Orthogonality criteria for refinable functions and function vectors

The Journal of Fourier Analysis and Applications Volume 6, Issue 2, 2000

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors Jeffrey C. Lagarias and Yang Wang Communicated by John J. Benedetto ABSTRACT. A refinable function φ(x) : Rn → R or, more generally, a refinable function vector 8(x) = [φ1 (x), . . . , φr (x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φj (x − α) : α ∈ Zn , 1 ≤ j ≤ r} form an orthogonal set of functions in L2 (Rn ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L2 (Rn ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.

1.

Introduction

Let A be an expanding matrix in Mn (Z), that is, one with integer entries and all eigenvalues |λ| > 1. A refinable function φ(x) : Rn → R is a solution to a refinement equation with dilation matrix A, X cα φ(Ax − α) , (1.1) φ(x) = α∈Z

in which {cα : α ∈ Z} are complex coefficients. More generally, a vector valued function 8(x) = [φ1 (x), . . . , φr (x)]T is called a refinable function vector, if it satisfies a vector refinement equation with dilation A, X Cα 8(Ax − α) , (1.2) 8(x) = α∈Zn

Math Subject Classifications. 42C10, 42C15. Keywords and Phrases. orthogonal refinable function, orthogonal refinable function vector, orthogonality criteria, wavelet, multiwavelet. Acknowledgements and Notes. Second author: Research supported in part by the National Science Foundation, grant DMS-97-06793.

c 2000 Birkh¨auser Boston. All rights reserved

ISSN 1069-5869

154

Jeffrey C. Lagarias and Yang Wang

where each Cα is a matrix in Mr (C). We call n the dimension and r the vector-multiplicity of the refinable function vector. We only consider the case that such functions and vector-valued functions have all components in L1 (Rn ). Refinable function vectors are natural generalizations of refinable functions (r = 1). The latter have been studied extensively due to their applications in constructing compactly supported orthonormal wavelet bases and in approximation theory, see Daubechies [9], [10]. General constructions are based on multiresolution analysis, for which see Mallat [28] and Jia and Shen [21]. More recently, refinable function vectors have been used to construct orthonormal multiwavelet bases, see for example Cohen et al. [4], Donovan et al. [12], Goodman and Lee [14], and Goodman et al. [15]. Multiwavelets can be made to combine smoothness with small support, an advantage that may be important in applications. In constructing orthonormal wavelet or multiwavelet bases, one requires that all integer translates of refinable functions or function vectors be orthogonal. A fundamental question in constructing orthonormal wavelet or multiwavelet bases is thus: under what conditions does a refinable function or function vector 8(x) have the property that all its integer translates {8(x − α) : α ∈ Zn } are orthogonal? This paper addresses the above question by giving a collection of necessary and sufficient conditions for orthogonality, derived in terms of the coefficients of the refinement equations and the dilation matrix A. We treat only the case where the vector refinement equation has finitely many nonzero coefficients. In this case, if the equation has a solution in L1 (Rn ), then it must be compactly supported.1 Also in this case, there is in principle a finite algorithm to determine whether a given vector refinement equation has a nonzero solution which is orthogonal in the sense of Definition 1 below. The criteria of this paper typically do not make sense in the case of infinitely many nonzero coefficients, but some sufficient conditions have been obtained by Conze et al. [7] in the infinite coefficient case. Various results regarding the orthogonality of compactly supported refinable functions and function vectors are known, especially for r = 1 and n = 1. Many (but not all) of these results generalize to higher dimensions (r = 1 and n > 1), and to compactly supported refinable function vectors. However, few of these generalizations have been documented, and even in those papers which discuss higher dimensional cases, the dilation matrix A was usually chosen to be 2In . As we see from Theorems 2 and 3 below, orthogonality conditions vary for different dilation matrices A. The object of this paper is to provide a comprehensive set of orthogonality criteria for compactly supported refinable functions and function vectors in the most general setting.

Definition 1.R Let 8(x) be a compactly supported refinable function vector. We say that 8(x) is orthogonal if

Rn

8(x)dx 6 = 0 and Z 8(x − α)8T (x − β)dx = δα,β 3, Rn

α, β ∈ Zn

(1.3)

where δα,β denotes the standard Kronecker symbol, and 3 is a diagonal matrix with positive diagonal entries. R The condition Rn 8(x)dx 6= 0 is necessary2 for the construction of multiwavelet bases associated to a multiresolution analysis. It is well known that for a compactly supported refinable function vector 8(x) to be orthogonal the coefficient matrices Cα of the corresponding vector refinement equation (1.2) must satisfy the necessary conditions encoded in (i) and (ii) of the following definition.

1 The converse is false, see Strang et al. [33]. Furthermore, a refinement equation with finitely many nonzero coefficients may also have a noncompactly supported L2 solution, see Malone [29]. 2 In fact this condition is automatically fulfilled under the orthogonality condition, see Lemma 2 (4) below.

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors

155

Definition 2. The vector refinement equation (1.2) with finitely many Cα 6= 0 satisfies the orthog-

onal coefficients condition (with respect to 3 where 3 is a diagonal matrix with positive diagonal entries) if the coefficients Cα satisfy the two properties P (i) 1 is an eigenvalue of the matrix | det(A)|−1 α∈Zn Cα . (ii) X ∗ Cα 3Cα+Aβ = δ0,β | det(A)| 3 . (1.4) α∈Zn

The necessity of condition (i) for orthogonality follows from Proposition 1 below. A proof of condition (ii) can be found in Flaherty and Wang [13]. Unfortunately, the orthogonal coefficients condition is not sufficient for the orthogonality of the corresponding refinable function vector 8(x), even for r = 1. The simplest counterexample, which has r = 1 and n = 1, is the refinement equation φ(x) = φ(2x) + φ(2x − 3) . It satisfies the orthogonal coefficients condition, but the solution φ(x) = χ[0,3) (x) has non-orthogonal integer translates. To ensure orthogonality of refinable functions and function vectors, additional conditions are needed. In the nonvector case r = 1, n = 1, these conditions were found by various authors, and the most prominent of these conditions is Cohen’s Criterion, due to Cohen [3]. We shall list them in Section 3. It should be pointed out that many of the criteria are given in the contrapositive form as conditions for 8(x) not being orthogonal. The contents of this paper are as follows: in Section 2 we state the orthogonality criteria for compactly supported refinable function vectors with arbitrary vector-multiplicity r, and in Section 3 we state a larger set of orthogonality criteria that are available for the special case r = 1, i.e., for compactly supported refinable functions. These criteria are then proved in Section 4 for arbitrary r and in Section 5 for r = 1. We add a comment on the novelty of the results. Many of the results for compactly supported refinable function vectors stated in Section 2 are new, as is the Generalized Cohen’s Criterion stated there. In particular criterion (d) in Theorem 2 is new and (c) is stated for the first time. The proofs extend some of the ideas of the r = 1 case stated as Theorem 3 (a)–(d) in Section 3, but there is extra complexity arising from products of matrices. The results in Section 3 for r = 1 and arbitrary dimension n have not all been stated before, but we do not claim significant novelty in the proofs. The most important idea leading to the criteria in Theorem 3 (e)–(f) is a result on transfer operators due to Cerveau et al. [2]. Other orthogonality criteria for the case r = 1 based on this result were derived by Conze et al. [7]. Further remarks on previous results appear at the end of Section 3. We are greatly indebted to K. Gröchenig for introducing us to this problem. The results and techniques in his paper [16] for the case r = 1 and n = 1 inspired our results. Several of his proofs generalize to dimension n > 1, see the discussion after Theorem 3. We are also indebted to Ingrid Daubechies, Andy Haas, Chris Heil, and Jianao Lian for helpful discussions and references. Finally, we would like to thank the anonymous referee for carefully reading the manuscript and providing valuable comments and suggestions.

