the properties stated. The space C(s1) is the vector space of all complex-valued continuous functions on s1. The family ’P = {P,,}F=~of semi-norms where
I
Pn(f) = SUP{lf(x) x E Icn}, satisfies clearly pl 5 pz vn
= 132,
* * *
>
5 . . . . The family {Vn}?=l defined by: 1
= { f ~ ~ ( ~ ) ) p n ( f ) < - - )n ,= 1 , 2 , . . . ,
forms a convex local base for C(s1).
Topological Properties of C(s1). C(s1) is a Frkchet space, in which a set E is bounded if and only if every p E and hence is not normable.
P
is bounded on E . This space is not locally bounded
Appendiw 1
212
Example 2.
The space C"(S2).
As in Example 1, let R be a non-empty open set in
Rn. We shall consider the space C"(R) of complex-valued functions f defined in R 3 D" f E C(R) (cf. Example 1) for each multi-index a . Here a multi-index a is an ordered n-tuple: a = (a1, . . . , a,) of non-negative integers ai,i= 1, . . . , n , and with each multiindex a we associate the differential operator
D"
=
(z) . ." . (">an
dxn
1
of order la1 = a1
+. . . + a,;
if la1 = 0 we define D" f = f.
This space C"(R) is customarily endowed with a locally convex topology, as follows. Let {I(n}~?l be a sequence of compact sets in R selected as in Example 1, __
m
3
=
U
I(, and
C
n = 1,2,
n=l
. .. .
The semi-norms p N on C"(R),
N = 1,2, . . . , are defined by
By the general theorem mentioned earlier, this countable family of semi-norms defines a locally convex topology on C"(i2) which also makes it a FrCchet space with an invariant metric as explained above. This topology on C"(R) further has the following properties: (i) for each z E 0, the functional f
+ f(x),
is continuous;
(ii) Cm(R) has the Heine-Bore1 property; (iii) the space is not locally bounded, hence not normable.
Example 3.
Let K be a compact set in R" with non-empty interior. The support
of a complex function f (on any topological space) is defined as the closure of the set
Topological vector spaces
I
{x f ( x )
#
0). DK is defined as the space of
C"
213
functions with support contained in
I-. If R is an open set containing K , we see that D K is a subspace of C"(R), and hence has all the properties which C"(R) has (cf. Example 2). To see that D K satisfies property (iii) (cf. Example 2), we note that the restriction that K"
# 4 implies that
dim(DK) = 00 because of the following proposition:
Proposition. Cm(R") 3
Suppose
4 = 1 on El,
B1
and Bz are closed balls in R" 3 B1
20
outside
c
B,";then 34 E
Bz and generally 0 5 4 5 1 everywhere.
We shall end this survey of concepts and results concerning topological vector spaces with some explanation about linear functionals on a T.V.S. X . The dual space of a T.V.S. X is the space X' whose elements are the continuous linear functionals on X . The next theorems show that there exist lots of continuous linear functionals on a locally convex space.
Theorem.
Suppose
(i) M is a subspace of a real vector space X ,
R satisfies: p ( x
(ii) p : X
+ y) 5 p ( x ) + p(y), and p(tx) = tp(x) if 5,y E X
and t 2 0,
and (iii) f : M
-t
R is linear and satisfies: f(x)
5 p(x)Vx E M . Then 3 linear F : X
-t
R 3
F ( x ) = f(x)Vx E M and - p ( - x ) 5 F ( z ) 5 p ( x ) V x E X . Theorem. 10
$
Suppose M is a subspace of a locally convex T.V.S. X , and xo E X . If
then 3F E X' 3 F ( q ) = 1 but F ( x ) = 0 Vx E M .
This Page Intentionally Left Blank
APPENDIX 2 Differential Calculus in Banach spaces
We shall collect here the main ideas and results concerning the differential calculus in Banach spaces which are needed in this monograph. N o proofs are provided; for the proofs the reader should refer to DieudonnC: Foundations [9]or S. Lang: Real Analysis
[331. All Banach spaces dealt with are assumed to be real, and mappings are supposed to be from subsets of Banach spaces to Banach spaces which are denoted by letters
E , F,G, E l , Ez, etc. Suppose U c E is an open set. A mapping f : U differentiable at x E
U if 3 continuous linear map A : E
4
F is (Fre'chet)
+ F , and 3 map $ defined for
all h E E with sufficiently small norm with values in F , satisfying lim $ ( h ) = 0
h-0
and also
We shall assume here that $(O) is defined and equals 0. Alternatively we can replace the term Ilhll$(h) by a term # ( h ) where
4(.) is a mapping with
the property:
Clearly i f f is differentiable at x then it is continuous at x. Also if the continuous linear mapping A exists satisfying (1) then it is uniquely determined by f and the derivative of f at z and denoted by f'() or
of(.). We write (1) as
I, and
is called
Appendix 2
216
or more simply as
f(.
+ h)
=
f ( x ) + D f ( s ) ( h )+ 4llhll) .
