Banded matrices with banded inverses, II.. Locally finite decomposition

Constr. Approx. (1993) 9:263-281

CONSTRUCTIVE APPROXIMATION

9 1993 Springer-VerlagNew York Inc.

Banded Matrices with Banded Inverses, II: Locally Finite Decomposition of Spline Spaces W o l f g a n g D a h m e n a n d Charles A. Micchelli

Abstract. Given two function spaces Vo, V1 with compactly supported basis functions Ci, F , i ~ Z, respectively, such that Ci can be written as a finite linear combination of the F~'s, we study the problem of decomposing V1 into a direct sum of Vo and some subspace W of V1 in such a way that W is spanned by compactly supported functions and that each F~ can be written as a finite linear combination of the basis functions in Vo and W. The problem of finding such locally finite decompositions is shown to be equivalent to solving certain matrix equations involving two-slanted matrices. These relations may be reinterpreted in terms of banded matrices possessing banded inverses. Our approach to solving the matrix equations is based on factorization techniques which work under certain conditions on minors. In particular, we apply these results to univariate splines with arbitrary knot sequences.

1. Introduction The t h e m e we investigate here is the d e c o m p o s i t i o n of u n i v a r i a t e spline spaces on n o n u n i f o r m grids which are locally finite. T o explain w h a t we have in m i n d we review a result from [ M ] on spline functions which m o t i v a t e s o u r s u b s e q u e n t remarks. W e begin with the space Vo of c a r d i n a l spline functions of degree k > 1 with k n o t s at the integers. T h u s a function f is in Vo p r o v i d e d t h a t it is in C k- I(R) a n d on each interval [ j , j + 1), j ~ Z, it is a p o l y n o m i a l of degree at m o s t k. A c c o r d i n g to S c h o e n b e r g IS] every f ~ Vo has a u n i q u e r e p r e s e n t a t i o n in the form (1.1)

f(x) = ~ cjM(x--j), jez

x~R.

H e r e M is the kth degree B-spline given as M = Z * " " * Z, the (k + 1)fold c o n v o l u tion of the characteristic function of [0, 1), a n d c = (cj:j E Z) is some bi-infinite vector with real c o m p o n e n t s .

Date received: October 25, 1991. Date revised: September 10, 1992. Communicated by Ronald A. DeVote. AMS classification: 15A23, 15A24, 39B42, 41A15, 42C15, 47A62. Key words and phrases: Splines, Wavelets, Matrix equations, Hurwitz matrices, Toeplitz matrices, Two-slanted matrices, Matrix factorization, Total positivity.

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It is apparent that the space V1 = {f(2"): f e Vo} consists of all spline functions of degree k with knots on the fine grid 89 Hence 1/1 contains Vo. According to [M] there is an N s V1 of compact support and bi-infinite vectors b, c of finite support such that (1.2)

M ( 2 x - i)= ~ bi_2iM(x--j) + ~ ci_2jg(x-j), jeZ

x~R.

j~Z

This formula leads directly to a locally finite direct sum decomposition of the form

(1.3)

vl = Vo | w,

where W:= algebraic span{N(" - j): j ~ Z}. That is, not only does every f ~ 1/1 have a unique representation as a sum of elements in Vo and W, but also when f is of compact support, it is a finite linear combination of integer translates of M and N. Our goal is to present an analog of (1.2) for spline spaces on nonuniform grids. This will complement the results in [BM] for splin e prewavelets on arbitrary knot sequences. Our point of view is to take Vo as the space of all spline functions of degree k with knots at X.'= {xj:j~Z} where xj < xj+l,j~Z, limj_~_+~ xj = +__oe. To construct 1/1 we choose another such partition Y:= {yj: j e Z} of R such that xj < yj < x~+~,jeZ. The composite partition X w Y gives rise to a spline space 1/1 containing Vo. In this degree of generality we prove (1.3) and give a description of W. Our analysis of this problem focuses on certain matrix factorization issues for bi-infinite matrices which is of interest in its own right. We study this problem in sufficient generality to cover our desired applications to spline functions described above. Another consequence of our analysis is a convenient description of a sequence of functions {0i: i E Z} __ V1 with compact support which are biorthogonal to the B-spline basis of Vo. This is explained in detail later. As a guide to our general results about bi-infinite matrices we first describe how we view (1.2) as a matrix factorization problem. To explain this we need the refinement equation for M, namely (1.4)

M(x) = ~ aj M(2x - j), jsZ

x ~ R,

where (1.5)

a(z):= ~ a S : = 2-k(1 + Z)k+l,

zsC.

jeZ

For an extensive discussion of the solution of multivariate refinement equations and their relationship to stationary subdivision schemes see [CDM]. Since N described above is in V1 and of compact support we can express it in the form (1.6)

N(x) = ~ d i M ( 2 x - j ) ,

xeR,

jsZ

for some bi-infinite vector d = (dj:j e Z) of finite support. Now we substitute (1.4) and (1.6) into the right-hand side of (1.2). Since the integer translates of M are linearly independent I-S], (1.2) is equivalent to the formula (1.7)

6,,z = Z bn-2jal-2J-'}- ~ cn-2jdl-2j ' jeZ jeZ

n,l~Z.

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We need a bit more notation to put this in final form. Given a bi-infinite vector h = (hj: j ~ Z) we associate with it the Hurwitz matrix H = (Hi,j)i,i~ z = (hj_ 2i)i,j~z. Thus (1.7) becomes

B r A + CrD = I.

