Generalized low-pass filters and multiresolution analyses
Lawrence W. Baggett, Veronika Furst, Kathy D. Merrill, Judith A. Packer
aa r X i v : . [ m a t h . C A ] D ec GENERALIZED FILTERS, THE LOW-PASS CONDITION, ANDCONNECTIONS TO MULTIRESOLUTION ANALYSES
LAWRENCE W. BAGGETT, VERONIKA FURST, KATHY D. MERRILL,AND JUDITH A. PACKER
Abstract.
We study generalized filters that are associated to multiplicity func-tions and homomorphisms of the dual of an abelian group. These notions are basedon the structure of generalized multiresolution analyses. We investigate when theRuelle operator corresponding to such a filter is a pure isometry, and then use thatcharacterization to study the problem of when a collection of closed subspaces, whichsatisfies all the conditions of a GMRA except the trivial intersection condition, mustin fact have a trivial intersection. In this context, we obtain a generalization of atheorem of Bownik and Rzeszotnik. Introduction
Filters have historically been an essential tool used in both building and analyzingwavelets and multiresolution structures. In particular, filters traditionally called “low-pass” arise naturally from refinement equations for multiresolution analyses (MRAs)and generalized multiresolution analyses (GMRAs). Beginning with work of Mallat[16] and Meyer [17], the process of defining filters from a multiresolution structurewas also reversed; that is, functions that behave like low-pass filters have been usedto build the structures. This construction technique has been remarkably fruitful,producing, for example, the smooth and well-localized wavelets of Daubechies [12].In generalizing this procedure to allow less restrictive conditions on the filters as wellas on the setting, for example in [2], [4], and [10], properties of an operator associatedwith the filter, called a Ruelle operator, are used to justify this construction. Theessential ingredient is that the Ruelle operator be a pure isometry. A theorem givinggeneral conditions under which the Ruelle operator is a pure isometry in the case ofan integer dilation in L ( T ) appeared in [10].In this paper we derive a similar theorem (Theorem 3 in Section 2) in a quite generalcontext. We then exploit this theorem both in analyzing multiresolution structuresand in building them. Our central result of the first type addresses the question ofwhen a structure that satisfies all the properties of a GMRA except possibly the trivialintersection property, must satisfy that as well. This generalizes work of Bownik andcollaborators ([6], [9]). Our main result of the second type is to show that very littlein the way of a low-pass condition is needed when building GMRAs from filters usingdirect limits as in [4] and [5].Our general context is as follows: Let Γ be a countable abelian group (writtenadditively) with dual group b Γ (written multiplicatively), equipped with Haar measure µ (of total mass 1). Let α be an isomorphism of Γ into itself, and suppose that theindex of α (Γ) in Γ equals N > . Assume further that ∩ n ≥ α n (Γ) = { } . Write α ∗ for the dual endomorphism of b Γ onto itself defined by [ α ∗ ( ω )]( γ ) = ω ( α ( γ )) , andnote that the kernel of α ∗ contains exactly N elements and that α ∗ is ergodic withrespect to the Haar measure on b Γ . Write K = ∪ n> ker( α ∗ n ) , and note that, because ∩ n ≥ α n (Γ) = { } , K is dense in b Γ . Of course the standard example (e.g., from wavelet theory) of these ingredients iswhere Γ = Z , b Γ = T , and α ( k ) = 2 k. Or, more generally, Γ = Z d , and α ( ~x ) = A~x, where A is a d × d integer dilation matrix of determinant N. Let m : b Γ → { , , , . . . , ∞} be a Borel map into the set of nonnegative integersunion ∞ , and write σ i for { ω ∈ b Γ : m ( ω ) ≥ i } . Note that m ( ω ) = X i χ σ i ( ω ) . We remark that such functions m arise, via Stone’s Theorem on unitary representa-tions of abelian groups, as multiplicity functions associated to such representationsof Γ , and we will in fact invoke this relationship in the final section. In that context,we will make use of a unitary representation π of Γ, acting in a Hilbert space H , anda unitary operator δ on H for which δ − π γ δ = π α ( γ ) for all γ ∈ Γ . In this general setting, we define a filter as follows:
Definition 1.
A (possibly infinite) matrix H = [ h i,j ] of Borel, complex-valued func-tions on b Γ is called a filter relative to m and α ∗ if, for every j, h i,j is supported in σ j , and H satisfies the following “filter equation:” (1) X α ∗ ( ζ )=1 X j h i,j ( ωζ ) h i ′ ,j ( ωζ ) = N δ i,i ′ χ σ i ( α ∗ ( ω )) for almost all ω ∈ b Γ . In the standard situation described above, i.e., where Γ = Z , α ( k ) = 2 k, and where m is the identically 1 function, a filter relative to m and α ∗ is just a 1 × h, and the filter equation becomes | h ( z ) | + | h ( − z ) | = 2 , for almost all z ∈ T , which is the classical equation satisfied by a quadrature mirrorfilter. These are the filters that played a central role in the early theory of mul-tiresolution analyses and wavelets in L ( R ) . Indeed, in the classical case, where φ is a scaling function for an MRA in L ( R ) , we know that the integral translates T n ( φ ) = φ ( · − n ) form an orthonormal basis for the core subspace V , and we maydefine a unitary operator J from V onto L ( T ) by sending the basis vector T n ( φ ) tothe function z n . This correspondence between an orthonormal basis of V with thecanonical Fourier basis for L ( T ) is clearly a unitary operator. Furthermore, J sendsthe element φ ( x/ / √ P c n T n ( φ ) to the function P n c n z n = h ( z ) , where h is theassociated quadrature mirror filter. We notice that, in addition to the fact that h satisfies the quadrature mirror equation, it satisfies another condition. Namely, if δ ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 3 denotes the dilation operator on L ( R ) given by [ δ ( f )]( x ) = √ f (2 x ) , then one canverify that the operator J ◦ δ − ◦ J − on L ( T ) is given by[[ J ◦ δ − ◦ J − ]( f )]( z ) = h ( z ) f ( z ) = [ S h ( f )]( z )for every f ∈ L ( T ) . We will call such an operator S h a Ruelle operator.