If f is differentiable at each point x E
(3)
U,we say f is dzflerentiable on U.In this case D f
is a mapping:
Df : U + L ( E , F )
U into the space of continuous linear maps E
from
+
F ; thus with each x E U we asso-
ciate the linear mapping D f (x) E L ( E ,F ) . If D f is continuous, we say that f is of class
C1or simply: f
is C'. We can inductively define f to be of class CP if all derivatives
D kf exist and are continuous for 0 5 k 5 p understanding that Dof means f itself.
Properties of the derivative. (1)
Suppose
z E U. Then f
'
If
cE
U c E is an open set, f , g : U
-+
F are mappings differentiable at
+ g is differentiable at x and
R', then D ( c f ) ( x )= c D f ( x ) .
(2)
Chain Rule.
Suppose
U and V are open sets in the Banach spaces E and
F respectively, and f , g are mappings 3 f : U + V , and g a Banach space. Let z E
U ,and suppose f
:
V + G, where G is also
is differentiable at z,and g differentiable at
g(s). Then g o f is differentiable at z and
(3)
Suppose A : E + F is a continuous linear map. Then A is differentiable at
every point of E and D A ( z ) = A ( z ) for every x E E .
21 7
Differential Calculus in Banach spaces Also, if U
c E is an open set, f
:
U + F is differentiable, and A : F + G a
continuous linear mapping, then D ( A o f ) ( x ) = A o of(.)Vx E U , and for every v E E ,
D ( A o f)(z)v = A ( D f ( z ) v ) . (4)
If f is a differentiable mapping from a closed interval [ a ,b] to F with zero
derivative everywhere on [ a ,b] then f = constant on [a,b ] . Before stating further properties we shall explain a suitable theory of integration in one (real) variable for some of the formulae in differential calculus. The concept of integral here is arrived a t as follows. Let [ a ,b] be a closed interval, E a Banach space. Then a mapping f : [a,b] + E is called a s t e p map if 3 partition
and elements
01,
. . . , vn
E
E 3 if t lies in the open interval
( a i ,a i + l )
then f ( t ) =
vi+l,
i = 0,1, . . . , ( n - 1). We then say that f is step with respect to P. The concept of a refinement of a partition is the usual, and if partition P 3 both
f and
f,g are two step maps of
[ a ,b] into E then 3
g are step w.r.t P . We see that the step maps form a subspace
of the space of all bounded maps. We endow this space with the sup norm. The integral of a step map f w.r.t to a partition P is defined by
using the above notation. We see that this definition of the integral of a step map is independent of P, and we write simply linear and llI(f)ll
I(f) or /:(f) to indicate the interval [a,b]. I is
5 ( b - a)llfll. Hence I is continuous, with bound b
- a. Then
I is
extended to the cloeure of the space of step maps by linear extension. If f lies in this
218
Appendix 2
closure, we denote the integral I(f) by
Jab
f is integrable. If a
I:c 5 b, then we verify that
1'1 If a 5 c < d 5 b, then we define
Sed+ Jif
f , and call it the integral and we then say that
=
/cf+pf
Jif = - Scdf,and we verify that the formula J,"f =
holds for any three points c, d, e in any order lying in an interval on which f is
in the closure of the space of step maps
A continuous map is uniformly continuous on a compact set, hence one concludes that the continuous maps of [ a ,b] into E are in the closure of the space of step maps. Thus the integral is defined on continuous maps. Suppose E = El x Ez
X.
' . x En is a product of Banach spaces and a map f
: [a, b] -+
E is an n-tuple f = ( f ~. , . . , f n ) where f; : [ a ,b] -+ E;, i = 1, . . . , n, then
if these integrals exist. If E = R1, and f 2 0 then integrable, and A : E
-+
Jab
f 2 0. Also i f f : [a,b] + E is
F is a continuous linear map, then
1 ,b A o f = A o J,"f .
Now going back to the differentiable calculus, the Fundamental theorem of the calculus holds just as in the real valued case as we find in the next result. (5)
If f : [a, b] + E is integrable and f is continuous at a point c E [ a ,b]. Then
the mapping t -+ (6)
J: f
= d ( t ) is differentiable a t c and its derivative a t c is f ( c ) .
If I = [a, b], a(.): I
+ L ( E , F ) is a
Jbn(qydt =
continuous mapping and y E E , then
1"
cu(t)dt. y
219
Differential Calculus in Banach spaces where the right side means the application of the linear map
Jab
a ( t ) d t to the vector y E
E. (7)
Let U
c E be open and x
suppose the linear segment {x
E U . Suppose f : U -+ F is a C1-map, and
+ ty, 0 5 t 5 1) is contained in U . Then rl
rl
(8)
Let U c E be an open set and x , z E U 3 the line segment I = ((1 - t ) x
1
t z 0 5 t 5 1) c U . I f f : U
-+
+
F is C' then
Higher order derivatives Suppose U derivative
c E is an open set and let f : U
of(.) is a mapping of(.): U
-+
-+
F be differentiable in U . Then the
L ( E ,F ) the space of continuous linear maps
from E to F (which is again a Banach space). Thus the second derivative is defined as the mapping which assigns to each x E U , the derivative of
of, and if this derivative
, exists (denoted by D 2 f ( z ) ) then
The right side here is identified with the space L ( E ,E ; F), the space of continuous bilinear maps E x E (9)
If U
-+
F , and this space is for convenience denoted by L 2 ( E ;F ) .
cE
is open and f : U -+ F is twice differentiable 3 Dzf(.) is continu-
ous in U , then D z f ( x ) is symmetric for each z E U .
