Spectral properties of the transition operator associated to a multivariate re nement equation q

Transcription

1 Linear Algebra and its Applications 292 (1999) 155±178 Spectral properties of the transition operator associated to a multivariate re nement equation q Rong-Qing Jia *, Shurong Zhang 1 Department of Mathematical Sciences, University of Alberta, Edmonton, Canada T6G 2G1 Received 27 November 1998; accepted 25 January 1999 Submitted by R.A. Brualdi Abstract Given a nitely supported sequence a on Z s and an s s dilation matrix M, the transition operator T a is the linear operator de ned by T a v a :ˆ Pb2Zs a Ma b v b, where a 2 Z s and v lies in `0 Z s, the linear space of all nitely supported sequences on Z s. In this paper we investigate the spectral properties of the transition operator T a and apply these properties to the study of the approximation and smoothness properties of the normalized solution of the re nement equation / ˆ Pa2Zs a a / M a. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Re nement equations; Subdivision operators; Transition operators; Eigenvalues; Invariant subspaces; Approximation order; Smoothness 1. Introduction The purpose of this paper is to investigate the spectral properties of the transition operator associated to a multivariate re nement equation and their applications to the study of the approximation and smoothness properties of the corresponding re nable function. q Supported in part by NSERC Canada under Grant OGP * Corresponding author. jia@xihu.math.ualberta.ca 1 szhang@vega.math.ualberta.ca /99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S ( 9 9 )

2 156 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 A re nement equation is a functional equation of the form / ˆ a a / M a ; 1:1 where a is a nitely supported sequence on Z s, and M is an s s integer matrix such that lim n!1 M n ˆ 0. The matrix M is called a dilation matrix, and the sequence a is called the re nement mask. Any function satisfying a re nement equation is called a re nable function. If a satis es a a ˆ m :ˆ jdet Mj; 1:2 then it is well known that there exists a unique compactly supported distribution / satisfying the re nement equation (1.1) subject to the condition ^/ 0 ˆ 1. This distribution is said to be the normalized solution to the re nement equation with mask a. Throughout this paper we assume that the condition (1.2) is satis ed. In this paper, the Fourier transform of an integrable function f on R s is de ned by ^ f n ˆ Z R s f x e ixn dx; n 2 R s ; where x n denotes the inner product of two vectors x and n in R s. The domain of the Fourier transform can be naturally extended to include compactly supported distributions. A multi-index is an s-tuple l ˆ l 1 ;... ; l s with its components being nonnegative integers. De ne x l :ˆ x l 1 1 xl s s for x ˆ x 1 ;... ; x s 2 R s : We may regard x l 1 1 xl s s as a monomial of total degree jlj :ˆ l 1 l s. For a nonnegative integer k, let P k denote the set of all polynomials of (total) degree at most k. A sequence u on Z s is called a polynomial sequence if there exists a polynomial p such that u a ˆ p a for all a 2 Z s. The degree of u is the same as the degree of p. For j ˆ 1;... ; s, we use D j f to denote the partial derivative of f with respect to the jth coordinate. For a multi-index l ˆ l 1 ;... ; l s, D l stands for the di erential operator D l 1 1 Dl s s. If p ˆ Pl c lx l is a polynomial, then we use p D to denote the di erential operator P l c ld l. We denote by ` Z s the linear space of all sequences on Z s, and by `0 Z s the linear space of all nitely supported sequences on Z s. For c 2 Z s, we denote by d c the element in `0 Z s given by d c c ˆ 1 and d c a ˆ 0 for all a 2 Z s n fcg. In particular, we write d for d 0. For j ˆ 1;... ; s, let e j be the jth coordinate unit vector. The di erence operator r j on ` Z s is de ned by r j u :ˆ u u e j,

3 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± u 2 ` Z s. For a multi-index l ˆ l 1 ;... ; l s, r l is the di erence operator r l 1 1 rl s s. Let a be an element in `0 Z s. The transition operator T a is the linear operator on `0 Z s de ned by T a v a :ˆ a Ma b v b ; a 2 Z s ; 1:3 b2z s where v 2 `0 Z s. The subdivision operator S a is the linear operator on ` Z s de ned by S a u a :ˆ b2z s a a Mb u b ; a 2 Z s ; 1:4 where u 2 ` Z s. We introduce a bilinear form on the pair of the linear spaces `0 Z s and ` Z s as follows: hu; vi :ˆ u a v a ; u 2 ` Z s ; v 2 `0 Z s : 1:5 Then ` Z s is the dual space of `0 Z s with respect to this bilinear form. It is easily seen that hs a u; vi ˆ hu; T a vi 8u 2 ` Z s ; v 2 `0 Z s : Hence, S a is the algebraic adjoint of T a with respect to the bilinear form given in (1.5). When the space dimension s ˆ 1, Deslauriers and Dubuc [6] discussed the spectral properties of the transition operator and applied those properties to their study of interpolatory subdivision schemes. For the multivariate case (s > 1), the subdivision operator was introduced by Cavaretta et al. [3] in their investigation of stationary subdivision schemes. In [9], Goodman et al. established spectral radius formulas for subdivision operators. In [10], Han and Jia showed that the transition operator T a has only nitely many nonzero eigenvalues. The following is an outline of the proof. For a bounded subset H of R s, the set P 1 nˆ1 M n H is de ned as ( ) 1 M n h n : h n 2 H for n ˆ 1; 2;... : nˆ1 If H is a compact set, then P 1 nˆ1 M n H is also compact. By suppa we denote the set fa 2 Z s : a a 6ˆ 0g. Let! :ˆ 1 M n suppa \ Z s : 1:6 nˆ1