2.

Orthogonality Criteria for Refinable Function Vectors

Throughout this paper we will be concerned with compactly supported refinable function vectors. Therefore, we assume that the vector refinement equation X 8(x) = Cα 8(Ax − α) (2.1) α∈Zn

156

Jeffrey C. Lagarias and Yang Wang

where Cα ∈ Mr (C) has only finitely many nonzero coefficient matrices Cα . In this section we state orthogonality criteria; the proofs are given in Section 4.

Definition 3. For a given vector refinement equation (2.1) we define its symbol m(ξ ) to be m(ξ ) := | det(A)|−1

X

Cα e−i2πhα,ξ i .

(2.2)

α∈Zn

The symbol m together with the expanding integer matrix A specifies the vector refinement equation uniquely, where we view m as a formal object containing all the coefficients Cα . However, we also view the symbol as defining a matrix-valued function m(ξ ) : Rn → Mr (C). Suppose that 8(x) is a refinable function vector satisfying (2.1). Then the Fourier transform of 8(x) satisfies     b B −1 ξ , b(ξ ) = m B −1 ξ 8 (2.3) 8 where B := AT , and the Fourier transform is applied term-by-term to the vector 8(ξ ). Denote n

o (2.4) Lrp Rn := 8(x) = [φ1 (x), . . . , φr (x)]T : each φj (x) ∈ Lp Rn . The following is a necessary condition for the orthogonality of 8(x):

Proposition 1. Let 8(x) be a compactly supported orthogonal refinable function vector satisfying X Cα 8(Ax − α) 8(x) = α∈Zn

with finitely many Cα 6 = 0. Then 1 is a simple eigenvalue of the r × r matrix m(0), and all other eigenvalues λ of m(0) satisfy |λ| < 1. Proposition 1 is a corollary of a stronger result of Hogan [20], in which the orthogonality of 8(x) is replaced by the weaker condition of stability. We include an independent proof of Proposition 1 in Section 4 for completeness. To state the general orthogonality criteria we must introduce the transfer operator Cm associated to the symbol m and dilation matrix A [and hence to (2.1)]. Let r×r (Rn ) denote the linear space of r × r Hermitian matrices whose entries are trigonometric polynomials with complex coefficients, i.e., functions of the form g(e−2π iξ1 , . . . , e−2π iξn ) where g is a Laurent polynomial in n variables, with ξ = (ξ1 , . . . , ξn ) ∈ Rn . Note that each F (ξ ) ∈ r×r (Rn ) is Zn -periodic, so we may
r×r . For any trigonometric polynomial view r×r (Rn ) as a subspace of the Hilbert space L2 (Tn ) P hγ i −i2π ,ξ of matrix coefficients we define its support to be F (ξ ) = γ ∈Zn Fγ e  supp(F ) = γ ∈ Zn : Fγ 6 = 0 .

Definition 4. The transfer operator Cm is a linear operator on r×r (Rn ) defined by Cm F (ξ ) :=

X

      m B −1 (ξ + d) F B −1 (ξ + d) m∗ B −1 (ξ + d) ,

(2.5)

d∈E

in which B = AT and E is a complete set of coset representatives of Zn /B(Zn ). It is not hard to check, using the computations in Section 4, that Cm (F ) ∈ r×r (Rn ) for any F ∈ r×r (Rn ), and it is independent of the choice of the coset representatives E. Furthermore, if (2.1) satisfies the orthogonal coefficients condition with respect to 3, then Cm 3 = 3. The linear space r×r (Rn ) is infinite-dimensional, but we will show that when the vector refinement

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors

157

equation with symbol m has finitely many nonzero coefficients we can restrict the action of the transfer operator to certain finite dimensional invariant subspaces of r×r (Rn ) depending on the symbol m and on A which contains the crucial information for orthogonality. We call a nonempty set S ⊆ Zn (m, A)-invariant if for any γ 6 ∈ S the elements Aγ +α−β 6∈ S for all α, β ∈ supp(m). An important (m, A)-invariant set is 
(2.6) Sm,A := γ ∈ Zn : Tm,A ∩ Tm,A + γ 6 = ∅ where Tm,A is the attractor of the iterated function system {A−1 (x + γ ) : γ ∈ supp(m)}. Clearly Sm,A is finite if supp(m) is.

Proposition 2. (i) Sm,A is (m, A)-invariant. (ii) Let S be a finite (m, A)-invariant set. Then

r×r Rn , S := F (ξ ) ∈ r×r Rn : supp(F ) ⊆ S is a Cm -invariant finite dimensional subspace of r×r (Rn ). By results of Cohen et al. [5] or Heil and Collela [19], if 1 is a simple eigenvalue of m(0) and all other eigenvalues λ of m(0) have |λ| < 1, then for B = AT the infinite (right) product p(ξ ) :=

∞ Y

  m B −j ξ

(2.7)

j =1

converges uniformly on any compact set of Rn . This defines p(ξ ) : Rn → Mr (Rn ). We have:

Theorem 1. Let 8(x) be a compactly supported refinable function vector satisfying X Cα 8(Ax − α) 8(x) = α∈Zn

where A ∈ Mn (Z) is expanding and finitely many Cα 6= 0. Suppose that the vector refinement equation satisfies the orthogonal coefficients condition and that 1 is a simple eigenvalue of m(0) while all other eigenvalues λ of m(0) satisfy |λ| < 1. Then the following statements are equivalent: (a) 8(x) is not orthogonal. (b) There exists an F (ξ ) ∈ r×r (Rn ), F (ξ ) 6= a3 for any a ∈ C, such that Cm F = F . (c) Let S be a finite (m, A)-invariant set containing Sm,A . The eigenvalue 1 of Cm restricted to r×r (Rn , S) is a multiple eigenvalue. (d) There exist η ∈ Rn \ Zn and a nonzero vector u0 ∈ C r such that u∗0 p(η + α) = 0, all α ∈ Zn .