Appendix 2
220
And now we define derivatives of higher order than two by induction. If the ( p derivative Dp-'f(.) exists in U then the pth derivative is defined to be the mapping which assigns to each x E U the derivative of DP-'f
at x and if this exists (denoted by
D P f ( x ) ) ,then
-
D p f ( . ): U + L ( E , L ( E , . . .)) . The right side is identified with the space L ( E , . . . , E ; F ) and denoted by LP(E;F ) . P
If D P f ( x ) exists for each x E U and if DPf(.) : U
-+
LP(E;F ) is continuous for each
k = 0,1, . . . , p , then we say f is CP in U . The p t h derivative DP is linear, i.e.,
DP(f DP(cf)
=
+g)
= DPf
+ DPg
cDPf, c being a constant. Also if p = q
1
+
T,
and if D P f ( x ) exists then
D Q D ' f ( x ) = D P f ( x ) = D'D'f(x). (10)
If U c E is open, and f : U
-+
F is of class CP, then for each x E U , the
mapping D P f ( x ) is multilinear symmetric. (11)
Taylor's formula. Suppose U is open in E and
Let x E U , and y E E 3 the segment
(5
f :U
+
F is of class CP.
+ ty I 0 5 t 5 1) is contained in U . Then
where the remainder Rp is given by
Rp
=
1'
(1 - tp-1 DPf( 5 ( p - l)!
+ty)
*
y(P)dt .
We shall conclude this appendix with a result on differentiation of sequences of mappings.
221
Differential Calculus in Banach spaces (12)
Suppose U
cE
is open, and { fn} is a sequence of C'-mappings U
+
Suppose { fn} converges pointwise to f, and also the sequence { D f n } converges uniformly to a mapping g : U
+ L ( E ,F ) .
Then Df exists and = g.
F.
This Page Intentionally Left Blank
APPENDIX 3 Differentiable Banach manifolds We shall collect in this appendix the basic ideas, definitions and results concerning differentiable Banach manifolds which are needed in this book. For more details and proofs the reader should refer to Lang [32], or Palais [47]. Let X be a topological space. A chart in X is a homeomorphism
Definition.
fined on an open set D ( 4 ) = U
cX
4
de-
onto an open set in a Banach space V (or onto an
open set in a half-space of V ) . If
4, $ are charts in X defined on open sets D ( 4 ) = U l , D ( 4 ) = Uz respectively and
U = U1n Uz, then 4,
+ are Ck-related if II, o 4-l
is a Ck-isomorphism of $ ( U ) onto II,(U).
A Ck-atlas for X is a collection A of charts in X which are pairwise Ck-related such that X = (J D ( 4 ) . A complete Ck-atlas is one which is maximal in the ordering by 4EA
inclusion. We shall need the following simple facts which we shall state as Lemmas.
Lemma.
Let A be an atlas for X and
each chart in A. Then
Lemma.
4, 4 charts in X each of which as Ck-related to
4, II, are Ck-related.
Any Ck-atlas A for
X
is included in a unique complete Ck-atlas d viz. d =
{d is a chart in X Ck-related to each chart of A } . d in the preceding Lemma will be refered to as the Ck-completion of the atlas A. Definition.
A Ck-manifold M is a pair (X, A ) where X is a Hausdorffspace and A is
a complete Ck-atlas for X; the underlying topological space X will often be denoted by
M . If p E M , then an element 4 E A with p E D ( 4 ) will be called a chart for M at p .
Appendix 3
224
On the other hand if A is any Ck-atlas for X then for any integer m 5 k, by the
Cm-manifold defined by A we shall mean the manifold ( X ,
A(-)),
being the Cm-
completion of A. If M is a Ck-manifold ( k 2 1) the b o ~ n d a r yof M is denoted by d M , shall mean the set { pEM
I3
chart q5 for M at p mapping D(q5) onto an open
set of a half-space H
c V, with + ( p ) E d H } .
This leads to the next lemmas. Lemma.
If M is a Ck-manifold ( k 2 1) and p E d M , and q5 is any chart for M at p ,
then q5 maps D(q5) onto an open set of a half-space H , with + ( p ) E d H .
Lemma.
I $1
If A4 = (X, A) is a Ck-rnanifold ( k 2 1) then the set {$ $ =
8M
,6 E A}
is a @--atlas for d M .
Definition. If M , N are Ck-manifolds, and f : M p if 3 charts
4
+N
is a mapping, then f is C k n e a r
for M at p , 1c, for N at f ( p ) 3 li, o f o q5-l is
C k at 4 ( p ) ; and f is Ck o n M
i f f is C k near every point p E M . This definition then yields the following theorem.
Theorem.
Suppose M , N are Ck-manifolds, and
f :M
-+
N is a mapping. Then f is
C k on M if and only if 4 o f o 4-l is C k for every choice of charts 4 for M , $ for N. Corollary.