4 158 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 We use ` to denote the linear space of all sequences supported in. It is easily seen that ` is invariant under T a. Moreover, if v is an eigenvector of T a corresponding to a nonzero eigenvalue of T a, then v must lie in `. Consequently, any nonzero eigenvalue of T a must be an eigenvalue of the matrix a Ma b a;b2 : In particular, T a has only nitely many nonzero eigenvalues. The spectral radius of T a, denoted by q T a, is de ned as the spectral radius of the matrix a Ma b a;b2. In [16], using the subdivision and transition operators, Jia investigated the approximation properties of a re nable function in terms of its re nement mask. Let us review some basic results about approximation with shift-invariant spaces. For a compactly supported distribution / on R s and a sequence c 2 ` Z s, the semi-convolution of / with c is de ned by / 0 c :ˆ / a c a : Let S / denote the linear space f/ 0 c : c 2 ` Z s g. We call S / the shiftinvariant space generated by /. A compactly supported distribution / on R s is said to have accuracy k if S / contains P k 1 (see [11]). If / has accuracy k and ^/ 0 6ˆ 0, then for any polynomial sequence u of degree at most k 1, the semi-convolution / 0 u is a polynomial of the same degree. Conversely, for any p 2 P k 1, there exists a unique polynomial sequence u such that p ˆ / 0 u. See [1, Proposition 1.1] and [15, Lemma 8.2] for these results. Suppose 1 6 p 6 1 and / is a compactly supported function in L p R s such that ^/ 0 6ˆ 0. It was proved in [14] that S / provides approximation order k if and only if / has accuracy k. Let / be the normalized solution of the re nement equation (1.1) with mask a and dilation matrix M. We say that a satis es the sum rules of order k if a c Mb p c Mb ˆ a Mb p Mb 8p 2 P k 1 and c 2 Z s : b2z s b2z s It was proved in [16] that / has accuracy k provided that a satis es the sum rules of order k. Let ( ) V k :ˆ v 2 `0 Z s : p a v a ˆ 0 8p 2 P k : Then a satis es the sum rules of order k if and only if V k 1 is invariant under the transition operator T a.

5 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± In [17], Jia analyzed the smoothness of re nable functions in terms of their masks. For m > 0, we denote by W2 m Rs the Sobolev space of all functions f 2 L 2 R s for which Z jf ^ n j 2 1 jnj m 2 dn < 1: R s The smoothness order m f of a function f 2 L 2 R s is de ned by m f :ˆ supfm : f 2 W m 2 Rs g: Let / be the normalized solution of the re nement equation (1.1) with mask a and dilation matrix M. We assume that M is isotropic, i.e., M is similar to a diagonal matrix diagfr 1 ;... ; r s g with jr 1 j ˆ ˆ jr s j. Let b be the sequence given by b a :ˆ a a b a b =m; a 2 Z s ; 1:7 b2z s where m ˆ jdet Mj and a denotes the complex conjugate of a. Suppose a satis es the sum rules of order k. Then b satis es the sum rules of order 2k. Hence V 2k 1 is invariant under T b, the transition operator associated to b. Let q k denote the spectral radius of T b j V2k 1. Suppose / lies in L 2 R s. It was proved in [17] that m / P log m q k s=2: 1:8 If, in addition, k > log m q k s=2 and the shifts of / are stable, then equality holds in (1.8). Note that the shifts of / are stable if and only if, for any n 2 R s, there exists an element b 2 Z s such that ^/ n 2bp 6ˆ 0. Moreover, if the shifts of / are linearly independent, that is, c a / a ˆ 0 ) c a ˆ 0 8a 2 Z s ; then the shifts of / are stable. See [18] for these facts. The following is an outline of the paper. Section 2 is devoted to a study of the spectrum of the transition operator. Suppose the dilation matrix M has eigenvalues r 1 ;... ; r s. Write r for the s- tuple r 1 ;... ; r s. By convention, for a multi-index l ˆ l 1 ;... ; l s we have r l :ˆ r l 1 1 rl s s and r l :ˆ r l 1 1 r l s s : We shall show that the spectrum of the transition operator T a contains fr l : jlj < kg, provided / has accuracy k. This gives an upper bound for the accuracy of / in terms of the re nement mask a.

6 160 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 In Section 3 we shall investigate invariant subspaces of the subdivision and transition operators. We give a necessary and su cient condition for a subspace of polynomial sequences to be invariant under the subdivision operator. Furthermore, we clarify the relationship among the spectra of the transition operator restricted to di erent invariant subspaces. In particular, we establish the following formula spec T a j` ˆ spec T a [ fr l : jlj < kg; j` \Vk 1 where is given by (1.6). Thus, the spectral radius of T a j Vk 1 can be found from the spectrum of T a j`. This result is signi cant for calculating the smoothness order of a re nable function in terms of its mask. Section 4 is devoted to a study of the smoothness order of re nable functions which are convolutions of box splines with re nable distributions. Box splines are natural extensions of cardinal B-splines to multidimensional spaces. In the univariate case, a factorization technique can be used to compute the smoothness order of a re nable function by nding the dominant eigenvalue of a certain matrix. In the mutltivariate case, if a re nable function is the convolution of a box spline with a re nable distribution, we will give a method to compute its smoothness order by nding the dominant eigenvalues of certain transition matrices. 2. The spectrum of the transition operator The spectrum of a square matrix A is denoted by spec A, and it is understood to be the multiset of its eigenvalues. In other words, multiplicities of eigenvalues are counted in the spectrum of a square matrix. The transpose of a matrix A is denoted by A T. Suppose T is a linear mapping on a nite dimensional vector space V over C. Let fv 1 ;... ; v n g be an ordered basis of V. If T v i ˆ b 1i v 1 b ni v n ; i ˆ 1;... ; n; then b ij 1 6 i;j 6 n is called the matrix representation of T with respect to fv 1 ;... ; v n g. The spectrum of T is the same as the spectrum of the matrix b ij 1 6 i;j 6 n. Suppose / is the normalized solution of the re nement equation (1.1) with mask a and dilation matrix M. Let supp/ denote the support of /. From (1.1) we observe that / x 6ˆ 0 implies / Mx a 6ˆ 0 for some a 2 suppa. It follows that x 2 M 1 suppa M 1 supp/ : Hence we have