Because δ − is an isometry on V , and ∩ Range( δ − n ) = ∩ V n = { } , it follows that the operator S h has these same properties. That is, S h is a “pure isometry.”In the next section we define a Ruelle operator S H similarly associated with anabstract filter H as in Definition 1, and present our first main result, a characterizationof when this Ruelle operator is a pure isometry. The final two sections contain theapplications of this result.2. Filters and pure isometries
Let H be a filter relative to m and α ∗ . Whenever the formula[ S H ( f )]( ω ) = H t ( ω ) f ( α ∗ ( ω ))= M j X i H i,j ( ω ) f i ( α ∗ ( ω )) , defines a bounded operator from L i L ( σ i , µ ) into itself, we will call S H the Ruelleoperator associated to H . In the contexts we study in this paper, this will always bethe case.We now prove a generalization of the filter equation of Definition 1 that will providea crucial step in determining when the Ruelle operator S H is an isometry. If thefunction m associated to H is finite a.e., this proposition follows from the standardfilter equation by induction (see Lemma 9 in [4]). However, without this restriction, itrequires a more careful argument exploiting the fact that, in the situations we study, S H is an isometry. Proposition 2.
Let H be a filter relative to m and α ∗ , and assume that the associatedRuelle operator S H is an isometry. Then: N n X α ∗ n ( ζ )=1 X i " n − Y k =0 H t ( α ∗ k ( ωζ )) i,j " n − Y k ′ =0 H t ( α ∗ k ′ ( ωζ )) i,j ′ = δ j,j ′ χ σ j ( α ∗ n ( ω )) . Proof.
In the calculations below, certain sums and integrals have to be exchanged.These exchanges are justified by Fubini’s Theorem, where we rely several times onthe fact that for any f ∈ L j L ( σ j , µ ) the element S nH ( f ) is again in L j L ( σ j , µ ) , and therefore P j | [ S nH ( f )] j ( ω ) | is finite for almost every ω. L. W. BAGGETT, V. FURST, K. D. MERRILL, AND J. A. PACKER
For each f and g in L j L ( σ j , µ ) , we have X i Z b Γ X j " n − Y k =0 H t ( α ∗ k ( ω )) i,j f j ( α ∗ n ( ω )) X j ′ " n − Y k ′ =0 H t ( α ∗ k ′ ( ω )) i,j ′ g j ′ ( α ∗ n ( ω )) dω = X i Z b Γ [ S nH ( f )] i ( ω )[ S nH ( g )] i ( ω ) dω = h S nH ( f ) | S nH ( g ) i = h f | g i = X j Z b Γ f j ( ω ) g j ( ω ) dω = X j Z b Γ f j ( α ∗ n ( ω )) g j ( α ∗ n ( ω )) dω. Therefore,1 N n X α ∗ n ( ζ )=1 X i Z b Γ X j " n − Y k =0 H t ( α ∗ k ( ωζ )) i,j f j ( α ∗ n ( ω )) · X j ′ " n − Y k ′ =0 H t ( α ∗ k ′ ( ωζ )) i,j ′ g j ′ ( α ∗ n ( ω )) dω = X j Z b Γ f j ( α ∗ n ( ω )) g j ( α ∗ n ( ω )) dω. Write C j for the element of the direct sum space L j L ( σ j , µ ) whose j th coordinateis χ σ j and whose other coordinates are 0. Set f = χ E C j , for E ⊆ σ j , and g = χ E ′ C j ′ , for E ′ ⊆ σ j ′ . Then we have1 N n X ζ X i Z α ∗− n ( E ∩ E ′ ) " n − Y k =0 H t ( α ∗ k ( ωζ )) i,j " n − Y k ′ =0 H t ( α ∗ k ′ ( ωζ )) i,j ′ dω = 1 N n X ζ X i Z b Γ " n − Y k =0 H t ( α ∗ k ( ωζ )) i,j χ E ( α ∗ n ( ω )) C j ( α ∗ n ( ω )) · " n − Y k ′ =0 H t ( α ∗ k ′ ( ωζ )) i,j ′ χ E ′ ( α ∗ n ( ω )) C j ′ ( α ∗ n ( ω )) dω = δ j,j ′ h χ α ∗− n ( E ) C j | χ α ∗− n ( E ′ ) C j ′ i = δ j,j ′ Z b Γ χ E ∩ E ′ ( α ∗ n ( ω )) dω = δ j,j ′ Z α ∗− n ( E ∩ E ′ ) dω. ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 5
Since this is true for any Borel sets E and E ′ , the proposition follows. (cid:3) We now prove our first main result, establishing conditions under which a Ruelleoperator S H that is an isometry must in fact be a pure isometry. This theoremgeneralizes Theorem 3.1 in [10], which finds a similar conclusion in the setting ofinteger dilations in L ( T ) . Theorem 3.