Suppose V is a Banach space, H is a half-space; then the identity map-
pings IV and I H are @-atlases for V for V and H respectively, and thus V ,H are Ckmanifolds in the preceding sense, for any k
Definition.
1 1.
Suppose M is a Ck-manifold and N is a subspace of M . A chart
4 for M
Differentiable Banach manifolds
225
will be said to admit restriction to N if 3 closed linear subspace W
c V 3 41N
is a chart
mapping D ( 4 ) n N homeomorphically onto an open subset of W or a half-space of W .
Lemma.
Suppose M is a Ck-manifold, N a subspace of M , and
which admit restrictions to N , then
Lemma.
I
4
IN, $ I N
4,$
are charts for M
are Ck-related.
If N is a Ck-submanifold of the Ck-manifold M , and l? = {charts 4 for
A4 4 admits
a restriction to N } , then U =
{+I 1 4 N
6
B} is a Ck-atlas for N ; in such a
case we shall denote the manifold obtained by the Ck-completion of the atlas by N , also;
N is called a properly imbedded submanifold. Lemma. map i~ is f
IN
Suppose N is a Ck-submanifold of the Ck-manifold M . Then the inclusion
: N -+ =
M is a Ck-isomorphism. If further, f : M
f o IN. Iff
:
M
+
+
M' is a Ck map then so
M' is a Ck-isomorphism and 0 is open in M , then 0 is a
Ck-submanifold of M . a M is always a Ck-submanifold of M .
Lemma.
Suppose M , N are submanifolds, one of them without a boundary. If
spectively $) is a chart for M ( respectively N ) , then
4
x $ :
4
(re-
D($)x D($)+ (target
space of 4) @ (target space of $) is a chart in M x N . The set of such $ x $ is a C k -
atlas for M x N and the Ck-manifold this atlas defines will be denoted also by M x N .
Tangent spaces
.
Suppose M is a @-manifold (Ic 2 l), and consider triples (U, 4, a ) , where ( U ,4) E
A,
d, the set of charts at p
E
M . We shall
and a is an element of the vector space in
which 4U lies. Two such triples (U, 4, a ) , (V,$, b) are said to be equivalent (-) if 4 - ' ) ( 4 ( p ) ) u = b. This agreement defined an equivalence relation
-
D($ o
(by the chain rule).
Appendix 3
2 26
A tangent vector at p means an equivalence class of such triples under the equivalence relation
w,
and the set of all such tangent vectors at p is called the tangent space of M
at p and denoted by T p M . Each chart ( U , 4) at p induces a Banach space structure on
TpM,which is independent of the chart selected. Suppose U , V are open sets in Banach spaces, then to every mapping f : U which is say C'-smooth ( k 2 l), we associate its derivative D f ( z ) . I f f : X
-+
-+
V
Y is a
CP-smooth mapping of a manifold X into another manifold Y , then by means of charts we can meaningfully define the derivative o f f on each chart at p as a mapping
This map is the unique linear mapping with the property: suppose (U, 4) is a chart at p and (V, 4 ) is a chart at f(p) 3 f ( U ) c V and a is a tangent vector at p represented by u in the chart ( U , 4), then T p f ( a )is the tangent vector at f ( p ) represented by Dfu,v(p)u. This map Tpf is linear and continuous for the Banach space structure placed on T p ( X ) and Tf(P)(Y>. For convenience we shall sometimes write the properties: i f f : X
-+
Y , and g : Y
-+
frp
instead of Tpf . This mapping T has
2 are CP-mappings ( p 2 1) then
and
Tp(id)= id. Two major concepts to be noted are the concepts of i m m e r s i o n and submersion:
Differentiable Banach manifolds
221
Suppose X , Y are manifolds modelled on Banach spaces, and suppose f : X
-+
Y is
a mapping. Let p E X .
(a) f is an immersion at p if 3 open neighbourhood
XI
of p in X 3 the restriction o f f
to XI induces a homeomorphism of XI onto a submanifold of
Y.f
is an immersion
if it is an immersion at every point. (b) f is a submersion at p if 3 chart ( U , $) and a chart (V,$) at f ( p ) 3 $ gives a homeomorphism of U on a product Ul x U, where Ul,U, are open sets in some Banach spaces and 3 the mapping
is a projection. We see that the image of a submersion is an open subset.
The following is a useful criterion for immersions and submersions in terms of the derivative.
Theorem. suppose f : X
Suppose X , Y are CP-manifolds ( p 2 1) modelled on Banach spaces, and +Y
is a CP-mapping. Let x E X . Then
(a) f is an immersion at x if and only if T, f is one-one and its image has a comple-
ment; (b) f is a submersion at x if and only if T, f is onto and its kernel has a complement. The concept of a mapping which is transversal over a submanifold needs to be clarified. A mapping f : X
+
Y is said to be transversal over the submanifold W
cY
is the
following condition is satisfied: Let x E X 3 f ( x ) E W . Let (V,$) b e a c h a r t at f ( x ) 3 $ : V
-+
Vl x V2 is a
homeomorphism onto a product with $ ( f ( x ) ) = (0,O) and $(W n V ) = V, x 0. Then 3
Appendix 3
228
open neighbourhood U of x 3 the composite mapping ( p r o 1C, o f ) : U
---f
Vz
is a submersion.