(1.8)

Therefore our problem is: given a bi-infinite matrix A find bi-infinite matrices B, C, D which satisfy (1.8). Here it is crucial to specify the pattern of zero entries in these matrices. For the matrices A -- (ai,j)i,i~z that are important to us there exist positive integers p, m such that (1.9)

ai,21+pal,2i+m -~ O, iGZ,

and

ai. k = 0

if

k < 2i + p

or

k > 2i + m.

We call matrices which have this property two-slanted. The connection of this condition to Hurwitz matrices corresponding to a finitely supported vector should be clear. Note that by shifting the column index of A by p, we can restrict ourselves throughout our presentation to the case p = 0. We show, see Theorem 2.2, that for certain matrices satisfying (1.9) there exist matrices B, C, D that satisfy (1.8) and are two-slanted. The proof is based on a certain type of matrix factorization of A. Given that factorization of A, the matrices B, C, and D are explicitly constructed. The special case when A is Hurwitz is significantly easier to handle. It is instructive to see the details in this case as it provides some motivation for the general situation and also leads easily to (1.2). Let us unravel (1.7) by using Laurent series

h(z) = Z hJZj.

(1.10)

jez The natural context for what we want to say next suggests that h is in ll(Z), the space of absolutely summable sequences. Thus (1.10) represents a continuous function on A, the unit circle. When h is finitely supported, h(z) is a Laurent polynomial defined on C\{0}. In either case we frequently "split" h(z) as

h(z) -- ho(z 2) -I- Zhl(Z2),

(!.11) where (1.12)

he(z) := Z

j~Z

h2 +eZJ,

e

E := {0, 1}.

N o w we return to (1.7): assume that a, b, c, d are in ll(Z), pick any z ~ A, multiply both sides of (1.7) by z", and sum over n ~ Z. We get (1.13)

zZ= b(z) ~ a,_2jz2J § c(z) ~,, d,_zjZ 2j. j~Z j~Z

Next we choose e ~ E and set l = 2r + e where r ~ Z and also split b(z) and c(z). U p o n simplifying and reordering terms, we derive from (1.13) the following equivalent system of e q u a t i o n s :

(1.14)

(bo(z)

C0(Z)~(a0(z -1 ) al(z-1)~ = (~

\bl(Z)

Cl(Z)Jkdo(z -1)

all(z-l)/

0).

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As a consequence both of the 2 x 2 matrices above are nonsingular for z e A and inverses of each other. If we let

q(z) := ao(z)dx(z) - al(z)do(z),

(1.15)

then it is an easy matter to solve (1.14) for b and c. This gives us

b(z) (1.16) c(z) -

z d ( - z - 1) - - , q(z- Z) za(--z -1) q(z - 2)

If all the vectors a, b, c, d are finitely supported, then q(z) v~ 0 for z ~ C\{0} and, in particular, ao(z), al(z) have no c o m m o n zeros in C\{0}. There is a converse to these observations. Suppose a = ( a j : j ~ Z ) is finitely supported and ao(z) and al(z ) have no c o m m o n zeros for z c C\{0}, then there is a Laurent polynomial d(z) such that q(z) = 1, z ~ C\{0} (see [W]). Hence, in this case (1.16) does give vectors b, c of finite support which satisfy (1.7). Thus we see, given a finitely supported vector a = (aj:j ~ Z), a necessary and sufficient condition for the existence of finitely supported vectors b, c, d satisfying (1.7) is that ao(z) and al(z) have no c o m m o n zeros in C\{0}. G o i n g back to B-splines we see that, to establish (1.2), we must verify that a(z) given by (1.5) has the p r o p e r t y that the corresponding polynomials ao(z ) and al(z ) have no c o m m o n zeros in C. Suppose to the contrary that ao(z) = al(z) = 0 for some z ~ C . Then we write z = ~z and use our splitting formula (1.11) on a(z) to conclude that a(~) = 0. Hence z = - 1 . However, clearly ao(1)al(1) > 0 which is a contradiction. Let us m a k e another c o m m e n t a b o u t (1.8) in the case of Hurwitz matrices which is g e r m a n e to our analysis, We want to observe that (1.8) implies, even when a, b, c, d are in P(Z), the four relations (1.17)

A B r - I,

A C T = O,

DB r

DC r = I.

= O,

TO see this observe that all matrix products appearing in (1.17) have the form R S T where R, S are Hurwitz matrices. It is straightforward to see that RS r is a Toeplitz matrix when R, S are Hurwitz matrices. In fact,

RST = (g~-j)~,j~z, where

gk := F, r~sj+2k,

k~Z.

jEZ

Moreover, since

g(z) = ro(z-1)So(Z ) + rl(z-1)sl(z), for z ~ A we see that (1.17) is equivalent to the identity (1.18)

oo z,,

\do(z_1)

d,(z_l),j\b,(z )

cl(z)]=

(10 o)

Banded Matrices with Banded Inverses, II

267

valid for all z ~ A. However, this is merely (1.14) where we have reversed the order of multiplication of the 2 x 2 matrices. Thus we see that (1.8) is equivalent to the system of equations (1.17) when A, B, C, D are Hurwitz matrices and a, b, c, d are in ll(Z). Our final comment in this section concerns the Toeplitz matrix R S T when R, S are Hurwitz matrices. Suppose cp is a given function in L2(R) of compact support. Let s ktp(2x - k),

O(x) = ~

x ~ R,

keZ

and consider the inner product i, j e Z.

f R cp(x - j)O(x - i) dx,

This is the (i,j)th entry of the matrix R S r where rk := fR r

k e Z.