(2.8)

The equivalence of (a) and (b) in Theorem 1 was established by several authors in the one dimension for the dilation 2, see Plonka [31] and Lian [27]. It was established in all dimensions for the dilation matrix A = 2In in Shen [32], and his proof should generalize to work for an arbitrary dilation matrix A. In addition, it was shown in [32] that under the hypotheses of Theorem 1 the orthogonality of 8(x) is equivalent to the stability of 8(x) and is equivalent to the L2 -convergence of the cascade algorithm. Several variations of criterion (b) were also given in [27].

Remark. We shall see in Section 4 that the equivalence of (a) and (c) relies only on the orthogonal coefficients condition, not on the assumptions regarding the eigenvalues of m(0). The equivalence

158

Jeffrey C. Lagarias and Yang Wang

of (a) and (c) gives rise to an algorithm for checking the orthogonality of a refinable function vector 8(x), which is a generalization of the algorithm in Lawton [25] for n = 1 and r = 1. In fact, all we need is to find a finite (m, A)-invariant set S containing Sm,A and check the multiplicity of the eigenvalue 1 for Cm restricted to r×r (Rn , S). Such a set is quite easy to find. Since A is expanding, there exists a norm k.k on Rn such that kAk ≥ s > 1. Let L be the diameter of supp(m). One such S is   L S = α ∈ Zn : kαk ≤ . (2.9) s−1 The drawback with this S is that it is often much larger than Sm,A , making the dimension of r×r (Rn , S) much larger than necessary. Fortunately there is a simple algorithm to find Sm,A . Here we skip the details; they can be found in Strichartz and Wang [34]. A corollary of Theorem 1 is the following generalization of Cohen’s Criterion. Recall that a set K ⊂ Rn is a fundamental domain of Zn if K is congruent to [0, 1)n modulo Zn .

Corollary 1 (Generalized Cohen’s Criterion).

Under the assumptions of Theorem 1, suppose that for each u0 ∈ C r there exists a fundamental domain Ku0 of Zn such that u∗0 p(ξ ) 6= 0, all ξ ∈ Ku0 . Then 8(x) is orthogonal. This corollary differs in appearance from the original Cohen’s Criterion in the case r = 1. This is due to the occurrence of infinite products of matrices which do not commute in general. For the special case r = 1, u∗0 p(ξ ) 6= 0 is equivalent to p(B −j ξ ) 6= 0 for all j ≥ 1. In this case the condition of Corollary 1 is equivalent to p(B −j ξ ) 6 = 0 for all j ≥ 1, where B = AT , on some fundamental domain of Zn . This is precisely the original form of Cohen’s Criterion, see Cohen [3].

3.

Orthogonality Criteria for Refinable Functions

More detailed criteria are available for orthogonality in the case r = 1, i.e., of refinable functions in Rn . In this section we state such criteria; the proofs are given in Section 5. The criteria of Theorem 1 can be strengthened for r = 1, especially when the dilation matrix A is irreducible over Z. A matrix A ∈ Mn (Z) is irreducible over Z if its characteristic polynomial fA (λ) is irreducible over Z. In particular, if A ∈ Mn (Z) is expanding and | det(A)| is a prime, then A is irreducible over Z. Note that if r = 1, then r×r (Rn ) = 1×1 (Rn ) is the space of all real trigonometric polynomials over Rn , and we set (Rn ) := 1×1 (Rn ). Let the invariant set Sm,A be as in (2.6) and set (Rn , S) := {F (ξ ) ∈ (Rn ) : supp(F ) ⊆ S}.

Theorem 2. Let A ∈ Mn (Z) be an expanding matrix that is irreducible over Z. Suppose that the compactly supported nontrivial φ(x) ∈ L2 (Rn ) satisfies the refinement equation X φ(x) = cα φ(Ax − α) , α∈Zn

which satisfies the orthogonal coefficients condition and has finitely many cα 6= 0. Let m(ξ ) be its symbol and B = AT . Then the following statements are equivalent: (a) (b)

The refinable function φ(x) is not orthogonal. There exists a nonconstant f (ξ ) ∈ (Rn ) such that Cm f = f .

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors

159

Let S be a finite (m, A)-invariant set containing Sm,A . The eigenvalue 1 of Cm restricted to (Rn , S) is a multiple eigenvalue. has the property: for each α ∈ Zn there exists a j (α) ≥ 1 (d) There exists η0 ∈ Rn \ Zn that
such that m B −j (α) (η0 + α) = 0.
(e) There exists ξ0 ∈ Rn \Zn such that B N ξ0 ≡ ξ0 (mod Zn ) for some N > 0, and m B j ξ0 = 1 for all j ≥ 0. (f) There exists ξ0 ∈ Rn \ Zn such that B N ξ0 ≡ ξ0 (mod Zn ) for some N > 0, and m B j ξ0 +
B −1 l = 0 for all j ≥ 0 and all l ∈ Zn \ B(Zn ). (c)

We derive Theorem 2 as a special case of a more general result that applies to an arbitrary expanding integer matrix A, given below as Theorem 3, which requires a more complicated generalization of (e) and (f). To state it, for each l ∈ Zn we denote  −1 (ξ + l) . τl (ξ ) := AT A rational subspace of Rn is a linear subspace W having a basis consisting of rational vectors v ∈ Qn . A set of vectors {zj : 0 ≤ j < N} in Rn is a periodic orbit of AT (mod Zn ) if
AT zj +1 ≡ zj mod Zn , 0 ≤ j < N , where zN := z0 . We let E denote an arbitrarily chosen complete set of coset representatives of Zn /AT (Zn ).

Theorem 3. Let A ∈ Mn (Z) be an expanding matrix. Suppose that the compactly supported nontrivial φ(x) ∈ L2 (Rn ) satisfies the refinement equation X cα φ(Ax − α) , φ(x) = α∈Zn

which satisfies the orthogonal coefficients condition and has finitely many cα 6= 0. Let m(ξ ) be its symbol and B = AT . Then the following statements are equivalent: (a) The refinable function φ(x) is not orthogonal. (b) There exists a nonconstant f (ξ ) ∈ (Rn ) such that Cm f = f . (c) Let S be a finite (m, A)-invariant set containing Sm,A . The eigenvalue 1 of Cm restricted to (Rn , S) is a multiple eigenvalue. n (d) There exists η0 ∈ Rn \ Zn that
has the property: for each α ∈ Z there exists a j (α) ≥ 1 −j (α) (η0 + α) = 0. such that m B (e) There exists a proper B-invariant rational subspace W of Rn and a periodic orbit {zj : 0 ≤ j < N } of B (mod Zn ) with every zj 6 ∈ W + Zn , such that X

|m (τl (ξ ))|2 = 1

l∈E τl (ξ )∈zj +1 +W +Zn

(3.1)

for all ξ ∈ zj + W , where 0 ≤ j < N with zN := z0 and E is a set of complete coset representatives of Zn /B(Zn ). (f) There exists a proper B-invariant rational subspace W of Rn and a periodic orbit {zj : 0 ≤ j < N } of B (mod Zn ) with zj 6 ∈ W + Zn , such that m (τl (ξ )) = 0

if l ∈ Zn and τl (ξ ) 6 ∈ zj +1 + W + Zn

for all ξ ∈ zj + W , where 0 ≤ j < N and zN := z0 .