Assume that m is finite on a set of positive measure. If S H is anisometry on L j L ( σ j , µ ) , then S H fails to be a pure isometry if and only if it hasan eigenvector. Specifically, S H fails to be a pure isometry if and only if there existsa nonzero element f ∈ L j L ( σ j , µ ) , and a scalar λ of absolute value 1, such that S H ( f ) = λf. Moreover, if f is a unit eigenvector for S H , then k f ( ω ) k = 1 a.e.Proof. Write R n for the range of the isometry S nH , and write R ∞ for the intersection ∩ R n of the R n ’s. By definition, S H is a pure isometry if and only if R ∞ = { } . If S H has an eigenfunction f, say S H ( f ) = λf, with λ = 0 , then clearly f belongsto the range of each operator S nH , and hence f ∈ R ∞ . Therefore, R ∞ = { } , and S H is not a pure isometry.Conversely, suppose S H is not a pure isometry. We now adapt an argument in [4]that was based on the reverse martingale convergence theorem (See Theorem 10.6.1in [13].) For each n ≥
1, let M n be the σ -algebra of Borel subsets of b Γ that areinvariant under multiplication by elements in the kernel of α ∗ n . Let f and g be twononzero vectors in R ∞ , and define a sequence of random variables { X n } ≡ { X f,gn } on b Γ by X n ( ω ) = 1 N n X α ∗ n ( ζ )=1 h f ( ωζ ) | g ( ωζ ) i . Then it follows directly that X n is M n -measurable, and the conditional expectationof X n , given M n +1 , equals X n +1 . Therefore, the sequence { X n , M n } is an integrable,reverse martingale. Hence, using the reverse martingale convergence theorem, wehave that the sequence { X n ( ω ) } converges almost everywhere and in L norm to anintegrable function L on b Γ . Clearly, L ( ωζ ) = L ( ω ) for almost every ω and every ζ ∈ K = ∪ n> ker( α ∗ n ) . Hence,the Fourier coefficient c γ ( L ) satisfies c γ ( L ) = γ ( ζ ) c γ ( L ) for every ζ ∈ K, implyingthat c γ ( L ) = 0 unless γ ( ζ ) = 1 for all ζ ∈ K. Since K is dense in b Γ , it then followsthat c γ ( L ) = 0 for all γ except γ = 0 . Consequently, L ( η ) is a constant function, and L. W. BAGGETT, V. FURST, K. D. MERRILL, AND J. A. PACKER we have, from the L convergence of the sequence { X n } , L ( η ) = Z b Γ L ( ω ) dω = lim n ≥ Z b Γ X n ( ω ) dω = lim n ≥ N n X α ∗ n ( ζ )=1 Z b Γ h f ( ωζ ) | g ( ωζ ) i dω = Z b Γ h f ( ω ) | g ( ω ) i dω = h f | g i . Therefore, the reverse martingale X n converges almost everywhere to the constant h f | g i . For each ω, write N ω for the set of all natural numbers n for which m ( α ∗ n ( ω )) < ∞ . Since m is finite on a set of positive measure, the ergodicity of α ∗ implies that N ω is infinite for almost all ω. We show next that, for each n ∈ N ω , there is a differentexpression for X n ( ω ) . To wit, for each n ∈ N ω , define f n = S ∗ H n ( f ) and g n = S ∗ H n ( g ) . Since S H is a unitary operator on R ∞ , we have X n ( ω ) = 1 N n X α ∗ n ( ζ )=1 h f ( ωζ ) | g ( ωζ ) i = 1 N n X ζ * n − Y k =0 H t ( α ∗ k ( ωζ )) f n ( α ∗ n ( ω )) | n − Y k ′ =0 H t ( α ∗ k ′ ( ωζ )) g n ( α ∗ n ( ω )) + = 1 N n X ζ X j X i " n − Y k =0 H t ( α ∗ k ( ωζ )) j,i X i ′ " n − Y k ′ =0 H t ( α ∗ k ′ ( ωζ )) j,i ′ f ni ( α ∗ n ( ω )) g ni ′ ( α ∗ n ( ω )) . When interchanging the sums in the previous expression is justified, we may continuethis computation; then, using Proposition 2, we would obtain1 N n X i X i ′ f ni ( α ∗ n ( ω )) g ni ′ ( α ∗ n ( ω )) X ζ X j " n − Y k =0 H t ( α ∗ k ( ωζ )) j,i " n − Y k ′ =0 H t ( α ∗ k ′ ( ωζ )) j,i ′ = X i f ni ( α ∗ n ( ω )) g ni ( α ∗ n ( ω ))= h f n ( α ∗ n ( ω )) | g n ( α ∗ n ( ω )) i . This gives the different expression for X n ( ω ) that we want, whenever we can justifythe interchanges of sums in the previous computations:(2) X n ( ω ) = h f n ( α ∗ n ( ω )) | g n ( α ∗ n ( ω )) i . The following calculation, which again uses Proposition 2 and the Cauchy-SchwarzInequality, shows that the interchange of sums above is justified whenever the sumson i and i ′ are finite sums. Because of Proposition 2, the sums on i and i ′ will befinite if m ( α ∗ n ( ω )) < ∞ , and this is the case when n ∈ N ω . Hence, the computation
ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 7 below will complete the derivation of Equation (2). Note also that the sums on i and i ′ will be finite sums if the vectors f n ( α ∗ n ( ω )) and g n ( α ∗ n ( ω )) only have a finitenumber of nonzero coordinates. We will use this later on.1 N n c X i =1 c ′ X i ′ =1 X ζ X j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" n − Y k =0 H t ( α ∗ k ( ωζ )) j,i " n − Y k ′ =0 H t ( α ∗ k ′ ( ωζ )) j,i ′ f ni ( α ∗ n ( ω )) g ni ′ ( α ∗ n ( ω )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N n c X i =1 c ′ X i ′ =1 | f ni ( α ∗ n ( ω )) g ni ′ ( α ∗ n ( ω )) |· X ζ,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" n − Y k =0 H t ( α ∗ k ( ωζ )) j,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ˜ ζ, ˜ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" n − Y k ′ =0 H t ( α ∗ k ′ ( ω ˜ ζ )) ˜ j,i ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / = c X i =1 | f ni ( α ∗ n ( ω )) χ σ i ( α ∗ n ( ω )) | c ′ X i ′ =1 | g ni ′ ( α ∗ n ( ω )) χ σ i ′ ( α ∗ n ( ω )) |≤ c X i =1 | f ni ( α ∗ n ( ω )) | c X ˜ i =1 | χ σ ˜ i ( α ∗ n ( ω )) | / c ′ X i ′ =1 | g ni ′ ( α ∗ n ( ω )) | c ′ X ˜ i ′ =1 | χ σ ˜ i ′ ( α ∗ n ( ω )) | / ≤ √ cc ′ k f n ( α ∗ n ( ω )) kk g n ( α ∗ n ( ω )) k < ∞ , for almost every ω. The first conclusion we can draw from Equation (2) is that for almost all ω ,lim n ∈ N ω h f n ( α ∗ n ( ω )) | g n ( α ∗ n ( ω )) i = lim n →∞ X n ( ω )= h f | g i , or, setting g = f, for f a unit vector in R ∞ , lim n ∈ N ω k f n ( α ∗ n ( ω )) k = k f k = 1 . A second conclusion we may draw is that we must have σ = b Γ , i.e., m ( ω ) ≥ m ( ω ) = 0 for all ω in a set F of positive Haar measure, then from theergodicity of α ∗ , we must have α ∗ n ( ω ) ∈ F infinitely often for almost all ω, so that k f n ( α ∗ n ( ω )) k = 0 infinitely often. But, since each such integer n belongs to N ω , thiscontradicts the first claim above.Now let i satisfy σ i = b Γ and σ i +1 be a proper subset of b Γ of measure strictly lessthan 1. (Of course σ i +1 could be the empty set, if m ( ω ) ≡ i . ) Then, by a similarkind of ergodicity argument as was used above, we know that for almost all ω, andfor infinitely many values of n , [ f ( α ∗ n ( ω ))] i = 0 for all i > i and all f ∈ R ∞ . Indeed,this is true whenever α ∗ n ( ω ) / ∈ σ i +1 , and this occurs infinitely often for almost all ω. Moreover, each such n belongs to N ω . L. W. BAGGETT, V. FURST, K. D. MERRILL, AND J. A. PACKER
Let f , . . . , f k be orthonormal vectors in R ∞ . Then, for infinitely many sufficientlylarge n, we must have that the k i -dimensional vectors { [ f pn ( α ∗ n ( ω ))] , . . . , [ f pn ( α ∗ n ( ω ))] i } are nearly orthogonal and nearly of unit length. Consequently, k must be ≤ i . Hence R ∞ is finite dimensional, and therefore S H (a unitary operator on R ∞ ) must have aneigenvector.To prove the final part of the proposition, let f be a unit vector in R ∞ . Fromthe second claim above, we know that the coordinates f i of f are all 0 for i > i . Therefore, the interchanges of summations in the calculations above are justified, andwe obtain X f,fn ( ω ) = k f n ( α ∗ n ( ω )) k , so that lim n →∞ k f n ( α ∗ n ( ω )) k = k f k = 1for almost all ω. Finally, let f be a unit eigenvector for S H . We have then thatlim n →∞ k f ( α ∗ n ( ω )) k = lim n →∞ k [ S nH ( f n )]( α ∗ n ( ω )) k = lim n →∞ k f n ( α ∗ n ( ω )) k = 1 . By the ergodicity of α ∗ , it follows that k f ( ω ) k = 1 almost everywhere. (cid:3) Pure isometries and the low pass condition
In this section, we use Theorem 3 to eliminate the need for a restrictive low-passcondition when building GMRAs from filters via the direct limit construction of [4]and [5]. First we recall the definition:
Definition 4.