+ k) dx,

Thus the questions of finding a 0 ~ 1/1 which is orthogonal to Vo or such that the 0(" - i), i E Z, are biorthogonal to the cp(. - k), k s Z, are embodied in the matrix equation R S r = 0 or R S T = I, respectively. This interpretation persists for spline spaces with arbitrary knots as described above and will later lead us to some useful consequences in that context.

2. Banded Matrices with a Banded Inverse

We are concerned with bi-infinite matrices A = (ai, j)i,j~z with the following pattern of zero entries: (2.1)

ai, k = O

if k < 2i

or

k > 2i + m,

for some m s N. According to the remarks in the previous section, we are interested in finding matrices B, C, D such that the relations (1.8) and (t.17) hold. We accomplish this through a factorization procedure which is described next. Let us assume that A satisfies (2.1) and a~, 2i # 0. The idea is to multiply A from the right by an appropriate one-banded matrix so that the support of the rows of A is reduced by at least one. This is indeed possible if we require in addition (2.2)

ai.zi+ 1 ~ O.

In fact, defining the matrix G~I} = r~,ffi, j : i , j E Z by

(2.3)

~tl)

ai, 2i

(I, k) = (2i + 1, 20,

ai,21+ 1

otherwise,

i~Z,

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W, D a h m e n a n d C. A. Micchelli

and setting A(1) : = A G 0 ) = ; ( a i(1) ,j)i,.izz ,

we readily conclude from (2.3) that indeed (2.4)

t~i, ,(1)j =

0,

j<2i+l,

j>2i+m.

Thus, to eliminate the leading element ai, zi from the ith row of A, we have subtracted a suitable multiple of the (2i + 1)st column of A from the 2ith column while the (2i + 1)st column is left the same. Accordingly, G(1) is a one-banded block diagonal matrix with 2 x 2 lower triangular blocks. It is clear that G(~) viewed as a bi-infinite lower triangular matrix has a lower triangular inverse with entries fl,

1= k,

(Gti))[kl:= laa2i'~2i+l,

(2.5)

~0,

(l,k)=(2i + l,2i),

i~Z,

otherwise.

Of course, to repeat this elimination procedure, we have to be sure that now the entries aI')2i+2 are different from zero. The following result describes exactly those circumstances under which successive factorizations are possible. To this end, it is convenient to adopt the following standard notation for minors of a matrix, namely

A(i~ . . . . , i t / = d e t ( a h , j 1 "" \Jl ..... Jl/ \al. jl 9

ai,d,)a -i,,j,

for all i~ < --- < it, j~ < "'" < j~. Theorem 2.1. Suppose the matrix A satisfies (2.1). Then, for every 0 <_ n < m, the followin9 two conditions are equivalent:

(i) i A 2i+r+1

(2.6)

i+1 2i+r+2

--" "-

i+r ) 2i+2r+1 #0

for i e Z and 0 < r < n. (ii) There exist matrices G~ = (g}!)k)l.k~Z,j = 1. . . . . n + 1, such that

,~(J)

(2.7)

where

(2.8)

. - -

~l,k .-

{11 a(j a(j-

- 1) i, 2i + j - 1 1) ' i,2i+j

l=k,

(l, k) = (2i + j, 2i + j - 1),

otherwise,

the matrices A (~ : = A ,

A (r) : = A G 1 1 ) ' ' " G (r) = ( a l ~ } ) i , j e z

Banded Matrices with Banded Inverses, II

269

satisfy (2.9)

for

a!~)=0

j < 2i + r

and j > 2i + m,

forO
The proof is based on the following identification of the pivot entries in

A(r). Lemma 2.1.

Condition (i) in Theorem 2.1 implies condition (ii), and (

(2.10)

i

i+l

A 2i+l+r

,,(r)

"i'21+~+1=

(

.'. i+r "" 2 i + 2 r + 1 i+ r )

2i+2+r

i+ l

...

A 2i+2+r

"'"

2i+2r+

)

1

for r <_ n. Proof. We have already confirmed the case r = 0 through the discussion at the beginning of this section. Suppose therefore that for some r > 1 the matrices G~) are well defined f o r j < r, that the corresponding matrices A (j) satisfy (2.9) f o r j < r, and that the representation (2.10) for the pivot entries is valid for the matrices A(~),j < r. Let us now denote the right-hand side of (2.10) 1,,, ui, t,(r)2 i + r + i" By condition (i) this is well defined. So as soon as we have verified that t~i, 2i+r+l'(r) = ui, 2i+r+l'h(r) we have advanced the induction step and thereby completed the proof. Now note that, by definition of G o), u

al,ti)2z+e(i) = a!Jftl+)e(i),

(2.11)

y

i, / e Z ,

where e(j) :=

1, O,

j odd, j even.

Thus, denoting by A~j) the lth column of A (~), we observe first that (2.12)

='A2i+2r+l,

AO) 2i+2r+1

icE.