160

Jeffrey C. Lagarias and Yang Wang

Remark. A transfer operator applied to wavelet bases apparently first appears in the appendix of Daubechies [9], and such operators were analyzed in Conze and Raugi [8]. The orthogonality criteria in Theorem 3 in dimension n = 1 for the case r = 1 were found by Cohen [3], Lawton [25], Conze and Raugi [8], and Cohen and Sun [6], and an elegant summary can be found in Gröchenig [16]. The equivalence of (a), (b), and (d) in dimension n > 1 is proved here by generalizing the arguments of Gröchenig in one dimension. In higher dimensions, Lawton et al. [26] gave an orthogonality criterion similar to (b), using the wavelet-Galerkin operator defined on l 2 (Zn ) instead of the transfer operator. Criteria (e) and (f) in Theorems 2 and 3 are much harder to prove. The proof given here uses as a principal ingredient a recent result of Cerveau et al. [2] concerning the structure of the set of zeros of eigenfunctions of transfer operators in the multidimensional case. The paper of Conze et al. [7], Section II, applies this result to give various orthogonality criteria, some of which apply even when an infinite number of cα 6= 0 in (1.2).

4.

Proof of Orthogonality Criteria for Function Vectors

For a given Hermitian matrix Q ∈ Mr×r (C) we define the norm k.kQ on Cr √ ∗positive definite ∗ by kxkQ := x Qx where x = x¯ T . This norm induces a matrix norm in Mr×r (C), which we also denote by k.kQ . Throughout this section, 3 denotes a diagonal matrix with positive diagonal entries.

Lemma 1. Suppose that the vector refinement equation (2.1) has finitely many Cα 6 = 0 and satisfies the orthogonal coefficients condition with respect to the diagonal matrix 3. Let E be any set of complete coset representatives of Zn /B(Zn ) where B = AT . Then Cm 3 = 3. (2) km∗ (ξ )k3 ≤ 1 for all ξ ∈ Rn . (3) Let v be Pa left λ-eigenvector of m(0) with |λ| = 1. Then v is a left λ-eigenvector of 1α := β∈Zn Cα+Aβ for all α ∈ Zn . (4) For any 1-eigenvector u0 of m(0), the vector refinement R equation (2.1) has a unique compactly supported solution 8(x) ∈ Lr2 (Rn ) such that Rn 8(x) dx = u0 .

(1)

Proof. (1) Let q = | det(A)| and B = AT . Then Cm 3

=

X d∈E

=

q −2

    m ξ + B −1 d 3m∗ ξ + B −1 d XX X d∈E α∈Zn β∈Zn

=

q −2

X X

α∈Zn γ ∈Zn



Cα 3Cβ∗ e−i2π α−β,ξ +B

∗ Cα 3Cα+γ

X



−1 d

e−i2π γ ,ξ +B



−1 d

d∈E

It follows from X d∈E

e

 −i2π γ ,ξ +B −1 d

 =

qe−i2πhγ ,ξ i 0


if γ ∈ A Zn , otherwise



.

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors

161

that Cm 3

q −1

=

X X α∈Zn β∈Zn

q −1

=

X

∗ Cα 3Cα+Aβ e−i2πhAβ,ξ i

e−i2πhAβ,ξ i

β∈Zn

q −1

=

X

X α∈Zn

∗ Cα 3Cα+Aβ

e−i2πhAβ,ξ i qδ0,β 3

β∈Zn

=

3.

(2) Choose E so that 0 ∈ E. By part (1), for any v ∈ Cr , X

    v ∗ m ξ + B −1 d 3m∗ ξ + B −1 d v = v ∗ 3v .

d∈E

Thus, km∗ (ξ )vk3 ≤ kvk3 for all ξ by taking d = 0, proving (2). (3) Let D be a complete set of coset representatives of Zn /A(Zn ). Then one easily checks that X α∈D

P

α∈D

1α = qm(0), and

1α 31∗α = q3 .

The above together with the Schwarz inequality yield

2

X X

kv1α k23 = q 2 kvk23 , v1α ≤ q

α∈D

3

α∈D

and the equality holds if and only if all v1α are equal. Now

2

X

v1α = kqvm(0)k23 = kqλvk23 = q 2 kvk23 .

α∈D

3

P So v1α = v0 for all α ∈ D, and α∈D v1α = qvm(0) = qλv implies that v0 = λv. Finally, for any β ∈ Zn there is an α ∈ D such that 1β = 1α . This proves (3). (4) For n = 1 and r = 1 this is a well-known result of Mallat [28]. Mallat’s proof generalizes easily to the general case. A proof of this partQcan be found in Flaherty and Wang [13]. We remark that −j b(ξ ) = ( ∞ the solution 8(x) is given by 8 j =1 m(B ξ ))u0 . A proof of Proposition 1 can be found in Hogan [20]. Here we present a different proof.

Proof of Proposition1. Let λ be an eigenvalue of m(0) and u0 be a left P λ-eigenvector of m(0).

 By (2) of Lemma 1 we have |λ| ≤ 1. Suppose that |λ| = 1. Define g(x) = α∈Zn 8(x + α), u∗0 .

162

Jeffrey C. Lagarias and Yang Wang

We view g(x) as a function in L1 (Tn ). Observe that X X

 Cβ 8(Ax + Aα − β), u∗0 g(x) = α∈Zn β∈Zn

=

X X

α∈Zn γ ∈Zn

=

X

γ ∈Zn

=

1−γ 8(Ax + γ ), u∗0

XD

γ ∈Zn

=

CAα−γ 8(Ax + γ ), u∗0





8(Ax + γ ), 1∗−γ u∗0

E

λ¯ g(Ax) ,

P where 1−γ = α∈Zn CAα−γ and 1∗−γ u∗0 = λu∗0 by (3) of Lemma 1. So |g(x)| = |λ||g(Ax)|. It c for some constant c, so g(x) ∈ L2 (Tn ). follows from the ergodicity of A on Tn that P |g(x)| =i2πhα,xi . The equality g(x) = λg(Ax) yields Consider the Fourier expansion of g(x) = α∈Zn bα e bα = 0 for all α 6 = 0 and b0 = 0 if λ 6 = 1, by comparing the Fourier coefficients of g(x) and λg(Ax). If λ 6 = 1, then g(x) = 0 almost everywhere. But this is impossible because 8(x) is orthogonal. So λ = 1. In this case, the ergodicity of A on Tn implies that g(x) = c almost everywhere for some constant c. We show that 1 is a simple eigenvalue of m(0). If not, because km(0)k3 ≤ 1 for some positive definite diagonal matrix 3, m(0) must have two independent left 1-eigenvectors u1 , u2 ∈ Cr . Therefore, there exists a nonzero linear combination u of u1 , u2 such that X

 8(x − α), u∗ = 0 a.e. . α∈Zn

Again this contradicts the orthogonality of 8(x).