A collection { V j } ∞−∞ of closed subspaces of H is called a g eneralizedmultiresolution analysis (GMRA) relative to π and δ if (1) V j ⊆ V j +1 for all j. (2) V j +1 = δ ( V j ) for all j. (3) ∩ V j = { } , and ∪ V j is dense in H . (4) V is invariant under the representation π. The subspace V is called the core subspace of the GMRA { V j } . In order to use the theorems from the previous section to build GMRA’s fromfilters, we need to know that associated Ruelle operators are isometries. The proofrequires the additional assumption that the multiplicity function m is finite a.e. Thishypothesis is standard in much of the literature. Proposition 5.
Assume m ( ω ) < ∞ for almost all ω, and let H be a filter relative to m and α ∗ . Then the Ruelle operator S H is an isometry of L i L ( σ i , µ ) into itself. ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 9
Proof.
Note that, because m ( ω ) < ∞ almost everywhere, the filter equation, togetherwith the very definition of h i,j , implies that h i,j ( ω ) = 0 if j > m ( ω ) or i > m ( α ∗ ( ω )) . Therefore, all the sums in the following calculation, that are inside integrals, arefinite, so that interchanges of these sums is allowed. k S H ( f ) k = X j Z σ j | [ S H ( f )] j ( ω ) | dω = X j Z σ j | [ H t ( ω ) f ( α ∗ ( ω ))] j | dω = X j Z σ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i H i,j ( ω ) f i ( α ∗ ( ω )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω = X j Z b Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i h i,j ( ω ) f i ( α ∗ ( ω )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω = Z b Γ X j "X i h i,j ( ω ) f i ( α ∗ ( ω )) i ′ h i ′ ,j ( ω ) f i ′ ( α ∗ ( ω )) dω = 1 N X α ∗ ( ζ )=1 Z b Γ X j "X i h i,j ( ωζ ) f i ( α ∗ ( ω )) i ′ h i ′ ,j ( ωζ ) f i ′ ( α ∗ ( ω )) dω = 1 N Z b Γ X i X i ′ X ζ X j h i,j ( ωζ ) h i ′ ,j ( ωζ ) f i ( α ∗ ( ω )) f i ′ ( α ∗ ( ω )) dω = Z b Γ X i χ σ i ( α ∗ ( ω )) | f i ( α ∗ ( ω )) | dω = X i Z σ i | f i ( ω ) | = k f k , as claimed. (cid:3) In earlier works (e.g., [2], and [4]), a so-called “low-pass condition” on the filter H was used to guarantee that S H was a pure isometry. This condition had various forms,but they all required something like H being continuous at the identity 1 in b Γ andthe matrix H (1) being diagonal with √ N ’s at the top of the diagonal and 0’s at thebottom. Results from [18] and more recently [9] loosened these assumptions somewhatby separating out a phase factor. We will show below that such assumptions on H imply that S H can have no eigenvector, and so by Theorem 3, S H must be a pureisometry. The following theorem gives quite general conditions on H under which S H is a pure isometry, subsuming the conditions on H (1) mentioned above as well as theresults for a 1 × H given in [5]. In particular, note that this theorem blursthe distinction between classical low-pass and high-pass filters by not requiring H totake on specific values near the identity. Theorem 6.
Let H be a filter relative to m and α ∗ . Suppose there exists a positivenumber δ and a set F ⊆ b Γ of positive measure, such that for all ω ∈ F the matrix H ( ω ) is in block form H ( ω ) = (cid:18) A ( ω ) B ( ω ) C ( ω ) D ( ω ) (cid:19) , where the four blocks satisfy the following: (1) A ( ω ) is a square expansive matrix with the property that k A ( ω ) − k ≤ δ . (2) max( k B ( ω ) k , k C ( ω ) k , k D ( ω ) k ) < ǫ = min( , δ ) . (The norm here can ei-ther be the operator norm of a matrix or the Euclidean norm.)Finally, assume that F ∩ α ∗ ( F ) also has positive measure. Then S H is a pure isometry,i.e., S H has no eigenvector.Remark . The hypothesis of this theorem clearly covers the previously cited caseswhere H ( ω ) is continuous and has the relevant diagonal at ω = 1 . Proof.