7 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± supp/ M 1 suppa M 1 supp/ : A repeated use of the above relation yields supp/ n M j supp a M n supp/ ; n ˆ 1; 2;... : jˆ1 Consequently, we obtain supp / 1 nˆ1 M n suppa : 2:1 It follows that Z s \ supp/, where is the set given in (1.6). Let T a and S a be the transition operator and the subdivision operator given in (1.3) and (1.4), respectively. It was pointed out that S a is the algebraic adjoint of T a with respect to the bilinear form given in (1.5). Let be a nonempty nite subset of Z s. Suppose ` is invariant under T a. By we denote the set f a : a 2 g. Clearly, ` is the dual space of ` with respect to the bilinear form hu; vi :ˆ u a v a ; u 2 ` ; v 2 ` : 2:2 a2 Let Q :ˆ Q be the linear mapping from ` Z s to ` given by u a for a 2 ; Q u a ˆ 0 for a 62 : 2:3 Then QS a maps ` to `. We claim that QS a j` is the algebraic adjoint of T a j`. Indeed, for u 2 ` and v 2 ` we have hqs a u; vi ˆ hqs a u; vi ˆ hs a u; vi ˆ hu; T a vi ˆ hu; T a vi : This justi es our claim. Consequently, the spectra of QS a j` and T a j` are the same. Moreover, for u 2 ` Z s and v 2 ` we have hqs a Qu u ; vi ˆ hqu u; T a vi ˆ 0; since T a v 2 ` and Qu u vanishes on `. Thus, QS a Qu u a ˆ 0 for all a 2. But, by the de nition of Q, we have QS a Qu u a ˆ 0 for all a 2 Z s n. This shows QS a Qu u ˆ 0. In other words, QS a Q ˆ QS a : For u 2 ` Z s, we have u a / a ˆ S a u a / M a : 2:4 2:5

8 162 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 Indeed, since / satis es the re nement equation (1.1), we obtain u a / a ˆ u a a b / M Ma b b2z s ˆ w c / M c ; c2z s where w c ˆ a c Ma u a ; c 2 Z s : Hence w ˆ S a u. This veri es (2.5). By K / we denote the linear space given by K / :ˆ fu 2 ` Z s : / 0 u ˆ 0g: It follows from (2.5) that K / is invariant under the subdivision operator S a. Lemma 2.1. Let Q :ˆ Q be the linear mapping from ` Z s to ` given by (2.3), where ˆ Z s \ P 1 nˆ1 M n H for some compact set H suppa. If u is a sequence on Z s such that p :ˆ / 0 u is a nonzero polynomial, then Qu 62 Q K /. Proof. Set G r :ˆ f x 1 ;... ; x s 2 R s : jx 1 j jx s j < rg; r > 0: By (2.1), the compact set supp / is disjoint from the closed set Z s n ; hence there exists some r > 0 such that supp/ G r \ Z s n ˆ ;: Suppose x 2 G r and a 2 Z s. Then / x a 6ˆ 0 implies x a 2 supp/. It follows that a 2 supp / G r. Consequently, a 2 Z s \ supp/ G r : In other words, x 2 G r and a 62 imply / x a ˆ 0. Therefore, p x ˆ u a / x a ˆ u a / x a ˆ u a / x a ; x 2 G r : a2 If Qu 2 Q K /, then there would exist some w 2 K / such that Qu ˆ Qw. It follows that u a ˆ w a for all a 2. Thus, for all x 2 G r,

9 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± p x ˆ u a / x a ˆ w a / x a a2 a2 ˆ w a / x a ˆ 0; which is impossible, because p is a nonzero polynomial. This veri es Qu 62 Q K /. We are in a position to establish the main result of this section. Theorem 2.2. Let / be the normalized solution of the re nement equation with mask a and dilation matrix M. If / has accuracy k, then the spectrum of the transition operator T a contains fr l : jlj < kg, where r ˆ r 1 ;... ; r s is the s- tuple of the eigenvalues of M and r l :ˆ r l 1 1 r l s s for a multi-index l ˆ l 1 ;... ; l s. Proof. Let be the set given in (1.6), and let Q :ˆ Q be the linear mapping from ` Z s to ` as de ned in (2.3). Since the spectra of QS a j` and T a j` are the same, it su ces to show that the spectrum of QS a j` contains fr l : jlj < kg. For this purpose, we introduce the set W :ˆ fu 2 ` Z s : / 0 u 2 P k 1 g: By (2.5), W is invariant under S a. Clearly, Q W is a subspace of `. The theorem will be proved by nding the matrix representation of QS a with respect to a suitable basis of Q W. There exists an invertible matrix H ˆ h ij 1 6 i;j 6 s such that HMH 1 is a triangular matrix: 2 3 HMH 1. ˆ 6 4. r : r s1 r ss For i ˆ 1;... ; s and x ˆ x 1 ;... ; x s 2 R s, let l i x :ˆ h i1 x 1 h is x s. Then Hx can be represented as l 1 x ;... ; l s x Š T. It follows that l 1 Mx r 11 r 11 l 1 x ˆ HMx ˆ Hx ˆ : l s Mx r s1 r ss r s1 r ss l s x 2:6 For simplicity, we write r j for r jj, j ˆ 1;... ; s. Thus, r 1 ;... ; r s are the eigenvalues of the matrix M. For two multi-indices l ˆ l 1 ;... ; l s and

10 164 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 m ˆ m 1 ;... ; m s, we write l m if there exists some j, 1 6 j 6 s, such that l j < m j, and l j 1 ˆ m j 1 ;... ; l s ˆ m s. For a multi-index l ˆ l 1 ;... ; l s, let p l be the polynomial given by p l :ˆ l l 1 1 ll s s : Clearly, p l (jlj < k) are linearly independent. With the help of (2.6) we obtain p l Mx ˆ l 1 Mx Š l 1 l s Mx Š ls ˆ r l p l x q l x ; x 2 R s ; where q l is a linear combination of p m with jmj ˆ jlj and m l. It follows that p l x ˆ r l p l Mx r l q l x : A repeated use of the above relation yields p l x ˆ r l p l Mx r l Mx ; 2:7 where r l is a linear combination of p m with jmj ˆ jlj and m l. By the assumption, / has accuracy k. Thus, for each l with jlj < k, the polynomial p l lies in S /. Since ^/ 0 6ˆ 0, there exists a unique polynomial sequence u l 2 ` Z s such that p l ˆ u l a / a : 2:8 It follows from (2.7) and (2.8) that p l x ˆ r l p l Mx r l Mx ˆ r l u l a v l a / Mx a ; 2:9 where v l is a linear combination of u m with jmj ˆ jlj and m l. On the other hand, we deduce from (2.5) and (2.8) that p l x ˆ S a u l a / Mx a : Comparing this equation with (2.9), we obtain S a u l ˆ r l u l v l w l ; where w l 2 K /. By (2.4) it follows that QS a Qu l ˆ r l Qu l Qv l Qw l : 2:10 Let U :ˆ U 0 U k 1, where each U j (j ˆ 0; 1;... ; k 1) is the linear span of u l, jlj ˆ j. Then W ˆ U K /. By Lemma 2.1, Q U \ Q K / ˆ f0g. Hence Q W is the direct sum of Q U and Q K /. Moreover, Q U is the direct sum of Q U 0 ;... ; Q U k 1. Choose a basis Y for Q K /. For each j, the set Y j :ˆ fqu l : jlj ˆ jg is a basis for Q U j. The order of this basis is