Proof of Proposition 2. (i) By definition A(Tm,A ) = Tm,A + supp(m). For any γ 6∈ Sm,A we have ∅



= A Tm,A ∩ Tm,A + γ

= Tm,A + supp(m) ∩ Tm,A + Aγ + supp(m) [

Tm,A ∩ Tm,A + Aγ + α − β + β . = α,β∈supp(m)

So Aγ + α − β 6 ∈ Sm,A for all α, β ∈ supp(m). Therefore, Sm,A is (m, A)-invariant. P (ii) Let F (ξ ) = γ ∈S Fγ e−i2πhγ ,ξ i ∈ r×r (Rn , S). It is straightforward to check that (Cm F ) (ξ ) =

X γ ∈Zn

Gγ e−i2πhγ ,ξ i , where Gγ =

X α,β∈Zn

Cα FAγ +β−α Cβ∗ .

Suppose that Gγ 6 = 0. Then there exist α, β ∈ Zn such that Cα FAγ +β−α Cβ∗ 6 = 0, so α, β ∈ supp(m) and Aγ + β − α ∈ S. It follows that γ ∈ S. Hence Cm F ∈ r×r (Rn , S). We now prove the orthogonality criteria for refinable function vectors. We first introduce some notation to simplify our exposition. For any k > 0 we let mk (ξ ) denote the (right) product mk (ξ ) =

k Y j =1

        m B k−j ξ := m B k−1 ξ m B k−2 ξ · · · m B 0 ξ

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors

163

where B = AT . Given a complete set of coset representatives E of Zn /B(Zn ) let EB,k := E + BE + · · · + B k−1 E . Observe that k F (ξ ) = Cm

      mk B −k (ξ + d) F B −k (ξ + d) mk B −k (ξ + d) .

X

(4.1)

d∈EB,k

Proof of Theorem 1. The standard Possion Summation Formula gives X Z α∈Zn

Rn

 X b(ξ + α)8 b∗ (ξ + α) . 8 8(x)8∗ (x + α) dx ei2πhα,ξ i =

(4.2)

α∈Zn

(a) ⇒ (b). The proof here is a generalization of the proof in Gröchenig [16] for the case n = 1, r = 1. Suppose that 8(x) is not orthogonal. Then F (ξ ) :=

X Z α∈Zn

Rn

 8(x)8∗ (x + α) dx ei2πhα,ξ i

is in r×r (Rn ) and F (ξ ) 6 = a3 for any a ∈ C. We show that Cm F = F . Let E be any complete set of coset representatives for Zn /B(Zn ). Denote ξd := B −1 (ξ + d). Then Cm F (ξ )

=

X

m (ξd ) F (ξd ) m∗ (ξd )

d∈E

= = = = =

XX

d∈E

α∈Zn

d∈E

α∈Zn

d∈E

α∈Zn

XX

XX X

b (ξd + α) 8 b∗ (ξd + α) m∗ (ξd ) m (ξd ) 8 b (ξd + α) 8 b∗ (ξd + α) m∗ (ξd + α) m (ξd + α) 8 b(ξ + d + Bα)8 b∗ (ξ + d + Bα) 8

b(ξ + α)8 b∗ (ξ + α) 8

α∈Zn

F (ξ ) .

(b) ⇒ (c). Since supp(8) ⊆ Tm,A , we see that supp(F ) ⊆ Sm,A . Therefore, F ∈ r×r (Rn , S) since S contains Sm,A . Observe that 0 ∈ Sm,A , so G(ξ ) := 3 ∈ r×r (Rn , S), and is also a 1-eigenvector of Cm . So 1 is a multiple eigenvalue of Cm restricted to r×r (Rn , S), proving (c). (c) ⇒ (b). Since 1 is a multiple eigenvalue of Cm restricted to r×r (Rn , S), either Cm has k is two independent 1-eigenvectors in r×r (Rn , S), in which case we complete our proof, or Cm n unbounded in r×r (R , S) as k → ∞. We show that the latter is impossible. Assume that it did, then there exists a F (ξ ) ∈ r×r (Rn , S) such that C k F is unbounded as k → ∞. By adding a3 to F for a sufficiently large a > 0 we may without loss of generality√assume that F (ξ ) is positive definite for all ξ . Let 0 be the positive definite diagonal matrix 0 := 3. Then for any ξ ∈ Rn and

164

Jeffrey C. Lagarias and Yang Wang

u ∈ Cr , (0u)∗ 0 −1 (Cm ) F (ξ )0 −1 (0u) =

u∗ (Cm F ) (ξ )u X u∗ m (ξd ) F (ξd ) m∗ (ξd ) u = d∈E

=

X

  u∗ m (ξd ) 0 0 −1 F (ξd ) 0 −1 0m∗ (ξd ) u

d∈E

≤

ρ0 (F )

X

u∗ m (ξd ) 00m∗ (ξd ) u

d∈E ∗

= ρ0 (F )u 3u = ρ0 (F )(0u)∗ (0u) , where ξd := B −1 (ξ + d) and ρ0 (F ) is the supreme over all ξ of the spectral radii ρ(0 −1 F (ξ )0 −1 ). Therefore, the spectral radius of 0 −1 (Cm F )(ξ )0 −1 is bounded by ρ0 (F ). This implies that for k F )(ξ )0 −1 is bounded by ρ (F ). But this would mean that all k the spectral radius of 0 −1 (Cm 0 k −1 −1 0 (Cm F )(ξ )0 is bounded for all ξ and k because it is Hermitian. This is a contradiction. (b) ⇒ (d). Since F (ξ ) is bounded and periodic (mod Zn ), there exist a+ , a− ∈ R such that  a+ = inf a ∈ R : a3 − F is positive definite for all ξ ∈ Rn ,  a− = sup a ∈ R : F − a3 is positive definite for all ξ ∈ Rn . Let F+ (ξ ) = a+ 3 − F (ξ ) and F− (ξ ) = F (ξ ) − a− 3. Then both F+ and F− are nonnegative definite but neither is positive definite for all ξ ∈ Rn . To simplify our notation we let 1 := m(0). The hypotheses of the theorem implies that 1∞ := limk→∞ 1k exists and is a rank one matrix whose columns are 1-eigenvectors of 1.

Claim 1.

Suppose that F+ (ξ ) (resp., F− (ξ )) is singular for ξ ∈ Zn only. Then F+ (0)v0 = 0 (resp., F− (0)v0 = 0) where v0 6 = 0 is a 1-eigenvector of m∗ (0).

Proof of Claim 1. We prove the claim for F+ (ξ ), the proof is identical for F− (ξ ). Let v ∈ Cr k F = F that such that kvk3 = 1, v ∗ F+ (0) = 0. Then it follows from Cm + +       X 0 = v ∗ F+ (0)v = v ∗ mk B −k d F+ B −k d m∗k B −k d v .