Hence (2.13)

i+p A 2i+r+l+p

"" ""

i+r

t

2i+2r+1

Atl) {, i+ p \2i + r + l + p

"'" ...

i + r "~ 2i+2r+lJ'

p=0,

1.

p=0,

1.

On the other hand, (2.9) says that ai+r.j (1) = 0 f o r j < 2i + 2r so that (2.14)

Ao)(

i+ p 2i+r+l+p = A(') (

"" ...

i+ r

i+ p

\2i+r+l

)

2i+2r+1 ""

+p

--.

i + r-

l)

(1)

2i+2r Jai+"2i+2"+"

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W. Dahmen and C. A. Micchelli

(1) ~i+2r+ 1 must be nonzero whence we conclude that By condition (i), ai+~,~ A(x) f

(2.15)

1,(~) c,i, 2 i + r +

i

\ 2i+1+r

l --~

A
i+ l

""

2i+2+r

...

i+l

"'" -"

\ 2i+2+r

i + r - - l'~. 2i+2rJ

i+r-l'~. 2i+2r J

~ ( 1 ) + 1 w e recognize (2.15) as a representation of type (2.10) relative Defining ~i,j"= -i,j to A. Since A ~) = .4(~- l) = AG (2)-." G (r) this confirms the claim by induction. 9

To complete the proof of T h e o r e m 2.1 it remains to show that (ii) implies (i). Again note that the existence of G ~) implies ai,2i+l ~ O, which is (i) for r = 0. Suppose, now, that (2.6) holds for j < r. Thus the right-hand side of (2.10) is well defined for r. Denote it again by h~r) n + 1, u i , 2 i + r + 1 . Since. the factors . . G ~ j = 1, are assumed to exist we may repeat the arguments in the proof of L e m m a 2.1 to show again .,(r) Moreover, since G ~'+~) exists, as long as t~i,2i+r+l = ~.(~) ~/,2i+r+lr -<- n < m, the pivot entry ~'~') must be different from zero. Hence ~i, 2i+r+ 1 i A 2i+1+r

i+1 2i+2+r

... ...

i+r

2i+2r+1

)

50.

This completes the proof of T h e o r e m 2.1. Corollary 2.1. (2.16)

I f (2.6) holds for n = m - 1 we have A G (l)... G ('~ = E ~m),

where the matrices G u) are given by (2.7) and

(2.17)

E(m) := (al,(m) 21+m~k, 2l+m)t, keZ"

Furthermore,

(2.18)

. a(m) i, 2i-rm

.~(m- i)

= t*i,2i+m ~

O,

ieZ.

N o t e that Corollary 2.1 has the following interesting consequence. Recall that a matrix A is called totally positive if all its minors are nonnegative. Corollary 2.2.

L e t A satisfy (2.1) and (2.6) with n = m - 1 where in addition all the minors appearing in (2.6) are positive. Then A is totally positive.

Proofi Since the entries in the subdiagonal of G u) are quotients of pivot entries (2.10) readily implies, in view of the hypotheses, that the entries in the subdiagonal of G u) are nonpositive. It is easy to conclude from (2.5) that (G(J))- 1 is totally positive. Since E (m) is trivially totally positive, (2.16) means that A is a product of totally positive matrices and is therefore itself totally positive. 9

Banded Matrices with Banded Inverses, II

271

The relation (2.16) now leads us to the construction of the matrices B, C, D. To this end, set ~(m)

(2.19)

(m)

= ((at, 21+m)

- 1

{~k,2t+m)t, keZ,

and observe that E(m)(s

(2.20)

T = I.

Thus by Corollary 2.1 B r := G~I)... G(m)(~(m))T

(2.21) satisfies

A B r = I.

(2.22) Next consider the matrix

Itm) : = (6k, 21 + re)l, ke Z

and note that (2.23)

gCm)(itm+ 1))T = 0.

Again, in view of Corollary 2.1 and (2.23), the matrix (2.24)

C r := G~I)... G~,,)(i~m+ 1))r

satisfies (2.25)

A C r = Etm)(i(,,+ 1))r = 0.

Moreover, let D := I tin+ 1)(Gem))-1... (Gm) - 1

(2.26) so that (2.27)

D C r = I.

Finally, because of (2.23), (2.28)

D B r = O.

The interrelation of the four matrices A, B, C, D can now be summarized as follows. Theorem 2.2.

Suppose A satisfies the assumptions of Corollary 2.1. Then the matrices B, C, D given by (2.21), (2.24), and (2.26), respectively, have the followin 9 properties:

(i) A B r = I,

A C r = O,

D C r = I,

(ii) B T A + C r D = I.

D B r = O.

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W. D a h m e n a n d C. A. Micchelli

(iii) bi,~=0

if j < 2 i + m

cld=0

if j < 2 i + m + l

di,j=O

if j < 2 i + 2

or

j>2i+2m--1, j>2i+2m+l,

or

or j > 2 i + m + l .

Furthermore, for m >>_2,

bi, zi+mbi, 2i+zm_ 1 # O, Ci, 2i+m+lCi, 2i+2m+lCi, 2i+2 m ~ O~

and

di,2i+ 2di,2i+m+ i --/=O.

Proof. The formulas in (i) have been established above. As for (ii), note that (2.16), (2.21) and (2.24), (2.26) yield (2.29)

B r A + C r D = G(1) . . . G~m)((ff(m))TE(,~ + (1tin+ 1))rI0~+ i>)(G0~))- i... (G(1))- i.