11 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± arranged in such a way that Qu m precedes Qu l whenever m l. Consequently, Y [ Y 0 [ [ Y k 1 is a basis for Q W. With respect to this basis, (2.10) tells us that QS a has the following matrix representation: 2 3 E F 0 F 1 F k 1 E E 1 0 ; E k 1 where each E j (j ˆ 0;... ; k 1) is a triangular matrix with r l (jlj ˆ j) being the entries in its main diagonal. We conclude that the spectrum of QS a j Q W contains fr l : jlj < kg, as desired. We emphasize that the conclusion of Theorem 2.2 is valid without any assumption of stability of /. Example 2.3. Let M be the matrix 1 1 ; 1 1 and let a be the sequence on Z 2 such that a a ˆ 0 for a 2 Z 2 n 2; 2Š 2 and a a 1 ; a a1 ;a ˆ : Let / be the normalized solution of the re nement equation (1.1) with mask a and dilation matrix M given as above. Then / has accuracy 4 but does not have accuracy 5. It can be easily checked that a satis es the sum rules of order 4. Hence / has accuracy 4. Let us show that / does not have accuracy 5. The matrix M has two eigenvalues r 1 ˆ 1 i and r 2 ˆ 1 i, where i denotes the imaginary unit. We have suppa 2; 2Š 2 and 1 nˆ1 M n 2; 2Š 2 ˆ x 1 ; x 2 2 R 2 : jx 1 j 6 6; jx 2 j 6 6; jx 1 x 2 j 6 8; jx 1 x 2 j 6 8 :

12 166 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 The set :ˆ Z 2 \ P 1 nˆ1 M n 2; 2Š 2 has exactly 129 points. Among the 129 eigenvalues of the matrix A :ˆ a Ma b a;b2 the following are of the form r l 1 1 r l 2 2 for some double-index l 1 ; l 2 with l 1 l 2 6 4: 1; 0:5 0:5i; 0:5 0:5i; 0:5i; 0:5; 0:5i; 0:25 0:25i; 0:25 0:25i; 0:25 0:25i; 0:25 0:25i; 0:25; 0:25i; 0:25i: Since / has accuracy 4, we expect that A has eigenvalues r l 1 1 r l 2 2 for all double-indices l 1 ; l 2 with l 1 l The above computation con rms our expectation. But A has only three eigenvalues of modulus 0:25. Therefore, by Theorem 2.2, / does not have accuracy Invariant subspaces of the transition operator In this section we investigate invariant subspaces of the subdivision and transition operators. We are particularly interested in invariant subspaces of the subdivision operator which consist of polynomial sequences. The results are then applied to smoothness analysis of re nable functions in terms of their masks. Let P denote the linear space of all polynomials of s variables. For a compactly supported distribution / on R s, the intersection S / \ P is not of the form P k in general. But S / \ P is always shift-invariant, i.e., p 2 S / \ P implies p a 2 S / \ P for all a 2 Z s. It is easily seen that a shift-invariant subspace P of P is D-invariant, that is, p 2 P implies all its partial derivatives belong to P. Suppose / is a compactly supported distribution on R s such that ^/ 0 6ˆ 0. Let P be a nite dimensional D-invariant subspace of P. Then P S / if and only if p id ^/ 2pb ˆ 0 8p 2 P and b 2 Z s n f0g: Suppose P S /. Then u 2 Pj Z s implies p :ˆ / 0 u lies in P. Conversely, for each p 2 P, there exists a unique polynomial sequence u 2 Pj Z s such that p ˆ / 0 u. See [1] for these results. Now let / be the normalized solution of the re nement equation with mask a and dilation matrix M, where P s a2z a a ˆ m ˆ j det Mj. Let C be a complete set of representatives of the distinct cosets of Z s =MZ s, and let H be a complete set of representatives of the distinct cosets of Z s =M T Z s. Recall that a satis es the sum rules of order k implies / has accuracy k. The converse of this statement is valid under the additional condition that

13 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± N / \ 2p M T 1 H ˆ ;; 3:1 where N / :ˆ fn 2 R s : ^/ n 2pb ˆ 0 8b 2 Z s g: These results can be extended to shift-invariant subspaces of P. Let P be a nite dimensional shift-invariant subspace of P. If a c Mb p c Mb ˆ a Mb p Mb b2z s b2z s 8p 2 P and c 2 C; 3:2 then P S /. Conversely, if P S / and (3.1) is valid, then a satis es the conditions in (3.2). The proof is similar to the one given in [16]. It was proved in [16] that a satis es the sum rules of order k if and only if P k 1 j Z s is invariant under the subdivision operator S a. In order to extend this result to shift-invariant subspaces of P, additional work is needed. Theorem 3.1. Let M be a dilation matrix, and let a be an element in `0 Z s such that P s a2z a a ˆ m ˆ j det Mj. Suppose P is a nite dimensional shift-invariant subspace of P. Then Pj Z s is invariant under S a if and only if a satis es the conditions in (3.2) and p 2 P implies p M 1 2 P. Proof. Suppose U :ˆ Pj Z s is invariant under S a. Let us rst show that S a j U is one-to-one. For this purpose, choose an element u in U such that S a u ˆ 0. Then b2z s a a Mb u b ˆ 0 8a 2 Z s : 3:3 Suppose u 6ˆ 0. Since u is a polynomial sequence, there exists a multi-index l and a complex number c 6ˆ 0 such that r l u b ˆ c for all b 2 Z s. It follows from (3.3) that b2z s a a Mb r l u b ˆ 0 8a 2 Z s : Hence P b2z a a Mb ˆ 0 for all a 2 s Zs. This contradicts the assumption that P a2z a a ˆ m 6ˆ 0. Therefore, S aj s U is one-to-one. But U is nite dimensional. Hence S a j U is one-to-one and onto. Next, we show that p 2 P implies p M 2 P. Let p 2 P. Since S a j U is onto, there exists f 2 P such that pj Z s ˆ S a f j Z s, that is, p a ˆ s Pb2Z a a Mb f b for all a 2 Z s. It follows that p Ma ˆ a Ma Mb f b ˆ a Mb f a b 8 a 2 Z s : b2z s b2z s