(4.3)

d∈EB,k

Since Cm is independent of the choice of E we choose 0 ∈ E. Now all F+ (B −k d) are positive definite unless B −k d ∈ Zn , which holds only for d = 0. We thus have m∗k (B −k d)v = 0 for all d ∈ EB,k , d 6 = 0. Note that the orthogonal coefficients condition gives  2  X

∗

mk B −k d v = kvk23 = 1 .

d∈EB,k

3

Hence km∗k (0)vk3 = k(1∗ )k vk3 = 1. It follows by letting k → ∞ that v0 := (1∞ )∗ v 6 = 0. Clearly v0 is the unique (up to scalar multiples) 1-eigenvector of 1∗ . By (4.3) v0∗ F+ (0)v0 = 0, and hence F+ (0)v0 = 0 by the nonnegative definiteness of F+ (0), proving the claim.

Claim 2.

Either G(ξ ) = F+ (ξ ) or G(ξ ) = F− (ξ ) has the property that 1∞ G(0) 6= 0 and G(η) is singular for some η ∈ Rn \ Zn .

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors

165

Proof of Claim 2. First we observe that F+ (ξ ) + F− (ξ ) = (a+ − a− )3 is always nonsingular, so Claim 1 implies that at least one of F+ (ξ ) and F− (ξ ) is singular for some η ∈ Rn \ Zn . Assume that Claim 2 is false. Then either 1∞ G(0) = 0 or G(η) is nonsingular for all η ∈ Rn \ Zn , where G(ξ ) is either F+ (ξ ) or F− (ξ ). Now 1∞ (F+ (0) + F− (0)) 6 = 0 because F+ (0) + F− (0) is nonsingular, so either 1∞ F+ (0) 6 = 0 or 1∞ F− (0) 6 = 0. If both are nonzero then we have a contradiction. So without loss of generality we assume that 1∞ F+ (0) = 0 and thus F− (η) is nonsingular for all η ∈ Rn \ Zn . By Claim 1 we have F− (0)v0 = 0, where v0 is a 1-eigenvector of 1∗ . Now, v0∗ 1∞ = v0∗ . So v0∗ (F+ (0) + F− (0)) v0 = v0∗ 1∞ (F+ (0) + F− (0)) v0 = 0 . This contradicts the positive definiteness of F+ (0) + F− (0), proving Claim 2. To finish proving (b) ⇒ (d), let G(ξ ) be F+ (ξ ) or F− (ξ ) such that 1∞ G(0) 6 = 0 and G(η) is singular for some η ∈ Rn \ Zn . Let G(η)u0 = 0 for some nonzero u0 ∈ Cr . We show that u∗0 p(η + α) = 0 for all α ∈ Zn . For a given α ∈ Zn , we write α = B l β for some β ∈ Zn \ B(Zn ). k G = G that Choose E so that 0, β ∈ E. Then for all k > l we have α ∈ EB,k . It follows from Cm 0 = u∗0 G(η)u0 =

X d∈EB,k

      u∗0 mk B −k (η + d) G B −k (η + d) m∗k B −k (η + d) u0 .

In particular we have       u∗0 mk B −k (η + α) G B −k (η + α) m∗k B −k (η + α) u0 = 0 . It follows by letting k → ∞ that u∗0 p(η + α)G(0)p∗ (η + α)u0 = 0 , and the nonnegative definiteness of G(0) yields u∗0 p(η + α)G(0) = 0 . Observe that p(ξ ) = p(ξ )1∞ . So p(ξ )G(0) = p(ξ )1∞ G(0). Since 1∞ G(0) 6 = 0 and 1∞ has rank 1, there exists a nonzero column v1 in 1∞ G(0), which is clearly a 1-eigenvector of 1. Hence all columns of p(ξ ) are scalar multiples of p(ξ )v1 . Thus u∗0 p(η + α) = 0. n b(ξ ) = p(ξ )8 b(0) that u∗ 8 b (d) ⇒ (a). It follows from 8 0 (η + α) = 0 for all α ∈ Z . Hence by the Poisson Summation Formula,  Z X X b(η + α)8 b∗ (η + α)u0 = 0 u∗0 8(x)8∗ (x − α) dx u0 ei2πhα,ηi = u∗0 8 α∈Zn

Therefore,

Rn

α∈Zn

X Z α∈Zn

Rn

 ˜ 8(x)8∗ (x − α) dx ei2πhα,ηi 6= 3

˜ with positive diagonal entries, and so 8(x) cannot be orthogonal. for any diagonal matrix 3

Proof of Corollary 1. It follows easily from the fact that for any fundamental domain K one of η + α in (d) of Theorem 1 is in K.

166

Jeffrey C. Lagarias and Yang Wang

5.

Proof of Orthogonality Criteria for Refinable Functions Let Tn be the n-dimensional torus Tn := Rn /Zn , and πn : Rn → Tn be the canonical covering

map.

Lemma 2.

Let V be a subspace of Rn . Then πn (V ) is closed in Tn if and only if V is a rational subspace of R . n

Proof. We first show that if V is a rational subspace of Rn , then πn (V ) is closed in Tn . Let

w1 , w2 , . . . , wr ∈ Zn form a basis of V . Suppose that z∗ ∈ Tn is in the closure of πn (V ). Then we may find a sequence {xj } in V such that limj →∞ πn (xj ) = z∗ . Write xj =

r X

bj,k wk .

k=1

Since all wk ∈ Zn , we may choose all bj,k ∈ [0, 1). Therefore we can find a subsequence {jm } of {j } such that all 1 ≤ k ≤ r . lim bjm ,k = bk∗ , Let Tn .

x∗

=

Pr

∗ k=1 bk wk .

m→∞

Clearly, πn (x ∗ ) = z∗ . Hence z∗ ∈ πn (W ). Therefore, πn (V ) is closed in

We next prove the following fact: If v ∈ Rn , then the closure of πn (Rv) in Tn is a rational subspace. To see this, let v = [β1 , . . . , βn ]T .PWithout loss of generality we assume that β1 , . . . , βr are linearly independent over Q while βk = rj =1 ak,j βj with ak,j ∈ Q for all 1 ≤ k ≤ n. The set    β1     m  ...    βr

mod Zr




  

: m∈Z

 

is dense in Tr (see Cassels [1], Theorem I, p. 64). Now let V0 = {Ax : x ∈ Rr } where A = [ak,j ]. Then V0 is a rational subspace of Rn , and πn (V0 ) is contained in the closure of πn (Rv). But πn (V0 ) is closed and V0 ⊇ Rv. Hence the closure of πn (Rv) is πn (V0 ), proving the fact. Finally, let v1 , . . . , vr be a basis of V . Suppose that W¯ j is the closure of πn (Rvj ) in Tn . Then the closure of πn (V ) contains W¯ 1 + · · · + W¯ r . But W¯ 1 + · · · + W¯ r is closed in Tn because it is a rational subspace, and it contains πn (V ). Hence the closure of πn (V ) is W¯ 1 + · · · + W¯ r , proving the lemma.

Corollary 2.