It can be easily verfied that ((ffJm))TE(m))i'J =

0, 1,

((I(m + t)lrrO,+ 1)].. = ~0, """J II,

i Cj, i=j

and

i # j, i=j and

m--ieven,

m-iodd,

so that (2.30)

(~t,,))rEtm) + (i(m+ 1))tit,,+ i) = I.

Hence, (ii) follows from (2.29) and (2.30). The proof of (iii) is based on the zero-entry pattern of the matrix [G] m:= G~l~... Gt,,). It follows by induction that (2.31)

[G]7,2~+~ = 0 tn

[G]j,2i+m+ ~ = 0

if j < 2i + m if j < 2 i + m + l

or j > 2i + 2 m - 1 , or j > 2 i + 2 m + 1 .

Also we note that (2.32)

[G]~',+,,.2,+,,[G]~',+2,,- 1,2i+m ~ 0

and (2.33)

[G]n~i+m+l,2i+m+l[G]n~i+2m+l,2i+m+l[G]r~i+2m, 2i+m+ 1 ~ O.

Moreover, we infer from (2.21) that bi,j = (a,,2i+=) (m) - 1 [G'lj, m zi+m.

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273

Likewise we obtain m

ci.~ = [G]j,21+m+x

so that the first two assertions of (iii) follow by (2.31). Furthermore, since (GU))- ~ has the same zero-entry pattern as G tj) it follows inductively that H m .'= ([G]m) - 1 satisfies (2.34)

HT~+~+I,j=0

ifj<2i+2

or j > 2 i + m + l ,

while m

(2.35)

ttl

H2i+m+ l,21+m+ lH2i+m+ l,2i+ 2 ~ O.

Since (2.26) gives di,j = H'~i+m+ 1,j

the rest of the claim in (iii) follows from (2.34) and (2.35). It is clear from the above construction of B, C, and D that (ii) has other solutions which are likewise two-slanted. In fact, we easily verify the following observation. R e m a r k 2.1. Suppose that M and K are any banded matrices such that K has a banded inverse. Then, for A, B, C, and D as above, the matrices/~, (~, a n d / ~ defined by

/~T := B T + C r M ,

(~T := C T K ,

/~ := K - I(D - - M A ) ,

are two-slanted and also solve (i) and (ii) in Theorem 2.2. The relations in (i) above may be interpreted as follows: Let -4 = (~i.j)i,j~z be defined as the matrix whose (2i)th row is the ith row of A and whose (2i + 1)st row is the ith row of D. Likewise let/~ denote the matrix whose (2j)th column is the jth column of B r and whose (2j + 1)st column is the jth column of C r, i.e., (2.36)

yak,j,

i = 2k,

k~Z,

)~dk,j,

i = 2k + 1,

Sbk.i,

j = 2k,

lCk, i,

j = 2k + 1,

k ~ Z,

and (2.37)

k~Z, k e Z.

Then A and/~ are both banded matrices satisfying, in view of (i) and (ii) in Theorem 2.2, (2.38)

A B = B A = I.

Moreover, part (iii) of Theorem 2.2 says that A has only nonzero elements in its rightmost diagonal, i.e., a2i. 2i +m # 0 while the leftmost band of .4 has alternately zero and nonzero elements, i.e.,~2i+ 1, zi + 1 = O, a2~. 2~ # O. B r has a similar pattern, i.e., bi+m, i ~ O, b2i+ 2m,2i -~- O, bzi+ i + 2m, 2i+ l =/=0, i ~ Z . Furthermore, (2.16) and

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W. Dahmen and C. A. Micchelli

(2.26) imply that (2.39)

l~(rn)21+m, |"t, (.~G~I)'"G~m))i,j=~I,

if ( i , j ) = ( 2 1 , 2 l + m ) , l~Z, if ( i , j ) = ( 2 1 + l, 2 1 + m + l), otherwise.

0,

l~Z,

Again, since ( [ G ] " ) - 1 is totally positive if the minors in (2.6) are all positive, we arrive at the following conclusion.

Corollary 2.3. A is totally positive/f(2.6) holds with n = m - 1 and all the minors appearing in (2.6) are positive. 3. Hurwitz Matrices In this section we m a k e a brief c o m m e n t on the special case of a Hurwitz matrix (3.1)

i,j~Z.

ai, j = aj-2i,

In this case at, 2~ = ao, a~. zi + 1 = al, i ~ Z. We call a matrix T a 2 x 2 block Toeplitz matrix if T~,j = Ti+2,j+2. T c a n also be written as T = (T/_j)i.j~ z where each T~ is a 2 x 2 matrix. It is interesting to note that when H is a Hurwitz matrix and T is a 2 x 2 block Toeplitz matrix, then H T is a Hurwitz matrix. Also we note in passing that H r K is a 2 x 2 block Toeplitz matrix if b o t h H, K are Hurwitz matrices. U n d e r assumption (3.1) we see from (2.3) that G (1) is a 2 x 2 block Toeplitz matrix. Thus A ~1) = AG ~1) is a Hurwitz matrix. In fact, al.j (1) = a ~ 2 / where, according to (2.3), ~ai, i odd,

a! 1) = ~ala~ - ai+ l a o

i even.

al Hence, inducitively, G 12). . . . . G tin) are also 2 x 2 block Toeplitz matrices and therefore so too is [G] tin) .'= G (1).." G tin). Since/~(m) and I (m+ 1) are H u r w i t z matrices we conclude that B, C, D are all Hurwitz matrices, that is, bi,j = b j - 2 i ,

ci,j = c j - 2 i ,

di, j = dj_2i ,

i,j~Z.