14 168 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 Let q x :ˆ Pb2Z s a Mb f x b, x 2 Rs. Since P is shift-invariant, q belongs to P. Thus, q and p M agree on the lattice Z s. Therefore, we have p M ˆ q 2 P. For p 2 P let u c :ˆ b2z s a Mb c p Mb c ; c 2 Z s : We claim that u is a polynomial sequence. Indeed, by using Taylor's formula, we obtain p Mb c ˆ t l Mb c l ; l where t l :ˆ D l p=l!. Since P is D-invariant, t l 2 P for every multi-index l. For x 2 R s, set q l x :ˆ t l Mx. By what has been proved, we have q l 2 P. Let u l :ˆ q l j Z s. Then for c 2 Z s, u c ˆ a Mb c p Mb c b2z s ˆ a c Mb u l b c l ˆ S a u l c c l : b2z s l Since U is invariant under S a, S a u l 2 U. Hence u is a polynomial sequence. By the de nition of u, we have u c Mg ˆ u c for all g 2 Z s and c 2 Z s. In other words, u is a constant sequence on the lattice c MZ s for each c 2 Z s. Therefore, u itself must be a constant sequence. This shows that a satis es the conditions in (3.2). Consequently, P S /, where / is the normalized solution of the re nement equation with mask a and dilation matrix M. It remains to prove that p 2 P implies p M 1 2 P. Let p 2 P. Then there exists a unique u 2 U such that p ˆ / 0 u. From (2.5) we deduce that p M 1 x ˆ u a / M 1 x a ˆ S a u a / x a : Since U is invariant under S a, we have S a u 2 U. This shows p M 1 2 P. Now suppose a satis es the conditions in (3.2) and p 2 P implies p M 1 2 P. We wish to show that U ˆ Pj Z s is invariant under S a. Let p 2 P and u ˆ pj Z s. We rst show that S a u is a polynomial sequence. Set q x :ˆ p M 1 x, x 2 R s. By our assumption, q 2 P. An application of Taylor's formula gives q Mb ˆ q a Mb a ˆ q l a Mb a l ; l where q l ˆ D l p=l!. Since P is D-invariant, we have q l 2 P for all multi-indices l. It follows that l

15 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± S a u a ˆ a a Mb p b ˆ a a Mb q Mb b2z s b2z s ˆ " # a a Mb q l a Mb a l : l b2z s Since a satis es the conditions in (3.2), c l :ˆ Pb2Z a a Mb q l a Mb is s independent of a. Therefore, S a u a ˆ Pl c la l for all a 2 Z s. This shows that S a u is a polynomial sequence. To nish the proof, we observe that S a u Mc ˆ a M c b u b ˆ a Mb p c b ; c 2 Z s : b2z s b2z s Since P is shift-invariant, there exists f 2 P such that b2z s a Mb p c b ˆ f c 8c 2 Z s : Let g x :ˆ f M 1 x, x 2 R s. Then g 2 P and S a u Mc ˆ f c ˆ g Mc 8c 2 Z s : This shows that S a u and g agree on the lattice MZ s. But both S a u and gj Z s are polynomial sequences. Therefore, S a u ˆ gj Z s 2 U. We conclude that U is invariant under S a. The following theorem clari es the relationship among the spectra of the transition operator restricted to di erent invariant subspaces. Theorem 3.2. Let U be a nite dimensional subspace of ` Z s, and let ( V :ˆ v 2 `0 Z s : ) u a v a ˆ 0 8u 2 U : 3:4 Then U is invariant under the subdivision operator S a if and only if V is invariant under the transition operator T a. Let be a nite subset of Z s such that ` is invariant under T a, and let Q :ˆ Q be the linear mapping from ` Z s to ` as de ned in (2.3). If U is invariant under S a, and if Qj U is one-to-one, then spec T a j` ˆ spec T a j` \V [ spec S a j U : 3:5 In particular, the above relation is valid when ˆ Z s \ P 1 nˆ1 M n H for some compact set H suppa and U ˆ Pj Z s for some nite dimensional shift-invariant subspace P of P.

16 170 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 Proof. Let hu; vi be the bilinear form de ned in (1.5). Then v 2 V if and only if hu; vi ˆ 0 for all u 2 U. Suppose U is invariant under S a. Then for v 2 V we have hu; T a vi ˆ hs a u; vi ˆ 0 8u 2 U: Hence v 2 V implies T a v 2 V. This shows that V is invariant under T a. Choose a basis fu 1 ;... ; u n g for U. Then there exist v 1 ;... ; v n 2 `0 Z s such that hu j ; v k i ˆ d jk for j; k ˆ 1;... ; n, where d jk stands for the Kronecker sign. It is easily seen that `0 Z s is the direct sum of V and the linear span of v 1 ;... ; v n. Suppose V is invariant under T a. We wish to show that U is invariant under S a. Let u 2 U and w ˆ S a u. Then hw; vi ˆ hs a u; vi ˆ hu; T a vi ˆ 0 8v 2 V : Moreover, with c j :ˆ hw; v j i, j ˆ 1;... ; n, we have hw c 1 u 1 c n u n ; v j i ˆ 0 8j ˆ 1;... ; n: It follows that hw c 1 u 1 c n u n ; yi ˆ 0 for all y 2 `0 Z s. This shows that w ˆ c 1 u 1 c n u n 2 U. In other words, U is invariant under S a. This proves the rst statement of the theorem. Now suppose U is invariant under S a. Choose a basis fu 1 ;... ; u r g for U. Since Qj U is one-to-one, fqu 1 ;... ; Qu r g is a basis for Q U. We supplement elements u r 1 ;... ; u n in ` such that fqu 1 ;... ; Qu r ; u r 1 ;... ; u n g forms a basis for `. Clearly, Qu j ˆ u j for j ˆ r 1;... ; n. Suppose QS a Qu j ˆ n kˆ1 b jk Qu k ; j ˆ 1;... ; n: 3:6 Let B :ˆ b jk 1 6 j;k 6 n. Then B T, the transpose of B, is the matrix of the linear mapping QS a j` with respect to the basis fqu 1 ;... ; Qu n g. Since U is invariant under S a, Q U is invariant under QS a in light of (2.4). Therefore, b jk ˆ 0 for j ˆ 1;... ; r and k ˆ r 1;... ; n. In other words, B is a block triangular matrix: B ˆ E 0 G F ; where E ˆ b jk 1 6 j;k 6 r and F ˆ b jk r 1 6 j;k 6 n. Since ` is invariant under T a, by (2.4) we have QS a Q ˆ QS a. By our assumption, Qj U is one-to-one. Thus, it follows from (3.6) that S a u j ˆ r kˆ1 b jk u k ; j ˆ 1;... ; r: Therefore, E T is the matrix of S a j U with respect to the basis fu 1 ;... ; u r g.