Let f : Rn → C be continuous and periodic (mod Zn ) and V be a subspace of Rn . If v0 + V is contained in the zero set of f (x) for some v0 ∈ Rn , then so is v0 + W where W is the smallest rational subspace of Rn containing V .
T T Proof. First, letT{Vα } be a set of rational subspaces of Rn . Then πn α Vα = α πn (Vα ) is n n closed in T , so α Vα must be a rational subspace of R . This implies that the minimal rational subspace W containing V exists. Since f (x) is periodic (mod Zn ) we may view it as a continuous function defined on Tn . Now, πn (v0 ) + πn (W ) is the closure of πn (v0 ) + πn (V ) in Tn . Hence πn (v0 ) + πn (W ) is in the zero set of f : Tn → C. Thus, v0 + W ⊆ Zf . We derive the following key lemma from a result of Cerveau et al. [2]. First, we define the notion of τ -invariance in Rn . Let m(x) be the symbol of a given dilation equation that satisfies the orthogonal coefficients condition. Let E be a given complete set of coset representatives of

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors

167

Zn /AT (Zn ). A closed set Y ⊆ Rn is τ -invariant if for any l ∈ E, ω∈Y

|m (τl (ω))| > 0

and

H⇒

τl (ω) ∈ Y .

(5.1)

A compact τ -invariant set is minimal if it contains no smaller nonempty compact τ -invariant set.

Proposition 3.

Let f (ξ ∈ (Rn ) and Y be a minimal compact τ -invariant set contained in the zero set of f (ξ ). Then there exist a subspace V of Rn and a periodic orbit {zj : 0 ≤ j ≤ N − 1} of AT (mod Zn ) such that N[ −1
Y ⊆ zj + V . j =0

Proof. This is Theorem 2.8 of Cerveau et al. [2]. The theorem of Cerveau et al. is actually valid in a more general setting, where f (ξ ) and m(ξ ) are allowed to be any real analytic functions.

Lemma 3.

Let f (ξ ) ∈ (Rn ) such that Cm f = f , and let Ef− := {ξ ∈ Rn : f (ξ ) = inf ω∈Rn f (ω)}. Then there exists a rational subspace W and a periodic orbit {zj : 0 ≤ j < N} of AT (mod Zn ) S − such that F := N−1 j =0 (zj + W ) ⊆ Ef and F is τ -invariant.

Proof. We first observe that Ef− is τ -invariant. This follows from Cm f (ξ ) = P

X

|m (τl (ξ ))|2 f (τl (ξ )) = f (ξ ) .

l∈E

Since l∈E |m(τl = 1, if ξ ∈ Ef− then all f (τl (ξ )) ≥ f (ξ ) so equality can hold above only if |m(τl (ξ ))| > 0 implies τl (ξ ) ∈ Ef− . We construct a nonempty minimal compact τ -invariant set Y in Ef− as follows. Take any point ξ0 ∈ Ef− and set X0 = {ξ0 } and recursively define the finite sets {Xj : j ≥ 0} by letting Xj consist of all points ξj such that ξj = τl (ξj −1 ) with ξj −1 ∈ Xj −1 and S l ∈ E such that |m(ξj )| > 0. Then then τ -invariance of Ef− gives Xj ⊆ Ef− for all j ≥ 0. The set ∞ j =0 Xj lies in a bounded region in R because the mappings τl are uniformly contracting with respect to a suitable norm in Rn (cf. Lagarias S∞ and Wang [23], Section 3). Thus the closure Y0 of j =0 Xj is compact, and Y0 ⊆ Ef− because Ef− is a closed set. We show that Y0 is τ -invariant. If ω ∈ Y0 and |g(τl (ω))| > 0 where l ∈ E, take a subsequence ξjk ∈ Xjk that converges to ω, so that τl (ξjk ) → τl (ω). Now |m(τl (ξjk ))| > 0 for k sufficiently large, hence τl (ξjk ) ∈ Xjk +1 ; so we may construct a sequence having τl (ω) as a cluster point, proving τl (ω) ∈ Y0 . The existence of a nonempty minimal compact τ -invariant set Y contained in Y0 follows by Zorn’s Lemma argument. It follows now from Proposition 3 that there exists an AT -invariant subspace V and a periodic orbit {zj ∈ Y : 0 ≤ j < N} such that (ξ ))|2

Y ⊆ SN−1

N[ −1 j =0


zj + V ⊆ Ef− ,

with the property that the set j =0 (zj + V ) is τ -invariant. Now let W be the smallest rational subspace of Rn containing V . Since AT (W ) is also a rational subspace containing V and it has the same dimension as W , AT (W ) = W . Because Ef− is the zero set of f˜(ξ ) := f (ξ ) − inf ω f (ω), Corollary 2 applies to f˜ to give Y ⊆

N[ −1 j =0


zj + W ⊆ Ef− .

168

Jeffrey C. Lagarias and Yang Wang



S SN −1 n Finally, since πn N−1 j =0 (zj + W ) is the closure of πn j =0 (zj + V ) in T , we conclude that SN−1 j =0 (zj + W ) is τ -invariant.

Proof of Theorem 3. Observe that for r = 1 criterion (d) of Theorem 1 is equivalent to criterion (d) of Theorem 3. Therefore, the equivalence of (a)–(d) of this theorem has already been established in Theorem 1. (b) ⇒ (e). Let the nonconstant f (ξ ) ∈ (Rn ) satisfy Cm f = f . Without loss of generality we assume that f (0) 6 = minω f (ω), or else we can replace f (ξ ) by −f (ξ ). By Lemma 3 there exists an AT -invariant rational subspace W and a periodic orbit {zj : 0 ≤ j < N} of AT (mod Zn ) such − that ∪N−1 j =0 (zj + W ) ⊆ Ef is τ -invariant. We prove the following claim: Let ξ ∈ zj + W . Suppose that |m(τl (ξ ))| > 0 for some l ∈ Zn . Then τl (ξ ) ∈ zj +1 + W + Zn , where zN := z0 . S −1 Assume that the claim is false. Then the τ -invariance of N j =0 (zj + W ) implies that τl (ξ ) ∈ n n T zk+1 + W + Z 6 = zj +1 + W + Z . Hence ξ ∈ A (zk+1 + W ) + Zn = zk + W + Zn . But this could happen only if zk + W + Zn = zj + W + Zn . Applying the operator (AT )N −1 to both sides of the above equality yields  N −1  N −1

Zn = zj +1 + W + AT Zn , zk+1 + W + AT and adding Zn to both sides then gives zk+1 + W + Zn = zj +1 + W + Zn , which is a contradiction. It now follows from the claim that for any ξ ∈ zj + W , X X |m (τl (ξ ))|2 = |m (τl (ξ ))|2 . 1= l∈E

l∈E τl (ξ )∈zj +1 +W +Zn

Finally, zj 6 ∈ W + Zn because otherwise we would have zj + W + Zn = W + Zn ⊆ Ef− , contradicting 0 6 ∈ Ef− . (e) ⇒ (f). It follows from (e) that m(τl (ξ )) = 0 for ξ ∈ zj + W and l ∈ E such that τl (ξ ) 6 ∈ zj +1 + W + Zn , where zN := z0 . Now for any l ∈ Zn there exists an l 0 ∈ E such that τl (ξ ) ≡ τl 0 (ξ ) (mod Zn ); hence (f) follows. (f) ⇒ (d). Choose any η ∈ z0 + W . Then η 6∈ Zn because z0 6 ∈ W + Zn . For any α ∈ Zn consider the sequence  −k (η + α), k = 0, 1, 2, · · · . ωk = AT SN −1