Suppose, by way of contradiction, that f is a unit eigenvector for S H witheigenvalue λ , | λ | = 1. For each ω, write the vector f ( ω ) in the form f ( ω ) =( f ( ω ) , f ( ω )) , where f ( ω ) is a -dimensional. Because f is an eigenvector for S H , we have λf ( ω ) = [ S H ( f )]( ω ) = H t ( ω ) f ( α ∗ ( ω )) . It follows from this, and the fact that k f ( ω ) k = 1 by the final conclusion of Theorem3, that for ω ∈ F we must have k f ( ω ) k = k B t ( ω ) f ( α ∗ ( ω )) + D t ( ω ) f ( α ∗ ( ω )) k < ǫ. Hence, again because k f ( ω ) k = 1 for ω ∈ F , we must have k f ( ω ) k > − ǫ. Since condition (1) on the matrix A ( ω ) implies that k A ( ω ) v k ≥ (1 + δ ) k v k for every a -dimensional vector v , we must have, for ω and α ∗ ( ω ) both in F, ≥ k f ( ω ) k = k A t ( ω ) f ( α ∗ ( ω )) + C t ( ω ) f ( α ∗ ( ω )) k > (1 + δ ) k f ( α ∗ ( ω )) k − ǫ > (1 + δ )(1 − ǫ ) − ǫ ≥ δ − δ − δ − δ
4= 1 + δ . We have arrived at a contradiction, and the theorem is proved. (cid:3)
Theorem 6 can be used to build generalized multiresolution analyses with moregeneral filters, using approaches that do not require an infinite product construction,such as the direct limit construction in [4] and [5].
Example . For another application of Theorem 6, consider the Journ´e filter systemgiven by H ( ω ) = (cid:18) h , h , h , h , (cid:19) . ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 11
In the classical Journ´e example described by Baggett, Courter, and Merrill in [1], h , = √ e πiχ E , h , = 0 ,h , = √ e πiχ E , h , = 0 , where E = (cid:20) − , − (cid:19) ∪ (cid:20) − , (cid:19) ∪ (cid:20) , (cid:19) ,E = (cid:20) − , − (cid:19) ∪ (cid:20) , (cid:19) . These are the classical filters in L ( R ) for dilation by 2 that can be associated tothe GMRA in L ( R ) coming from the Journ´e wavelet.We now apply a device very similar to that first used on pp. 259–260 of [3]. Choosea very small δ > , and an even smaller ε > . Define q : T → R by q ( e πix ) = √ √ − r , if x = 0 , , if − ε < x < + ε,C ∞ monotone decreasing , if 0 < x < − ε, √ , if − ε < x < + ε,C ∞ monotone increasing , if + ε < x < − ε, , if − ε < x < + ε, √ · r, if x = ,C ∞ monotone increasing , if + ε < x < , q − [ q ( e πi ( x + ) )] , if − < x < . Here r ∈ (0 ,
1) is a number yet to be determined. We now define generalized filters( h qi,j ) by: h q , ( e πix ) = q ( e πix ) e πiχ [ − ,
27 ) ( x ) ,h q , ( e πix ) = √ e πiχ [ − , −
37 ) ∪ [ 37 ,
12 ) ( x ) ,h q , ( e πix ) = q ( e πi ( x + ) ) e πiχ [ − ,
17 ) ( x ) ,h q , ( e πix ) = 0 . Denote the matrix ( h qi,j ( z )) by H q . A routine calculation shows that the filter equa-tions of Baggett, Courter and Merrill are satisfied, i.e. X j =1 1 X k =0 h i,j (cid:16) e πi x + k (cid:17) h i ′ ,j (cid:16) e πi x + k (cid:17) = δ i,i ′ χ σi ( x ) , i = 1 , , where σ = [ − , − ) ∪ [ − , ) ∪ [ − , ) , and σ = [ − , ) . We now take A ( z ) = h q , ( z ) , B ( z ) = h q , ( z ) , C ( z ) = h q , ( z ) , and D ( z ) = h q , ( z )in the matrix H = H q . We want to determine a specific value r ∈ (0 ,
1) and a set F = { e πix : x ∈ F } , where F ⊂ [ − , ) such that the hypotheses of Theorem6 are satisfied. We let F = [ − n , n ] , where n ∈ N is chosen so that n ≥ q ( e πix ) > √ √ − r , for all x ∈ F . This can be done by applications of theIntermediate Value Theorem, since q is continuous. It’s clear that F ∩ α ∗ ( F ) = F has positive measure. Also we want to find δ > | h q , ( e πix ) | ≥ δ and max( | h q , ( e πix ) | , | h q , ( e πix ) | , | h q , ( e πix ) | ) < ǫ = min( , δ ) , ∀ x ∈ F . Notethat h q , ( e πix ) and h q , ( e πix ) are identically 0 on F , so we need only show that | h q , ( e πix ) | < ǫ on F . Since h q , ( e πix ) is continuous at x = 0 , where its value is equalto √ · r, we choose F = [ − n , n ] so that √ · r ≤ h q , ( e πix ) < · r, ∀ x ∈ F . Having chosen δ > , we thus must choose r so that √ √ − r ≤ δ and 2 r < ǫ =min( , δ ) . So we first choose r < min( , δ ) . For r , as long as δ < √ − , if we choose r ≤ q √ − (1+ δ )1+ δ , one can verify that1 √ p − r ≤
11 + δ .