17 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± Note that ` is the dual space of ` with respect to the bilinear form hu; vi de ned in (2.2). Let fv 1 ;... ; v n g be the basis of ` dual to fqu 1 ;... ; Qu n g, that is, hqu j ; v k i ˆ d jk for j; k ˆ 1;... ; n: Clearly, fv r 1 ;... ; v n g is a basis for ` \ V. It was proved in Section 2 that QS a j` is the adjoint of T a j` with respect to the bilinear form hu; vi. Consequently, by (3.6) we have hqu j ; T a v k i ˆ hqs a Qu j ; v k i ˆ b jk ; j; k ˆ 1;... ; n: This shows that B ˆ b jk 1 6 j;k 6 n is the matrix of T a j` with respect to the basis fv 1 ;... ; v n g. But fv r 1 ;... ; v n g is a basis for ` \ V. Hence F ˆ b jk r 1 6 j;k 6 n is the matrix of T a j` \V with respect to this basis. To summarize, we obtain spec T a j` ˆ spec B ˆ spec E [ spec F ˆ spec S a j U [ spec Ta j` \V : This veri es (3.5). Finally, suppose U ˆ Pj Z s for some shift-invariant subspace P of P and U is invariant under S a. Theorem 3.2 tells us that P S /, where / is the normalized solution of the re nement equation (1.1) with mask a and dilation matrix M. Suppose that ˆ Z s \ P 1 nˆ1 M n H for some compact set H suppa. Let u 2 U and p :ˆ / 0 u. If u 6ˆ 0, then p 6ˆ 0; hence Qu 6ˆ 0 by Lemma 2.1. This shows that Qj U is one-to-one. Therefore, (3.5) is valid for this case. The case U ˆ P k 1 j Z s is of particular interest. Suppose a satis es the sum rules of order k. Then U is invariant under S a, by Theorem 3.2. By Theorem 2.2 we have spec S a j U ˆ fr l : jlj < kg. Thus, (3.5) reads as follows: spec T a j` ˆ spec T a [ fr l : jlj < kg: 3:7 j` \Vk 1 For the univariate case (s ˆ 1), this formula was established by Deslauriers and Dubuc in [6, Theorem 8.2.] Theorem 3.2 has useful applications to smoothness analysis of re nable functions in terms of their masks. Let / be the normalized solution of the re nement equation (1.1) with a mask a and an isotropic dilation matrix M. Let b be the sequence given by (1.7). Suppose a satis es the sum rules of order k. Then b satis es the sum rules of order 2k. Hence V 2k 1 is invariant under the transition operator T b. Let q k denote the spectral radius of T b j V2k 1. It follows from (3.7) that

18 172 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 spec T b ˆ spec T j` \V1 b [ fr l : 2 6 jlj < 2kg; j` \V2k 1 where :ˆ Z s \ P 1 nˆ1 M n suppb. If q k < 1, then the above relation tells us that q 1 < 1, which implies that the subdivision scheme associated to mask a and dilation matrix M converges in the L 2 -norm (see [10]). In particular, q k < 1 implies / 2 L 2 R s. We can nd q k from spec T b j` by using the following formula: spec T b j` ˆ spec T b [ fr l : jlj < 2kg: j` \V2k 1 The following example illustrates this technique. Example 3.3. Let M be the matrix and let a be the mask given in Example 2.3. Let us determine the smoothness order of the normalized solution / of the re nement equation with mask a and dilation matrix M. Let b be the mask computed from a by using (1.7). Then suppb 4; 4Š 2 and the set P 1 nˆ1 M n 4; 4Š 2 is f x 1 ; x 2 2 R 2 : jx 1 j 6 12; jx 2 j 6 12; jx 1 x 2 j 6 16; jx 1 x 2 j 6 16g: The set :ˆ Z 2 \ P 1 nˆ1 M n 4; 4Š 2 has exactly 481 points. We use MATLAB to compute the eigenvalues of the matrix b Ma b a;b2. These eigenvalues are arranged in the order of descending absolute values. The following is a list of the rst 22 eigenvalues. 1; 0:5 0:5i; 0:5 0:5i; 0:5; 0:5i; 0:5i; 0:25 0:25i; 0:25 0:25i; 0:25 0:25i; 0:25 0:25i; 0:25; 0:25; 0:25i; 0:25i; 0:25; 0: ; 0:125 0:125i; 0:125 0:125i; 0:125 0:125i; 0:125 0:125i; 0:125 0:125i; 0:125 0:125i: Note that the matrix M has two eigenvalues r 1 ˆ 1 i and r 2 ˆ 1 i. In the above list, 21 eigenvalues are of the form r l 1 1 r l 2 2 for double indices l 1 ; l 2 with l 1 l Therefore, q 4 0: By (1.8) we obtain