Hence the matrices A,/~ of Section 2 are in this case 2 x 2 block Toeplitz matrices, i.e., they have the form

: (~i--j)i,jEZ'

B = (Bj_j)i,jez,

where Ai, Bi, i E Z, are 2 • 2 matrices. Define the symbol of ,4 as

A(~) =

Y, i S "

jzZ

Since ~ is a b a n d e d matrix, zi(z) is a 2 • 2 matrix of Laurent polynomials. The relation (2.38) means that (3.2)

A(z)B(z)=(10

01).

Banded Matrices with Banded Inverses, II

275

Thus /~(z)= A(z)-i = .~-l(z) is also a Laurent polynomial and det .4(z) is a constant multiple of some integer power of z. According to (3.1) and our notational convention (1.11) and (1.12), we have

A1, l(z) = ao(z),

A,, 2(z) = al(z ),

2~, ~(z) = do(Z),

"4z, 2(z) = dl(z).

B~, l(z) = bo(z),

BI,z(Z) = bl(z),

9~,,(z) = Co(Z),

9~,2(z) = el(Z).

Similarly,

In these terms the above results correspond to our discussion in the introduction. However, there is a difference concerning the hypotheses that we require. For an application of Theorem 2.2, we need the determinental inequalities (3.3)

A(

0,1...n n+l...2n+

) 1 #0,

n=0,1 ..... m-l,

while in the introduction we assumed that ao(z) and al(z) have no common zeros in C\{0}. In the ease that (aj:j~Z), aj = 0, j < 0, o r j > m, aoam ~ O, is a P61ya frequency sequence, i.e., the matrix (a~-j)~.j~z is totally positive, condition (3.3) is indeed satisfied. To see this, we just have to recall that a minor of (a~_j)/.j~z is positive if and only if the diagonal entries of that minor are positive. This easily allows us to verify (3.3). It is also instructive to observe that in this case ao(z) and al(z) do not have common zero in C. The proof of this uses the fact that ~j~z aj zj has only negative zeros (see [K]). Keeping this in mind, the proof we used for the special P61ya frequency sequence defined by (1.5) works in general.

4. Spline Spaces with Nonuniform Knot Sequences We now turn our attention to an application of our study of the matrix equation (1.8) to locally finite decompositions of spline spaces. Specifically, we consider the following situation. Let X = {xl}~z, Y = (Y/}~z be strictly increasing knot sequences such that xi
Vi(x) ' =

(t/+k+

- - Xi) 1 - 1 ] P [ x i , X i + 1 . . . .

1 --

ti) 1 - 1]P[ti, t i + 1 . . . .

, X i + k + 1"](" - - X) k ,

, gi+k + 1](" - - X) k ,

respectively, where xt+ := (max{0, x}) / and 1 < p < ~ . Thus Vo:= span{Ci: i ~ Z},

V1 := span{F/: i ~ Z},

W. Dahmen and C. A. Micchelli

276

and it is clear that Vo c V~. In particular, we may write

c,(~) = Z a,,jUx),

(4.1)

j~Z

where the coefficients aij satisfy (4.2)

a~j=0

iff j < 2 i

or

j>2i+k+l

(see Lemma 4.1 below). Here we are using the fact that the B-splines C~, F~ are nonzero only on (xi, xi +k + 1), (ti, ti + k+ 1), respectively. The matrix A = (a0i,~z

is known to be totally positive and, ~s we shall see, satisfies the conditions (2.6). To this end, we specialize a result from I-J] to our situation. For each i ~ Z, the coefficients {aij}jEz, as a function of j, are referred to as the kth degree discrete B-spline. Jia labels these coefficients {fli(J)}j~z. He deals with two arbitrary knot sequences X and T where X _ T (called t and z, respectively in [J]) and allows for splines with multiple knots. Even in this most general case he proves, see Theorem 1 in [J], that the matrix (fl~(J))~,j~z is totally positive and identifies all positive minors. See also [Mo] for recent improvements and commentaries on the results in [J]. This result gives in the present situation Lemma 4.1 [J]. For X , T, and A = (aij)ij~z as above we have for every il < "'" < ir and j l < "'" <•, A

,J,/> 0

if and only if the following conditions are satisfied: (i) ai,.j~ > O for all l = 1, 2 . . . . . r. (ii) I f jl is even and l > k, then . Jl-k-1 < Jl -- k -

1.

Consequently, we obtain (4.3)

( i A 2i+r+l

i+l 2i+r+2

... ...

i+r 2i+2r+1

) >0,

r = l . . . . ,k.

In fact; by (4.2) ai+t,2~+,+l+ 1 > 0 for l <_ r _< k, so that condition (i) in Lemma 4.1 is satisfied for l < r < k. Moreover, since for r < k we have 2i+ 2r-k-1

<2i+r+1,

condition (ii) in Lemma 4.1 does not impose any constraints on the minors appearing in (4.3) so that our claim follows from Lemma 4.1.