In particular i f f is transversal over W then f - ' ( W ) is a submanifold of X, since
the inverse image of 0 by the local composite ( p r o II, o f ) is equal to the inverse image of
w n v by$. The following is a characterisation of transversal maps in terms of tangent spaces.
Theorem.
Suppose X, Y are CP-manifolds ( p 2 1) modelled on Banach spaces, and
suppose f : X
--t
Y is a CP-mapping and W a submanifold of Y . The mapping f is
transversal over W if and only if for each x E X 3 w = f (x) E W the composite
is onto and its kernel has a composite.
APPENDIX 4 Probability theory It has been recognised that a satisfactory and rigorous presentation of probability theory can be given only in the setting of modern measure theory and abstract integration. Several excellent accounts of measure theory are now available, starting with S. Saks [53], Halmos [16], Lokve [37] and more recently, in the monograph of Wheeden and Zygmund [65]. We shall assume familiarity with basic measure theory and integration. In this appendix we shall review the basic concepts and results in probability theory which are needed in this monograph. For very elementary explanations of simple probabilistic distributions such as e.g., the binomial distribution arising in connection with a succession of coin-tossing experiments, we refer the reader t o e.g., M. Rosenblatt [51]. The basic setting for probabilistic results is a probability space, i.e., a triple (52, 23,) where 52 is a space of points, 23 is a Bore1 field (sometimes the word a-field is also used) of special subsets of 52 called measurable sets, and P is a probability measure on 23, i.e.,
P ( R ) = 1. The subsets of 23 are the “random events” on which a “probability” viz. P , is defined. A random variable (r.v.) is a function X ( w ) measurable w.r.t. 23; it is thought of as a possible observable in an experiment whose outcome is governed by the measure
P . The integral E(X) =
J
X(w)P(dw)
(if it exists) is called the mean or ezpectation of X . Suppose X is a random variable with a finite absolute moment: E(1x1) <
00.
A
Appendix 4
230
very useful inequality is Chebyshev's inequality: for c
> 0,
It is useful in obtaining bounds on the amount of probability or probability mass in the "tail" of distribution. A variant of Chebyshev's inequality is: if ct
> 0, c > 0, then
provided the expectation on the right side exists. Jensen's inequality is also a useful inequality. Suppose q5 is a real-valued continuous
convex function of a real variable. Convex means for every pair of points x,x E R1,
If q5 is continuous then q5 is convex if and only if for each xo E R1, there is a number X(xo) such that for all x E
W1, X(xo)(x-
50)
I
- d(x0)
.
The inequality of Jensen which we are referring to here, asserts that
Events (measurable sets) A l ,
. .., A ,
are independent if the following is true: writ-
ing A!') - Ai,AI1) = C ( A i ) (i.e., the complement of Ai), we have
Probability theory
23 1
where mi = 0,1, for i = 1, . . . n. An arbitrary family of events ( A a ) a is ~ ~independent
if every finite subset is independent. The Borel-Cantelli Lemma gives us the probability of the set { A i i . 0 . ) of points lying in an infinite number of sets A j , j = 1 , 2 , . . . . The set { A i i.o.} is defined to be the set
n,=, ,u; m
A j , or the superior limit of A , denoted by limsup A,. This lemma then,
is as follows.
Borel-Cantelli Lemma.
Let P(Ai) be the probability of the event Ai, i = 1 , 2 , . . .. If
00
C P(Ai) <
00
then P(Ai
i.0.)=
0. If the events Ail i = 1,2,
. . . are independent,
then
i= 1 00
C P(Ai) = M implies that P(Ai 2.0.)
i= 1
= 1.
There is a basic zero-one law also due to Kolmogorov, somewhat similar t o the Borel-Cantelli Lemma. Some explanation is necessary before this law can be stated. The terminology “An’s (occurring) infinitely often” corresponds to the fact that limsup A , is the set of all those events which belong to infinitely many A,, or equivalently, to some of “the A,,A,+l,
... ,
however large be n”, i.e. the so-called “tail” of the sequence { A , } .
To the “tail” of the sequence { A , } corresponds the “tail” of the sequence
{+A,,}
of their
characteristic functions. More generally the “tail” of a sequence of r.v.’s is explained as follows. Let
{X,} be a sequence of r.v.s, and let B(Xn),B(X,,X,+~), , . . , B(X,,X,+i,
. . .),
B(X,+1, Xn+2, . . . , ) be the sub-a-fields of events induced by the random functions within the brackets. The concept of limsup Z?(X,) is defined precisely as follows. The sequence B ( X n ) ,Z?(X,, X,+1)
. . . is a non-decreasing sequence of a-fields,
a-field over its supremum or union, is B(X,, X,+,
the minimal
, . . .), which is also loosely written as
Appendix 4
232
“sup B(Xm)”.Furthermore, the sequence B(Xn,Xn+l, . . .), B(Xn+l,Xn+2, . . .), ... is m>n
a non-increasing sequence of a-fields, its limit or intersection is a a-field contained in
f?(X,, Xn+l, . . .) however large n may be, and loosely denoted by “limsup B(X,)”. This a-field C is called the “tail a-field of the sequence {X,}”, or “the sub a-field of events induced by the tail of the sequence {X,}”, and its elements are called “tail events”, and functions (finite or not) which are C-measurable are called “tail functions”. Thus limsup n
X,, as also the limits inferior and superior of the sequence {
C Xi} are tail functions, i= 1
while the sets of convergence of these sequences, or the set of convergence of EX,, are tail events. Kolmogorov’s law is the following.