19 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± m / P log 2 0: : : From the results in [10] we know that the subdivision scheme associated to mask a and dilation matrix M converges uniformly. Moreover, the mask a is interpolatory, i.e., a 0 ˆ 1 and a Ma ˆ 0 for a 2 Z s n f0g. Consequently, / is continuous, / 0 ˆ 1, and / a ˆ 0 for a 2 Z s n f0g. Hence, the shifts of / are linearly independent. We conclude that m / 2: Re nable functions induced by box splines In the univariate case, a factorization technique can be used to compute the smoothness order (regularity) of a re nable function by nding the dominant eigenvalue of a certain matrix. In this regard, the reader is referred to the work of Daubechies and Lagarias [5], Eirola [8], and Villemoes [20]. In the multivariate case, if a re nable function is the convolution of a box spline with a re nable distribution, then it is still possible to compute its smoothness order by nding the dominant eigenvalues of certain transition matrices. For an element a 2 `0 Z s we use ~a z to denote its symbol: ~a z :ˆ a a z a ; z 2 C n f0g s : The convolution of two sequences a and b in `0 Z s is de ned by a b a :ˆ a a b b b ; a 2 Z s : b2z s If c ˆ a b, then ~c z ˆ ~a z ~ b z ; z 2 C n f0g s : For r ˆ 1; 2;..., let a r be the element in `0 Z de ned by its symbol: ~a r z ˆ 1 z r =2 r 1 : The cardinal B-spline B r of order r can be viewed as the normalized solution of the re nement equation / ˆ Pa2Z a r a / 2 a. Box splines are natural extensions of cardinal B-splines. The reader is referred to the monograph [2] by de Boor et al. for a comprehensive study of box splines. In this section we are particularly interested in box splines on the three-direction mesh on R 2. For r; s; t P 1, let a r;s;t be the element in `0 Z 2 de ned by its symbol: ~a r;s;t z 1 ; z 2 :ˆ 1 z 1 r 1 z 2 s 1 z 1 z 2 t =2 r s t 2 ; z 1 ; z 2 2 C 2 :

20 174 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 The box spline B r;s;t is de ned as the normalized solution of the re nement equation / ˆ a r;s;t a / 2 a : a2z 2 In [7], Dyn et al. analyzed convergence of the so-called butter y scheme which is induced by the box spline B 1;1;1. More generally, using convolutions of box splines with distributions, Riemenschneider and Shen [19] constructed a family of bivariate interpolatory subdivision schemes with symmetry. The following theorem provides a method to simplify the computation of the smoothness order of re nable functions which are convolutions of box splines B r;r;r with re nable distributions. In what follows we use T 2 to denote the torus f z 1 ; z 2 2 C 2 : jz 1 j ˆ 1; jz 2 j ˆ 1g: Theorem 4.1. Let c be an element in `0 Z 2 such that P 2 a2z c a ˆ 4, and let a be given by its symbol ~a z ˆ 1 z r r r 1 1 z 2 1 z 1 z 2 ~c z ; z ˆ z 1 ; z 2 2 T 2 ; where r is a positive integer. Let / be the normalized solution of the re nement equation / ˆ Pa2Z a a / 2 a. Write z 2 3 for z 1 z 2. Let a j (j ˆ 1; 2; 3) be given by ~a j z ˆ 1 z r j ~c z ; z 2 T 2 ; 2 and let b j (j ˆ 1; 2; 3) be given by bj ~ z ˆ ja j z j 2 =4, z 2 T 2. Let q :ˆ max 1 6 j 6 3 fq T bj g. If q > 1 and if the shifts of / are stable, then m / ˆ 2r log 4 q: 4:1 Proof. Let b 2 `0 Z 2 be given by ~b z ˆ j~a z j 2 =4; z 2 T 2 ; and let f be the normalized solution to the re nement equation with mask b. Then f ^ n ˆ j ^/ n j 2 for all n 2 R s. Thus, if the shifts of / are stable, then so are the shifts of f. Let P r;s;t :ˆ P \ S B r;s;t. It is known (see, e.g., [2]) that P r;s;t ˆ p 2 P : D r 1 Ds 2 p ˆ 0; Dr 1 D 1 D 2 t p ˆ 0; D s 2 D 1 D 2 t p ˆ 0 : In particular, P r;s;t P r s t 2.

21 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± For j ˆ 1; 2; 3, we use D j to denote the di erence operator on `0 Z 2 given by D j v :ˆ v e j 2v v e j ; v 2 `0 Z 2 ; where e 1 ˆ 1; 0, e 2 ˆ 0; 1, and e 3 ˆ 1; 1. Let V be the linear span of D r 1 Dr 2 d b, D r 2 Dr 3 d b, and D r 1 Dr 3 d b, b 2 Z 2, and let U :ˆ u 2 ` Z 2 : hu; vi ˆ 0 8v 2 V ; where hu; vi is the bilinear form given in (1.5). Then u belongs to U if and only if u satis es the following system of partial di erence equations: D r 1 Dr 2 u ˆ 0; Dr 1 Dr 3 u ˆ 0; Dr 2 Dr 3 u ˆ 0: By [4, Proposition 2.1] we have U ˆ P 2r;2r;2r j Z 2. Also see [12, 5] for properties of partial di erence equations associated to box splines. Note that ` Z 2 is the dual space of `0 Z 2 with respect to the bilinear form hu; vi. Suppose w 2 `0 Z 2 n V. Then there exists an element u 2 ` Z 2 such that hu; wi ˆ 1 and hu; vi ˆ 0 for all v 2 V. This shows V ˆ v 2 `0 Z 2 : hu; vi ˆ 0 8u 2 U : Since U is invariant under the subdivision operator S b, V is invariant under the transition operator T b, by Theorem 3.2. Let U k :ˆ P k j Z 2. Then we have V k ˆ v 2 `0 Z 2 : hu; vi ˆ 0 8u 2 U k : Consequently, U 4r 1 U and V V 4r 1. We observe that P 1 nˆ1 2 n supp b is contained in the convex hull of supp b. Let be the intersection of Z 2 with the convex hull of suppb. By Theorem 3.2 we have spec T b j` ˆ spec T b j` \V [ spec S b j U 4:2 and spec T b j` ˆ spec T b j` \V4r 1 [ spec S b j U4r 1 : 4:3 Let us nd the di erence between spec S b j U and spec S b j U4r 1. For j ˆ 0; 1;..., by H j we denote the linear space of homogeneous polynomials of degree j. Let E j :ˆ fu 2 U : f 0 u 2 H j g: Then U is the direct sum of E j, j ˆ 0; 1;... ; 6r 2. Let u 2 E j and p :ˆ f 0 u 2 H j. Since f ˆ Pa2Z2 b a f 2 a, by (2.5) we obtain