Banded Matrices with Banded Inverses, II

277

Hence, Theorem 2.2 applies and provides the following result. Theorem 4.1. For A as above there exist matrices B, C, D satisfying (i), (ii), and (iii) in Theorem 2.2 for m = k + 1. Define (4.4)

~i(x) = ~_, di.jF,(x). jeZ

Then Ft(x) = ~ bj3 Cl(x) + ~ cja$j(x).

(4.5)

jEZ

\,

j~Z

Moreover, defining W:= span{~j:j ~ Z}, we have (4.6) Proof.

Vx = Vo@ W, We first verify (4.5). By (4.1) and (4.4) we obtain

bj, tCj(x) + ~ cj, z~j(x)= ~ I ~ bj, laj, r + ~ cjddj, rtFr(x) jcZ

r~Z [.jeZ

j~Z

jeZ

)

= ~,, (BTA + CrO)l,,Fr(x) r~Z

= Ft(x), where we have used (ii) in Theorem 2.2 in the last step. As for (4.6) suppose

o= E

E

icZ

i~Z

By (4.1) and (4.4) this means 0 = Y, (i,~z 2iai'j + ~ I't~di'j)FJ(x) jeZ

iEZ

which, in view of the linear independence of the B-splines F j, implies (4.7)

AT2 q- DT# = 0,

where 2 = {2i}i~z, # = {/~i}i,z. Now we can use (i) in Theorem 2.2 to conclude, upon multiplying (4.7) from the left by B, that 2 = 0. Likewise, multiplying (4.7) by C, yields # = 0. This completes the proof of Theorem 4.1. 9 We comment next on the stability of the functions ~kj. To this end, we let, for any bi-infinite matrix G, IlGl[p:= sup [Iafllie where IlflltP := Ej~z Ifjl p.

278

W. Dahmen and C. A. Micchelli

Proposition

4.1. Given any p, 1 < p < ~ , there exists a constant 7 independent of the knot sequence T such that

T [,f{,le
Ilclt~

Proof.

fetp.

Let S(x) = }-',i~z fiOi(x) and set g = Drf. Then, by (4.4),

S(x)-~ ~" gjFj(x) jez and the.well-known stability estimates for B-splines (see, e.g., [Sch]) assure that there exists a constant 7 independent of T and g such that

(4.8)

~llgll,, --- IISII, -< Ilgl[,~.

Clearly, we have

Ilgll~p<

(4.9)

[IDrllpllfilt~,

while, on the other hand, condition (i) in Theorem 2.2 says Cg = C D r f = f s o that

(4.10)

Ilfllz~ <

ILCllpllgll,~.

Hence, combining (4.8), (4.9), and (4.10) proves our claim.

9

It is interesting to note that we can reinterpret (4.6) as an orthogonal decomposition relative to some Hilbert space. For this purpose we assume that the matrices J , /3 act as bounded linear operators from 12(Z) into /2(Z). Let us define the matrices Ao = (ai,i)i,j~ Ao z and 13o = (d~.j)i.~z o by (4.11)

t~o..= ~ak,j,

"~"

(0,

i = 2k,

keZ,

i=2k+l,

k~Z,

and (4.12)

~oj := {0, dk.j,

i = 2k, k e Z, i = 2k + l, k e Z,

so that Ao + I3o = ~ and therefore, in view of (2.38), 2 r = ,~r3.~ + 2r13o/~

holds for any sequence 2. Thus (4.13)

Vo := {),rAo/~: 2 r e 12(Z)},

!~ := {2r13o/3:2 r s/2(Z)}

are subspaces if 12(Z) and we have (4.14)

Vo ~ I~ =/2(Z),

Vo j- !fie,

relative to the standard inner product on 12(Z). For every fa = ~ j , z 2~Fj we define the inner product = 2rBBr2.

Banded Matrices with Banded Inverses, II

279

Moreover, let us define by V~o2), W (2) the L2-span of the functions Ci, q/i, i t Z, respectively. Specializing Proposition 4.1 and (4.8) to p = 2, we observe that fz ~ V~o2) and f , ~ W ~z) if and only if 2 = A~'q and # = bor~ for some sequences r/, ~ ~/2(Z), respectively. Hence for such fx, fu we have (fz, fu> = ~/rAoBBr/5~'r = 0, by (4A3) and (4.14), which means V~o2) _1_<..>W ~2).

Our final remarks concern another application of Theorem 2.2. For this purpose we let (f, g):= [ - f(x)g(x) dx. .h~ We construct functions Bi(x) = ~ bidFj(x),

Ki(x) = Z c i , f , ( x ) ,

j~Z

j~Z

and Ri(x) = ~ didF~(x) jeZ

in V1 which satisfy

(4.15)

(Ci, Bj) = 61,i,

(Ci, K s) = O,

i, j 6 Z ,

(K,, Rj) = 6id,

i, j 6 Z .

and (4.16)

(Bi, R 9 = 0,

To this end, define the matrix F = (Yi.j)i.j~z by Yi,i := (Fi, F j) and note that, by (4.1), (4.15) and (4.16) are equivalent to the matrix equations (4.17)

A F B r = I,

A F C T = O,

CFD r = I,

B F D r = O.

We observe next that, again by (4.1), the entries of the matrix ~ := AF = (~i,j)i.j~z are given by ai,j "~- (Ci, F j).