Kolmogorov’s Zero-one Law.
On a sequence of independent r.v.’s, the tail events
have probability either 0 or 1 and the tail functions are degenerate. In our considerations in this monograph the concept of a stochastic process in an important one. Very simply stated, a stochastic process is a mathematical model of a process which occurs in nature. In other words, in non mathematical terms we can describe a stochastic process as a process evolving in time and subject to probabilistic laws. If we make numerical observations as the process continues in time, these observations give some idea of the evolution of the process. Thus it is customary to define a
I
stohcastic process as a family of r.v.’s {x,‘”’ t E
T }
(cf. [51]). Here
range and Xt(w) in practice is the observation at time t E
7 ;the
7
is the “time”
“w” serves to denote
the dependence of the tth observation (or “observable”) on random causes i.e. which are subject to a probabilistic law. In practice, if a concrete process in the natural world is being observed, we can only
233
Probability theory
observe it at a finite set of instants of time, one group of observations being made at instants t l , t z , . , . , t,, say, another group of observations perhaps at instants t:, t i , . . . , t:; and so on. We know the space in which the values of these observations lie. Also we may have good reason to conclude that each finite group of “random” observations, say at
T,
= (tl, t z , . . . , t,), is subject to a definite probabilistic law, associated with a prob-
ability measure pTn. And then we would like to conclude that all these observables observed at different instants of time in varying (finite) groups are actually random variables (as defined earlier) on one probability space in which these finite dimensional measures pTncan be realised. A basic theorem of Kolmogorov describes a condition under which this does really occur. This condition is called the “consistency condition”. Before stating Kolmogorov’s theorem we shall be more explicit about the space S in which our observables take values. S is assumed to be a a-compact Hausdofi space with the Borel field A generated by the topology on S.
T
is the time interval over which the
observables are observed. At each instant of time t E is taken to be the Borel field A. If
7,
= ( t l , ..
. , t,)
T,
S, is defined to be = S, and At
is a finite subset of
T,
then
B,
is
Ati,
the product Borel field ticln
Kolmogorov’s Extension Theorem.
Suppose the range space of an observable to
be a a-compact Hausdorff space S, with the Borel field A generated by the topology on
5’. Assume that for each finite subset
B,
T,
of
T
there is given a probability measure pTmon
regular with respect to the product topology on
n
Sti. Suppose the family { p T n }
t i ETn
of measures satisfies Kolmogorov’s consistency condition:
Consistency condition:
every pair of measures pTn,pT, corresponding t o finite sub-
Appendix 4
234
sets r,,, r,,, of by
7 ,must
agree on the Borel field
0 the empty set, 130 is understood
B,“,.
(Note: here, denoting as usual
to be the trivial Borel field consistency of only the
null set and the whole space.)
Contention:
Then there exists a certain probability space and there exists a stochas-
tic process { X t ( w ) ,t E r } on this probability space realising these finite dimensional measures pr,, ,r,,
c r.
A little more explanation regarding this extension theorem of Kolmogorov appears in order. Let S‘ = n:
St, the product space of points w = (wt,t E
7 ) whose
tth co-
tEr
ordinate is wt E S t . We identify a finite-diemnsional set B E B , on Srn=
II St with
tEr,
the set B x
Thus
B,
n:
tEr-r,
St = B x Sr-ln,and set
is identified with the Borel field of “cylinder sets” B x Sr-ln,B E
S’ which we shall still call B,.
B,
on
Then using the consistency restriction on the measures
{ p r , , } , the definition of p on all sets B x Sr-In,B E
B,
for all finite sets rn of
7,
is
possible because of the consistency condition on the set { p r , , } . Then p is a non-negative additive set function, with p(S’) = 1, on the field 3 =
U B,
obtained by taking the
r,C+
union of the Borel fields B,
for all finite subsets
T,,
of
7.
Next we invoke the following extension result due to Caratheodory.
Caratheodory’s extension theorem: Let u be a non-negative additive set function on a field 3 of sets of a space A, with v ( A ) = 1. If u is continuous at the null set 31 extension V of v(V(B)= v(B)VB E F),on the Borel field
B
0 then
generated by the field 3.
Probability theory
23 5
Using this extension theorem we finally conclude that the additive measure p on
.F can be extended to the Bore1 field
B
generated by
required stochastic process is obtained by defining
as the random variables of the stochastic process.