22 176 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 p x ˆ u a f x a ˆ S b u a f 2x a ; x 2 R 2 : a2z 2 a2z 2 On the other hand, since p is a homogeneous polynomial of degree j, we have p x ˆ 2 j p 2x ˆ a2z 2 2 j u a f 2x a ; x 2 R 2 : But the shifts of f are stable. Thus, the above two equations yield S b u ˆ 2 j u for all u 2 E j. In particular, E j is invariant under S b. Therefore, spec S b j U ˆ [ 6r 2 jˆ0 spec S bj Ej ˆ spec S b j U4r 1 [ [ 6r 2 jˆ4r spec S bj Ej : 4:4 Note that each element in spec S b j Ej is equal to 2 j. Write q V for q T b j` \V. Combining (4.2)±(4.4) together, we obtain q 2r ˆ q T b ˆ max q j` \V4r 1 V ; 2 4r : 4:5 For convenience, we set D j 3 :ˆ D j, j ˆ 1; 2; 3. In order to nd q V, let W j be the minimal invariant subspace of T b generated by the sequences D r j 1 Dr j 2 d b, b 2 Z 2. Then V ˆ W 1 W 2 W 3, so q V ˆ max 1 6 j 6 3 n o q T b j Wj : Let S a denote the subdivision operator associated to a as de ned in (1.4). It follows from [10, Theorem 4.1] and [17, Theorem 3.2] that q lim 1=n q T b j W3 : n!1 rr 1 rr 2 Sn a d 2 ˆ Since ~a z ˆ 2 2r 1 z 1 r 1 z 2 r ~a 3 z, by [13, Theorem 3.2] we have lim n!1 rr 1 rr 2 Sn a d 1=n ˆ 2 2r lim S n 2 n!1 a 3 d 1=n : 2 But b ~ 3 z ˆ j~a 3 z j 2 =4, z 2 T 2. Hence lim n!1 S n a3 d 1=n p ˆ q T 2 b3. The preceding discussion tells us that q T b j Wj ˆ 2 4r q T bj is true for j ˆ 3. Clearly, this relation is also valid for j ˆ 1 or j ˆ 2. It follows that n o q V ˆ max q T b j Wj ˆ 2 4r max q T bj ˆ 2 4r q: 1 6 j j 6 3 By our assumption, q > 1. Hence, (4.5) tells us that q 2r ˆ maxfq V ; 2 4r g ˆ q V > 2 4r. It follows that 2r > log 4 q 2r. If, in addition, the shifts of / are stable, then

23 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155± Table 1 r m u r m f r m / ˆ log 4 q 2r ˆ log 4 q V ˆ 2r log 4 q: This veri es (4.1). Example 4.2. For r ˆ 1; 2;..., let h r be the mask on Z 2 given by its symbol ~h r z 1 ; z 2 ˆ z r 1 z r 2 1 z 1 r 1 z 2 r 1 z 1 z 2 r =2 3r 2 : There exists a unique sequence c r supported in 1 r; r 1Š 2 such that q r :ˆ h r c r is an interpolatory mask. Let u r be the normalized solution of the re- nement equation associated with mask q r. The smoothness order m u r was computed in [19] for r ˆ 2;... ; 8. Theorem 4.1 enables us to simplify the computation signi cantly so that we obtain m u r for r ˆ 9;... ; 16 as shown in Table 1. In [6] Deslauriers and Dubuc showed that, for each r ˆ 1; 2;..., there exists a unique interpolatory mask b r supported on 1 2r; 2r 1Š such that b r is symmetric about the origin and its symbol b ~ r z is divisible by 1 z 2r. Let f r be the normalized solution of the re nement equation / ˆ Pa2Z b r a / 2 a. The smoothness order m f r was computed in [8] for r ˆ 1; 2;... ; 20. For the purpose of comparison, we have listed the values of m f r (r ˆ 9;... ; 16) in Table 1. References [1] C. de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx. 3 (1987) 199±208. [2] C. de Boor, K. Hollig, S. Riemenschneider, Box Splines, Springer, New York, [3] A.S. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary Subdivision, Memoirs of Amer. Math. Soc., vol. 93 (453) (1991). [4] W. Dahmen, C.A. Micchelli, On the solution of certain systems of partial di erence equations and linear dependence of translates of box splines, Trans. Amer. Math. Soc. 292 (1985) 305± 320. [5] I. Daubechies, J.C. Lagarias, Two-scale di erence equations: II. Local regularity in nite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992) 1031±1079.

24 178 R.-Q. Jia, S. Zhang / Linear Algebra and its Applications 292 (1999) 155±178 [6] G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes, Constr. Approx. 5 (1989) 49±68. [7] N. Dyn, D. Levin, C.A. Micchelli, Using parameters to increase smoothness of curves and surfaces generated by subdivision, Computed Aided Geometric Design 7 (1990) 129±140. [8] T. Eirola, Sobolev characterization of solutions of dilation equations, SIAM J. Math. Anal. 23 (1992) 1015±1030. [9] T.N.T. Goodman, C.A. Micchelli, J.D. Ward, Spectral radius formulas for subdivision operators, in: L.L. Schumaker, G. Webb, (Eds.), Recent Advances in Wavelet Analysis, pp. 335±360, Academic Press, New York, [10] B. Han, R.Q. Jia, Multivariate re nement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998) 1177±1199. [11] C. Heil, G. Strang, V. Strela, Approximation by translates of re nable functions, Numer. Math. 73 (1996) 75±94. [12] R.Q. Jia, Multivariate discrete splines and linear Diophantine equations, Trans. Amer. Math. Soc. 340 (1993) 179±198. [13] R.Q. Jia, Subdivision schemes in L p spaces, Advances in Comp. Math. 3 (1995) 309±341. [14] R.Q. Jia, The Toeplitz theorem and its applications to Approximation Theory and linear PDEs, Trans. Amer. Math. Soc. 347 (1995) 2585±2594. [15] R.Q. Jia, Shift-invariant spaces and linear operator equations, Israel J. Math. 103 (1998) 259± 288. [16] R.Q. Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998) 647± 665. [17] R.Q. Jia, Characterization of smoothness of multivariate re nable functions in Sobolev spaces, Trans. Amer. Math. Soc., to appear. [18] R.Q. Jia, C.A. Micchelli, On linear independence of integer translates of a nite number of functions, Proc. Edinburgh Math. Soc. 36 (1992) 69±85. [19] S.D. Riemenschneider, Z.W. Shen, Multidimensional interpolatory subdivision schemes, SIAM J. Numer. Anal. 34 (1997) 2357±2381. [20] L.F. Villemoes, Wavelet analysis of re nement equations, SIAM J. Math. Anal. 25 (1994) 1433±1460.