To construct solutions of (4.17) we verify that A satisfies the hypotheses of Theorem 2.2. First note that ~,j = 0 for j < 2 i - k or j > 2i + 2k + 1 and is positive otherwise, The Cauchy-Binet formula combined with a result from [K], which states that the B-spline collocation matrices M := (Ci(xj))i,j~z, F .'= (Fi(xj))i,j~z are totally positive when xl < X~+l, i ~ Z , proves that A is also totally positive. Furthermore, using the fact that any minor of M or F is positive if and only if the diagonal entries of this minor are positive [B], we see that .....

\J1 . . . . . Jp/

280

w. Dahmen and C. A. Micchelli

i 1 < -.. < ip,jl < ... <jp, if and only if a~r,jr ~ 0, r = 1. . . . . p. Thus, A satisfies the hypotheses of Theorem 2.2 for m = 3k + 1 and so there exist matrices Gu), j = 1. . . . . 3k + 1, such that ~(1)...

~(3k+ 1) =/~(3k + 1),

where/~(3k+ 1)is defined by (2.17) relative to .4. Thus the matrices (4.18)

B T

:= ~(1) ... ~(3k+ 1)(/~(3k + 1))T

C T

:= ~(1)... ~(3k+ 1)(1(3k+ 2))T

satisfy the first two relations in (4.17). Using a special case of a result of Demko [D] it follows from the stability estimate (4.8), when p = 2, that F has an /2-inverse F -1 = (F-ll.;)~.~z whose elements decay exponentially fast in the sense that, for some constants c > 0 and p 9 (0, 1), we have IF~,-jxI < cp li-jl,

i, j e Z .

Likewise, the matrix (4.19)

D := I (3k +2)(~(3k+ 1))- t . . . (~(1))- xF - 1

has elements which decay exponentially fast and more importantly satisfies the remaining relations in (4.17). Note that the rows of B and C, given by (4.18), contain finitely many nonzero entries while the matrix D has elements which decay exponentially fast. Thus the functions Ri:= ~ d~,jFi jeZ

(4.20)

decay exponentially fast as x ~ _ oo while B~ and Ki are of compact support. According to Theorem 2.2 our choice of the matrices B, C, and D also satisfy the equation

BTAF + CTDF = 1, which can be shown to be equivalent to the equations

Fj = ~ (Fj, Bi)C i + 2 (F j, Ki)Ri, ieZ

j ~ Z.

ieZ

Consequently, we have the decomposition (4.21)

f = ~. (f, B~)C, + ~ (f, K~)R~ ieZ

i~Z

valid at least for all f e V1 of compact support. Similarly, we can verify that (4.22)

f = ~ (f, C,)B, + ~" (f, R,)K, i~Z

ieZ

for the same f. Alternatively, we could factor the matrix A given by (4.1) as in Theorem 4.1

AG(1)... G(k+ 1)

=

E(k+ a)

Banded Matrices with Banded Inverses, II

281

and solve (4.17) with the matrices B T := G(1)... G(k+1)(/~(k+I))TF- I, C T : = G(1)... G(k+ 1)(i(k + 2))TF - 1,

and D := I(k+2)(G (k+ 1))- 1... (G(t))-,. In this case the function Ri is actually ~ki of Theorem 4.1 and so the functions Bi, Ki form a biorthogonal system relative to C~ and O~, that is, both (4.21) and (4.22) hold. However, this time R~ has compact support while B~, K~ decay exponentially fast. References rB] C. DE BOOR (1976): Totalpositivity of the spline collocation matrix. Indiana Univ. J. Math., 25:541-551.

[BM] M. D. BUHMANN, C. A. MICCHELLI (1992): Spline prewavelets for non-uniform knots. Numer. Math., 61:455~,74.

[CDM] A. S. CAVARETTA,W. DAHMEN,C. A. MICCHELLI(1991): Stationary subdivision. Mem. Amer. Math. Soc., 93, 4~453.

[D] S. DEMKO (1977): Inverses of band matrices and local convergence of spline projectors. SIAM J. Numer. Anal., 14:616-619.

[GM] T. N. T. GOODMAN, C. A. MICCHELLI (1992): On refinement equations determined by P6lyafrequency sequences. SIAM J. Math. Anal., 23:766-784.

[J] R. Q. JIA (1983): Total positivity of the discrete spline collocation matrix. J. Approx. Theory, 39:11-23.

[K] S. KARLm (1968): Total Positivity. Stanford, CA: Stanford University Press. [M] C. A. MICCHELLI(1991): Usin9 the refinement equation for the construction of prewavelets. Numer. Algorithms, 1:75-116.

[Mo] K. MORKEN(1992): Total positivity of the discrete collocation matrix 11. Preprint, University of Oslo.

IS] I. J. SCrfOENBERG(1973): Cardinal Spline Interpolation. CBMS, vol. 12. Philadelphia, PA: SIAM.

[Sch] L. L. SCHUMAKER(1981): Spline Functions: Basic Theory. New York: Wiley. [W] R. J. WALKER(1950): Algebraic Curves. Princeton, NJ: Princeton University Press. W. Dahmen Institut f/ir Geometric und Praktische Mathematik RWTH Aachen Templergraben 55 D-5100 Aachen F.R.G.

C. A. Micchelli IBM T. J. Watson Research Center P.O. Box 218 Yorktown Heights New York 10598 U.S.A.