F,as a probability
measure. The
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BIBLIOGRAPHY Abraham, R. and Robbin, J., Transversal mappings and flours, (1967), W.A. Benjamin, New York. Aron, R. and Prolla, J.B., Polynomial approximation of differentiable functions on Banach spaces, J. f i r Reine Angervendte Math., 313 (1980), 195-210. Balazs, Katherine, Approximation by Bernstein-type rational functions on the real line, Acta Math. Hung., 46 (1-2) (1985), 195-204. Bernstein, S.N., Dkmonstration du theorkme de Weierstrass fondke sur le calcul de probabilitk, Proc. Kharkov Math. SOC.XIII, 1912. Bonk, R. and Frampton, J., Smooth functions on Banach manifolds, J. Math. Mech., 15 (1966), 877-895. Butzer, P.L. and Behrens, H., Semi-groups of operators and approximation, (1967), Springer-Verlag, New York. Cisielski, Z., Holder conditions for realisations of Gaussian processes, Amer. Math. SOC., Transactions, Vol. 99 (1961), 403-413. Desolneux-Moulis, Nicole, Approximation de fonctions diffkrentiables sur certains espaces de Banach, Ann. Inst. Fourier, Grenoble, 21, 4 (1971), 293-345. Dieudonnk, J., Foundations of modern analysis, (1960), Academic Press, New York. Eells, J. and McAlpin, J., An approximate Morse-Sard theorem, J. Math. Mech., 17 (1967), 1055-1064. Enflo, P., Banach spaces which can be given a uniformly convex norm, Israel J. Math., 13 (1972), 281-288. Federer, H., Geometric measure theory, (1969), Springer-Verlag, New York. Feller, W., A n introduction to probability theory and its applications, Vol II, (1957), John Wiley, New York. Gnendenko, B.V. and Kolmogorov, A.N., Limit distributions for sums of independent random variables, (1954), Addison Wesley Publishing Co., Cambridge, Mass. Gohberg, I.C. and Krein, M.G., Basic properties of defect numbers and indices of linear operators, Uspehi Mat. Nauk. 12, 2 (74) (1957), 43-118.
Bibliography
Halmos, P., Measure theory, (1950), Van Nostrand, New York. Heble, M.P., Approximation of differentiable functions on a Hilbert space, C. R. Math. R e p . Acad. Sci. (Can.) V (1983), 179-183. Heble, M.P., Approximation of differentiable functions on a Hilbert space 11, Contemporary Mathematics, Amer. Math. SOC.,Providence, R.I., Vol. 54 (1986), 17-33. Heble, M.P., Approximation of differentiable functions on a Hilbert space 111, Math. Ann., Vol. 282 (1988), 473-484. Hille, E. and Phillips, R., Functional analysis and semi-groups, (Amer. Math. SOC. Colloq. Publ. Vol. 31, revised 1974 c. 1957, Providence, R.I.). Hirsch, M., Differential topology, (1976), Springer-Verlag, New York. Ito, K. and McKean, H.P., Diffusion processes and their sample paths, SpringerVerlag, Berlin, Heidelberg and New York. Ito, K. and Nisio, M., On stationary solutions of stochastic differential equations, J. Math. Kyoto Univ., 4 (1964), 1-75. James, R.C., Some self-dual properties of normed linear spaces, Symp. Infinite dimensional topology, Ann. Math. Studies, Vol. 69 (1972), 159-175. Johnson, W.B., Rosenthal, H.P. and Zippin, M., On bases, finite-dimensional decompositions and weaker structures in Banach space, Israel J. Math., 9 (1971), 488506. Kahane, J.-P., SBries de Fourier alkatoires, SBminaire Bourbaki, 12= annee, 1959/60, no. 200. Kallianpar, G., Stochastic filtering theory , (1980), Springer-Verlag, New York. Khan, R.A., Some probabilistic methods in the theory of approximation operators, Acta Math. Acad. Sci. Hung., Tomus 35 (1-2), (1980), 193-203. Kothe, G., Topological vector spaces I , (1969), Springer-Verlag, New York. Kurzweil, J., On approximation in real Banach spaces, Studia Math., 14 (1954), 214-231. Kurzweil, J., On approximation in real Banach spaces by analytic operations, Studiu Math., 17 (1957), 124-129.