(zj + W ) + Zn is locally compact and is disjoint from Zn , for S −1 sufficiently large k we must have ωk 6 ∈ N (zj + W ) + Zn . Now ω0 = η + α ∈ z0 + W , so there SN −1 j =0 S −1 n exists a k0 > 0 such that ωk0 −1 ∈ j =0 (zj + W ) + Zn but ωk0 6 ∈ N j =0 (zj + W ) + Z . n We show that m(ωk0 ) = 0. Assume that ωk0 −1 ∈ zj + W + Z . So ωk0 −1 = ξ0 + l for some ξ0 ∈ zj + W and l ∈ Zn . Now Then limk→∞ ωk = 0. Since

j =0

 −1  −1 ωk0 −1 = AT ωk0 = AT (ξ0 + l) = τl (ξ0 ) . But ωk0 = τl (ξ0 ) 6 ∈ zj +1 + W + Zn , where zN := z0 . So m(ωk0 ) = 0 by (f), proving (d).

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors

169

Proof of Theorem 2. AT is irreducible because it has the same characteristic polynomial as A does. So the only AT -invariant rational subspace W of Rn with dim(W ) < n is W = {0}, see Theorem III.12 of Newman [30]. Theorem 2 now follows immediately from Theorem 3.

References [1]

Cassels, J.W.S. (1957). An Introduction to Diophatine Approximations, Cambridge University Press.

[2]

Cerveau, D., Conze, J.P., and Raugi, A. (1996). Fonctions harmoniques pour un opérateur de transition en dimesion > 1, Bol. Soc. Bras. Mat., 27, 1–26.

[3]

Cohen, A. (1990). Ondelettes, analyses multirésolutions et filtres mirriors em quadrature, Ann. Inst. Poincaré, 7, 439–459. [4] Cohen, A., Daubechies, I., and Feauveau, J.C. (1992). Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., XLV, 485–560. [5] Cohen, A., Daubechies, I., and Plonka, G. (1997). Regularity of refinable function vectors, J. Fourier Anal. Appl., 3, 295–324. [6] Cohen, A. and Sun, Q. (1993). An arithmetic characterisation of conjugate quadrature filters associated to orthonormal wavelet bases, SIAM J. Math. Anal., 24, 1355–1360. [7]

Conze, J.P., Hervé, L., and Raugi, A. (1997). Pavages auto-affines, opérateur de transfert et critères de réseau dans Rd , Bol. Soc. Bras. Mat., 28, 1–42.

[8]

Conze, J.P. and Raugi, A. (1990). Fonctions harmoniques pour un operateur de transition et applications, Bull. Soc. Math. France, 118, 273–310. Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41, 909–996.

[9]

[10] Daubechies, I. (1992). Ten Lectures on Wavelets, SIAM, Philadelphia, PA. [11] Daubechies, I. and Lagarias, J.C. (1991). Two–scale difference equations I: existence and global regularity of solutions, SIAM J. Appl. Math., 22, 1388–1410. [12] Donovan, G., Geronimo, J., Hardin, D., and Massopust, P. (1996). Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal., 27, 1158–1192. [13] Flaherty, T. and Wang, Y. (1999). Haar-type multiwavelet bases and self-affine multi-tiles, Asian J. Math., 3(2), 387– 400. [14] Goodman, T.N.T. and Lee, S.L. (1994). Wavelets of multiplicity r, Trans. Am. Math. Soc., 342, 307–324. [15] [16] [17] [18] [19] [20] [21] [22]

Goodman, T.N.T., Lee, S.L., and Tang, W.S. (1993). Wavelets in wandering subspaces, Trans. Am. Math. Soc., 338, 639–654. Gröchenig, K. (1994). Orthogonality criteria for compactly supported scaling functions, Appl. Comp. Harmonic Anal., 1, 242–245. Gröchenig, K. and Hass, A. (1994). Self–similar lattice tilings, J. Fourier Anal., 1, 131–170. Hervé, L. (1994). Multi-resolution analysis of multiplicity d: applications to dyadic interpolation, Appl. Comput. Harmon. Anal., 1(4), 299–315. Heil, C. and Collela, D. (1996). Matrix refinement equations: existence and uniqueness, J. Fourier Anal. Appl., 2, 363–377. Hogan, T. (1998). A note on matrix refinement equations, SIAM J. Math. Anal., 29(4), 849–854. Jia, R.-Q. and Shen, Z. (1994). Multiresolution and wavelets, Proc. Edinburgh Math. Soc., 37, 271–300. Lagarias, J.C. and Wang, Y. (1995). Haar type orthonormal wavelet basis in R2 , J. Fourier Anal. Appl., 2, 1–14.

[23] Lagarias, J.C. and Wang, Y. (1996). Self-affine tiles in Rn , Adv. Math., 121, 21–49. [24] Lagarias, J.C. and Wang, Y. (1997). Integral self-affine tiles in Rn , part II: lattice tilings, J. Fourier Anal. Appl., 3, 83–102. [25] Lawton, W. (1991). Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys., 32, 57–61. [26] Lawton, W., Lee, S.L., and Shen, Z. (1997). Stability and orthonormality of multivariate refinable functions, SIAM J. Math Anal., 28, 999–1014. [27] Lian, J. (1998). Orthogonality criteria for multi-scaling functions, Appl. Comput. Harmon. Anal., 5(3), 277–311. [28] Mallat, S. (1989). Multiresolution analysis and wavelets, Trans. Am. Math. Soc., 315, 69–88. [29] Malone, D. L2 (R) solutions of dilation equations and fourier-like transformations, preprint.

170

Jeffrey C. Lagarias and Yang Wang

[30]

Newman, M. (1972). Integral Matrices, Academic Press, New York.

[31]

Plonka, G. (1997). Necessary and sufficient conditions for orthonormality of scaling vectors, in Multivariate Approx. and Splines, Nürnberger, G., Schmidt, J.W., and Walz, G., Eds., ISNM, Vol. 125, Birkhäuser, Basel.

[32] Shen, Z. (1998). Refinable function vectors, SIAM J. Math. Anal., 29, 235–250. [33] Strang, G., Strela, V., and Zhou, D. Compactly supported refinable functions with infinite masks, preprint. [34]

Strichartz, R. and Wang, Y. Geometry of self-affine tiles I, Indiana J. Math., to appear.

Received April 7, 1999 Revision received September 6, 1999 Information Science Research, AT&T Labs-Research, Florham Park, NJ 07932 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 e-mail: [email protected]