Finally, we choose r = min( r , r ) . Then,2 r < min (cid:18) , δ (cid:19) , and 1 √ √ − r ≤
11 + δ , so that the conditions of Theorem 6 are satisfied, and S H is a pure isometry acting on L ( σ ) ⊕ L ( σ ) . In fact, letting r → , we can construct a one-parameter family offilter systems giving rise to pure isometries; when r = 0 , we obtain exactly the filtersystem constructed in [3].In Theorem 5 of [4], it is shown that given a pure isometry S on a Hilbert space K together with a representation ρ of a countable abelian group Γ , such that δ − ρ γ δ = ρ α ( γ ) for all γ ∈ Γ , then it was possible to construct a generalized multiresolutionanalysis via a direct limit process. Taking S = S H , and Γ = Z , the desired hypotheseswill be satisfied, and it follows that a GMRA can be constructed from the above filtersystem. In a paper in preparation, the authors will present a more constructiveapproach to making the GMRA under the same hypotheses as in Theorem 5 of [4].4. Pure isometries and the trivial intersection property
The following “problem” was first noticed by Baggett, Bownik and Rzeszotnik.Suppose { ψ k } is a Parseval multiwavelet in L ( R d ); i.e., the functions { ψ j,n,k ( x ) } ≡{√ j ψ k (2 j x + n ) } form a Parseval frame for all of L ( R d ) . If V j is defined to be theclosed linear span of the functions { ψ l,n,k } for l < j, then these subspaces can be shownto satisfy all of the properties of a GMRA except for the condition ∩ V j = { } . Bownikand Rzeszotnik demonstrated the delicacy of this condition in [8] by constructing, forany δ >
0, a frame wavelet in L ( R ), with frame bounds of 1 and 1 + δ , that has anegative dilate space V equal to all of L ( R ). They showed in [9], however, that aParseval multiwavelet in L ( R d ) generates a GMRA (that is, the trivial intersectionproperty does hold) whenever the multiplicity function of the negative dilate space V is finite on a set of positive measure. In fact, Bownik proved in [6] that the conditionthat m is not identically ∞ a.e. implies ∩ ∞ j =1 D j ( V ) = { } in the more general settingwhere D j f ( x ) = f ( A j x ) for a sequence { A j } of invertible n × n real matrices that ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 13 satisfy k A j k → j → ∞ . For a history of the intersection problem in L ( R d ), see[7].This question about subspaces of L ( R d ) obviously generalizes to a collection { V j } of subspaces of a Hilbert space that satisfy all the conditions for a GMRA exceptthe trivial intersection condition. Below, we apply the results from Section 2 to showthat this intersection is { } if certain extra assumptions hold. In doing so, we extendsome of the results mentioned the previous paragraph.Let Γ, α , π , and δ be as in the previous sections. We recall some implications ofStone’s Theorem, whereby certain GMRAs give rise to an associated filter. Let { V j } be a GMRA in a Hilbert space H , relative to the representation π and the operator δ. Then, according to Stone’s Theorem on unitary representations of abelian groups,there exists a finite, Borel measure µ (unique up to equivalence of measures) on b Γ , unique (up to sets of µ measure 0) Borel subsets σ ⊇ σ ⊇ . . . of b Γ , and a (notnecessarily unique) unitary operator J : V → L i L ( σ i , µ ) satisfying[ J ( π γ ( f ))]( ω ) = ω ( γ )[ J ( f )]( ω )for all γ ∈ Γ , all f ∈ V , and µ almost all ω ∈ b Γ . In this paper, we assume themeasure µ is absolutely continuous with respect to Haar measure, in which case wemay assume that µ is the restriction of Haar measure to the subset σ . Write C i for the element of the direct sum space L j L ( σ j , µ ) whose i th coor-dinate is χ σ i and whose other coordinates are 0. Write L j h i,j for the element J ( δ − ( J − ( C i ))) . The following theorem displays a connection between GMRA struc-tures and filters and will allow us to apply Proposition 2 and Theorem 3.
Theorem 9.
Let the functions { h i,j } be as in the preceding paragraph. Then thematrix H = { h i,j } is a filter relative to m and α ∗ . Moreover, the operator J ◦ δ − ◦ J − on L i L ( σ i , µ ) is the corresponding Ruelle operator S H :[ J ◦ δ − ◦ J − ( f )]( ω ) = H t ( ω ) f ( α ∗ ( ω )) . Proof.
By definition we have µ ( σ i ) = k C i k = k J ( δ − ( J − ( C i ))) k = X j Z b Γ | h i,j ( ω ) | dω, which implies that P j | h i,j ( ω ) | is finite for almost all ω. Write F i,i ′ ( ω ) = X α ∗ ( ζ )=1 X j h i,j ( ωζ ) h i ′ ,j ( ωζ ) , and note, by the Cauchy-Schwarz Inequality, that F i,i ′ ∈ L ( µ ) , and that the Fouriercoefficient c γ ( F i,i ′ ) = 0 unless γ belongs to the range of α. We have that c α ( γ ) ( F i,i ′ ) = Z b Γ F i,i ′ ( ω ) ω ( − α ( γ )) dµ ( ω )= X ζ Z b Γ X j h i,j ( ωζ ) h i ′ ,j ( ωζ ) ω ( − α ( γ )) dω = N X j Z b Γ h i,j ( ω ) h i ′ ,j ( ω ) ω ( − α ( γ )) dω = N h J ( δ − ( J − ( C i ))) | J ( π α ( γ ) ( δ − ( J − ( C i ′ )))) i = N h J − ( C i ) | π γ ( J − ( C i ′ )) i = N h C i | γC i ′ i = N δ i,i ′ h C i | γC i ′ i = N δ i,i ′ Z b Γ χ σ i ( ω ) ω ( − γ ) dω = N δ i,i ′ Z b Γ χ σ i ( α ∗ ( ω )) α ∗ ( ω )( − γ ) dω, showing that the two L functions F i,i ′ ( ω ) and N δ i,i ′ χ σ i ( α ∗ ( ω )) have the same Fouriercoefficients, and hence are equal almost everywhere. This verifies Equation (1). Itfollows from the filter equation that h i,j is supported on α ∗− ( σ i ) . That is, h i,j ( ω ) = 0unless both ω ∈ σ j and α ∗ ( ω ) ∈ σ i . Next, for any γ, we have[ J ( δ − ( J − ( γC i )))]( ω ) = [ J ( δ − ( π γ ( J − ( C i ))))]( ω )= [ J ( π α ( γ ) ( δ − ( J − ( C i ))))]( ω )= ω ( α ( γ )) M j h i,j ( ω )= ω ( α ( γ )) χ σ i ( α ∗ ( ω )) M j h i,j ( ω )= H t ( ω )[ γC i ]( α ∗ ( ω )) . Then, by the Stone-Weierstrass Theorem, we must have[ J ( δ − ( J − ( f C i )))]( ω ) = H t ( ω )[ f C i ]( α ∗ ( ω )) ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 15 for every continuous function f on b Γ . Then, by standard integration methods, thisequality holds for all L functions f. Finally, if F = L i f i , then[ J ( δ − ( J − ( F )))]( ω ) = " J δ − J − X i f i C i !!! ( ω )= X i [ J ( δ − ( J − ( f i C i )))]( ω )= X i H t ( ω ) f i ( α ∗ ( ω ) C i ( α ∗ ( ω ))= H t ( ω ) F ( α ∗ ( ω ))= [ S H ( F )]( ω ) , proving the second assertion. (cid:3) We call this matrix function H a low-pass filter determined by the GMRA . Remark . The preceding proof works in a more general setting. That is, we donot use all of the GMRA structure, particularly the property that ∩ V j = { } . Inparticular, ∩ V j = { } if and only if S H = J ◦ δ − ◦ J − is a pure isometry.We introduce two more groups. Let D be the direct limit group determined byΓ and the monomorphism α of Γ into itself. (See for example [15].) For clarity, wemake this construction explicit as follows.Let e Γ be the set of all pairs ( γ, j ) for γ ∈ Γ and j a nonnegative integer. Define anequivalence relation on e Γ by ( γ, k ) ≡ ( γ ′ , k ′ ) if and only if α k ′ ( γ ) = α k ( γ ′ ) , and let D be the set of equivalence classes [ γ, k ] of this relation. Define addition on D by[ γ , k ] + [ γ , k ] = [ α k ( γ ) + α k ( γ ) , k + k ] . One verifies directly that this addition is well-defined, and that D is an abelian group.Next, define a map e α on D by e α ([ γ, k ]) = [ α ( γ ) , k ] . Again, one verifies directly that e α is well-defined and that it is an isomorphism of D onto itself. Indeed, the inverse e α − is given by e α − ([ γ, k ]) = [ γ, k + 1] . Define G to be the semidirect product D ⋊ Z , where the integer j acts on theelement d by j · d = e α j ( d ) . Explicitly, the multiplication in G is given by( d , j ) × ( d , j ) = ( d + e α j ( d ) , j + j ) . As before, π is a unitary representation of Γ , acting in a Hilbert space H , and δ aunitary operator on H for which δ − π γ δ = π α ( γ ) for all γ ∈ Γ . Define a representation e π on G by e π ( d,j ) = e π ([ γ,k ] ,j ) = δ k π γ δ − k − j . One verifies directly that this is a representation of G. Note also that e π e α ( d ) = δ − e π d δ and e π ( d,j ) = e π d δ − j . Finally, for | λ | = 1, define the (irreducible) unitary representation P λ of G actingin the Hilbert space l ( D ) by[ P λ ( d,j ) ( f )]( d ′ ) = λ j f ( e α − j ( d ′ − d )) . Remark . The representation P λ is equivalent to the induced representation Ind G Z χ λ , where χ λ is the character of the subgroup Z determined by λ . Theorem 12.
Suppose { V j } is a collection of closed subspaces of H that satisfy allthe conditions for a GMRA, relative to π and δ , except possibly the condition that ∩ V j = { } . Assume that the measure µ associated to the representation π restrictedto V is Haar measure, and that the multiplicity function m is finite on a set of positivemeasure. Then the following conditions are equivalent: (1) ∩ V j = { } . (2) δ has an eigenvector. (3) The representation e π of G contains a subrepresentation equivalent to the rep-resentation P λ for some | λ | = 1 .Proof. We define the functions { h i,j } as in Theorem 9 and reiterate that the matrix-valued function H is a filter relative to the space V and that the operator J ◦ δ − ◦ J − is the Ruelle operator. Then S H is a composition of isometries, and thus clearly anisometry from L j L ( σ j , µ ) into itself.Assume (1), and thus that S H is not pure. From Theorem 3, we know that S H hasa unit eigenvector f : S H ( f ) = λf, and | λ | = 1 . (We are using here the hypothesis that m ( ω ) < ∞ on a set of positive measure.) For such an eigenfunction f, v = J − ( f ) isan eigenvector for δ − . This proves (1) implies (2).Assume (2), and let v be a unit eigenvector for δ with eigenvalue λ . Because π isequivalent to a subrepresentation of some multiple of the regular representation ofΓ , we must have, from the Riemann-Lebesgue Lemma, that the function h π γ ( w ) | w i vanishes at infinity on Γ for every w ∈ H . But, for each γ, we have |h π α j ( γ ) ( v ) | v i| = |h δ − j π γ δ j ( v ) | v i| = |h π γ ( v ) | v i| , implying then that h π γ ( v ) | v i = 0 for all γ = 0 , or equivalently that h π γ ( v ) | π γ ′ ( v ) i =0 unless γ = γ ′ . But now, for d = [ γ, k ] ∈ D, we have h e π d ( v ) | v i = h e π [ γ,k ] ( v ) | v i = h δ k π γ δ − k ( v ) | v i = h π γ ( v ) | v i = 0unless γ = 0 . It follows then that the vectors { e π d ( v ) } form an orthonormal set. Thespan X of these vectors is obviously an invariant subspace for the representation e π of G. Moreover, we claim that the restriction of e π to X is equivalent to the representation P λ . Thus, let U be the unitary operator from X onto l ( D ) that sends the basis vector ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 17 e π d ( v ) to the point mass basis vector ǫ d in l ( D ) . We have U ( e π ( d,j ) ( e π d ′ ( v ))) = U ( e π d δ − j e π d ′ ( v ))= U ( e π d e π e α j ( d ′ ) δ − j ( v ))= λ − j U ( e π d + e α j ( d ′ ) ( v ))= λ − j ǫ d + e α j ( d ′ ) = P λ ( d,j ) ( ǫ d ′ )= P λ ( d,j ) ( U ( e π d ′ ( v ))) , showing the equivalence of the restriction of e π to the subspace X and the represen-tation P λ . This proves (2) implies (3).Assume (3). Because the operator e π (0 , is δ − , it follows that there is a nonzero vec-tor v ∈ H for which δ − ( v ) = e π (0 , ( v ) = λv. (Just note that P λ (0 , has an eigenvectorwith eigenvalue λ . This shows (3) implies (2).Finally, assume (2), and let v be an eigenvector for δ with eigenvalue λ . Write W j for the orthogonal complement of V j in V j +1 . Then H = ∞ M j = −∞ W j ⊕ R ∞ , where R ∞ = ∩ V j , so we may write v = ∞ X j = −∞ v j + v ∞ , where v j is the projection of v onto the subspace W j , and v ∞ is the projection of v onto the subspace R ∞ . Applying the operator δ gives X j v j + v ∞ = v = λ X j δ ( v j ) + λδ ( v ∞ ) , implying that v j +1 = λδ ( v j ) , whence k v j +1 k = k v j k for all j. Therefore v j = 0 forall j, and hence v = v ∞ . So, R ∞ = { } , and (2) implies (1). (This part of the proofuses neither the hypothesis on m nor the one on π. ) This completes the proof of thetheorem. (cid:3) Remark . We note that the proofs of Theorems 3 and 12 imply that δ has aneigenvector whenever ∩ V j is nontrivial but finite-dimensional. Therefore, ∩ V j mustbe infinite-dimensional whenever it is nontrivial. Example . Let H = L ( R ) , let Γ = Z and π be the representation