Bibliography
239
[32] Lang, S., Differential manifolds, (1972), Addison Wesley Publ. Co., Reading, Mass. [33] Lang, S., Real Analysis, (1969), Addison Wesley Publ. Co., Reading, Mass. [34] Lesmes, J., On the approximation of continuously differentiable functions in Hilbert spaces, Rev. Coll. M a t . 8 (1974), 217-233. [35] Levy, P., Processus stochastiques et m o u v e m e n t brownien, (1948), Gauthier-Villar, Paris. [36] Llavona, J.G., Notas de Matematica 190, (1986), North Holland Publ. Co. [37] Lokve, M., Probability theory, (1960), Van Nostrand, New York. [38] Lorentz, G.G., Bernstein polynomials, (1953), Univ. of Toronto Press, Toronto. [39] Lusternik, R.A. and Sobolev, V.J., Elements of functional analysis, (1986 c. 1961), Gordon and Breach, New York. [40] Malgrange, B., Ideals of differentiable functions, (1964), T.I.F.R. Studies in Mathematics; Oxford University Press, Bombay. [41] Mergleyan, S.N., Weighted approximation by polynomials, Amer. Math. SOC.Translations, Series 2, Vol. 10, (1958), 50-106. [42] Munkres, J.R., Elementary differential topology, Annals of Mathematics Studies, Vol. 54, Princeton 1966. [43] Nachbin, L., Elements of approximation theory, (1976), Robert E. Krieger Publ. Co., Huntington, New York. [44] Nachbin, L., Sur les algkbres denses de fonctions differentiables sur une vari8t8, C.R. Acad. Sci., Paris, 288 (1949), 1549-1551. [45] Nemirovski, A.M. and Semenov, S.M., On polynomial approximation of functions in a Hilbert space, Sbornik U.S.S.R. 21 (1973), 255-277. [46] Orlicz, W., Bertrage zur theorie der Orthogonal entwicklungen 11, Studia Math., 1 (1929), 241-255. [47] Palais, R.S., Lectures on the differential topology of infinite-dimensional manifolds, Brandeis Univ., 1964-1965. [48] Pollard, D., Convergence of stochastic processes, (1984), Springer-Verlag, New York.
Bibliography
Restrepo, G., An infinite-dimensional version of a theorem of Bernstein, Proc. Amer. Math. SOC.23 (1969), 193-198. Rosenblatt, M., Markou processes, structure and asymptotic properties, (1971), Springer-Verlag, New York. Rosenblatt, M., Random processes, Springer-Verlag, New York. Rudin, W., Fupctional analysis, (1973), McGraw Hill, New York. Saks, S., Theory of the integral, (1937), Hafner Publ. Co., New York. Schwartz, L., La fonction alCatoires du mouvement brownien, SCminaire Bourbaki, Fev. 1958, no. 161. Shiota, M., Approximation theorems for Nash mappings and Nash manifolds, Trans. Amer. Math. SOC.,Vol. 293, no. 1, Jan. 1986, 319-337. Skorokhod, A.V., Limit theorems for stochastic processes, I, Theory of Probability and its Applications, Vol. I, no. 3, 1956, 261-290. Skorokhod, A.V., Limit theorems for stochastic processes with independent increments 11, Theory of Probability and its Applications, Vol. 11, no. 2, 1957, 138-171. Smale, S., An infinite-dimensional version of Sard’s theorem, Amer. J. Math., 87 (1965), 861-866. Steinhaus, H., Uber die Wahrscheinlichkeit dafur, dass der Konvergenzkreis einer Potenzreihe ihre naturliche Grenze ist, Math. Z., t. 31 (1930), 408-416. Stone, M.H., The generalized Weierstrass theorem, Mat. Mag., 21 (1948), 167-184, 237-254. Sundaresan, K., Geometry and non-linear analysis in Banach spaces, Pac. J. Math., 1 0 2 , no. 1 (1982). Sundaresan, K. and Swaminathan, S., Geometry and non-linear analysis in Banach spaces, Lect. Notes Math., Vol. 1131, (1986), Springer-Verlag, New York. Wells, J., Differentiable functions on
CO,
Bull, Am e r . Math. SOC.,75 (1969), 117-118.
Wittman, R., Approximations - und Fursetzungssatze fur konvexe funktionen, Arch. Math. , 45 (1985), 463-467.
Bibliography
24 1
[65] Wheeden, R.L. and Zygmund, A., Introduction to real analysis, (1977), Marcel Dekker, New York. (661 Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. SOC.,36 (1934), 63-89. [67] Zygmund, A., Trigonometric Series, 2nd ed., Cambridge, University Press, 1959.
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INDEX
Analytic Approximation
59,78
Banach manifold Banach space Bernstein'sapproximation problem Bernstein polynomial Brownian motion
l05,221 78,102,108,113 14 170 183
CADLAG process Caratheodory's extension theorem Chebyshev'sinequality k C - fine approximation k C - fine topology Compact-open topology Com pletely regular space Critical point Critical value
189 232 228 63 63,128 4
Expectation
227
Feller's operator First category Frechet differentiable Fredholm map
172 104 63,128 104
Haar functions Hilbert space
184 125,128
4 73
73
244
Index
Independent Invariant under rotations
228 179
Jensen'sinequality J - tOpOlOgy
228 190
Lattice Localisable Locally convex space Locally finite covering
5, 1 1 15 204,210 68
Markov chain Module Multi-index
189 9
43
Nash manifold Nash mapping
155
Partition of unity Probability space Polynomial Proper map
50,65, 100 227 78 106
Quasi-analytic class
26
Random trigonometric series Random variable Reflexive Regular value
156
180
227
115 73
245
Index
Sard property Self-adjoint function algebra Seminorm Space:L’. I’, or CIO,II Stationary Stochastic process Stone-Weierstrass theorem Strong approxi mation Superreflexive
125 6 208 86
Tail of a sequence of r.v.’s Taylor polynomial Topological vector space Transversal mapping
230
Weighted approximation
14
Zero-one law
230
180 230 6 63
115
133 203 73
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