of Γ determinedby translation. Let α ( k ) = 2 k, and define δ on H by [ δ ( f )]( x ) = √ f (2 x ) . Then δ − π k δ = π k . Set V j equal to the subspace of H comprising those functions f whoseFourier transform is supported in the interval ( −∞ , j ) . Then the V j ’s satisfy all theconditions for a GMRA relative to π and δ except the trivial intersection condition.Indeed, the intersection is nontrivial, because ∩ V j is the subspace of functions whoseFourier transform is supported in the interval ( −∞ , . The subspace V comprises the functions whose transform is supported in the interval ( −∞ , , and it followsthat the multiplicity function associated to the restriction of π to this subspace isinfinite everywhere. Hence, since δ has no eigenvector, we see that we cannot dropthe hypothesis that m < ∞ on a set of positive measure from the theorem above. Example . Now let Γ = Z , let α ( n, k ) = (2 n, k ) , let H = L ( R ) , let π be therepresentation of Γ on H determined by translation, and let [ δ ( f )]( x, y ) = 2 f (2 x, y ) . Then, δ − π γ δ = π α ( γ ) . Let V j be the subspace of H comprising the functions whoseFourier transform is supported in the rectangle ( −∞ , j ) × ( − j , j ) . Then, the multi-plicity function associated to the restriction of π to V is infinite everywhere, but thistime ∩ V j = { } . Indeed, if f ∈ ∩ V j , then the support of the Fourier transform of f issupported on the negative x -axis, and such a function is the 0 vector in L ( R ) . Hence,the finiteness assumption of Theorem 12 on m is not necessary for the intersection tobe trivial.The following example shows that for a Hilbert space H 6 = L ( R d ), a collection ofsubspaces { V j } can satisfy all of the properties of a GMRA except the trivial inter-section property, even though the multiplicity function is finite almost everywhere.As Theorem 12 shows, this is made possible by the presence of an eigenvector forthe dilation δ . The implication (3) implies (2) in Theorem 12 suggests a method ofconstruction of such examples. Example . Let H = l ( D ), where D is the group of dyadic rationals. The groupΓ = Z acts on this space by π γ ( f )( x ) = f ( x − γ ), and if we define α ( γ ) = 2 γ and δf ( x ) = f (2 x ), we see that δ − π γ δ = π α ( γ ) . Note that f = χ is a fixed vector forthis dilation. Now take V to be the subspace l ( Z ), and V j = δ j V . Since π is theregular representation of Γ on V , the multiplicity function m is identically 1. We seethat the fixed vector χ is a nonzero element of the intersection of the V j .Finally, to clarify how Theorem 12 fits in with previous results about the inter-section problem, let H , π, and δ be as in the beginning of this section. If { ψ i } is a set of vectors in H for which the collection { δ j ( π γ ( ψ i )) } forms a frame for H , then δ can have no eigenvector. Indeed, if v were an eigenvector for δ, then thenumbers |h v | δ j ( π γ ( ψ i )) i| are constant independent of j. Hence, they must all be 0,contradicting the frame assumption. Therefore, in the original context of the trivialintersection problem, i.e., where the subspaces V j = span { δ k ( π γ ( ψ i )) : k < j, γ, i } are constructed from a Parseval wavelet { ψ i } , there is no eigenvector for δ. Hence, ifthe intersection is nontrivial, then it must be infinite-dimensional, and the multiplic-ity function m is infinite almost everywhere. Theorem 12 therefore extends Theorem6.1 of [9] from the classical case of H = L ( R d ) to an abstract Hilbert space.In the case when H = L ( R d ), the group Γ = Z d acts by translation, and δ ( f ) = δ A ( f ) = | det A | / f ( A · ) for an expansive integer matrix A , we need not requirethe subspaces { V j } to be constructed from a Parseval wavelet, as in the previousparagraph. For any bounded neighborhood E ⊆ R d of 0, there exists a positiveinteger n such that E ⊂ A n E . Since Z E | f ( x ) | dx = Z A k E | f ( x ) | dx ENERALIZED LOW-PASS FILTERS AND MULTIRESOLUTION ANALYSES 19 for any f that satisfies λf ( x ) = | det A | / f ( Ax ), we see that δ A can have no eigen-vector, and, by Theorem 12, the assumption that m is not identically ∞ a.e. impliesthe trivial intersection property. Thus, Theorem 12 provides an alternative proof forthe classical scenario of Theorem 1.1 of [6]. References [1] L. W. Baggett, J. E. Courter and K. D. 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Math.Ann. (2005), 705–720.[9] M. Bownik and Z. Rzeszotnik, Construction and reconstruction of tight framelets and waveletsvia matrix mask functions, preprint.[10] O. Bratteli and P. E. T. Jorgensen, Isometries, shifts, Cuntz algebras and multiresolutionwavelet analysis of scale N , Integral Equations Operator Theory (1997), 382–443.[11] J. Courter, Construction of dilation- d wavelets, in The Functional and Harmonic Analysis ofWavelets and Frames , Contemp. Math. , vol. 247, Amer. Math. Soc., Providence, 1999, pp. 183–205.[12] I. Daubechies, Ten Lectures on Wavelets,
CBMS-NSF Lecture Notes , no. 61, SIAM, 1992.[13] R. M. Dudley,
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Rev. Math. Iberoamericana (2006),131–180.[15] N. S. Larsen and I. Raeburn, From filters to wavelets via direct limits, in Operator Theory,Operator Algebras and Applications , Contemp. Math. , vol. 414, Amer. Math. Soc., Providence,2006, pp. 35–40.[16] S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L ( R ), Trans.Amer. Math. Soc. (1989), 69–87.[17] Y. Meyer, Wavelets and Operators,
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J. Geom. Anal. (2001), 311–342 Lawrence Baggett, Department of Mathematics, University of Colorado, Boul-der, Colorado 80309, USA
E-mail address : [email protected] Veronika Furst, Department of Mathematics, Fort Lewis College, Durango, Col-orado 81301, USA
E-mail address : furst [email protected] Kathy Merrill, Department of Mathematics, Colorado College, Colorado Springs,Colorado, 80903, USA
E-mail address : [email protected] Judith Packer, Department of Mathematics, University of Colorado, Boulder,Colorado 80309, USA
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