Normalized Tight Frame Wavelet Sets in Rd

January 26, 2001

X. Dai, Y. Diao, Q. Gu and D. Han

Abstract

Let A be a d×d real expansive matrix. We characterize the reduc-ing subspaces of L2(Rd) for A-dilation and the regular translation op-erators acting on L2(Rd). We also characterize the Lebesgue measur-able subsets E of Rd such that the function defined by inverse Fouriertransform of [1/(2π)d/2]χE generates through the same A-dilation andthe regular translation operators a normalized tight frame for a givenreducing subspace. We prove that in each reducing subspace, the setof all such functions is non-empty and is also path connected in theregular L2(Rd)-norm.

1991 Mathematics Subject Classification. Primary 42XXX, 47XXXKey words and phrases. Normalized tight frame wavelet set, reducingsubspace, connectivity.

1 Introduction

A sequence xn in a Hilbert space H is called a frame for H if there existconstants C1, C2 > 0 such that

C1‖x‖2 ≤

∑

n∈N

|〈x, xn〉|2 ≤ C2‖x‖

2,∀x ∈ H.

If C1 = C2 = 1, it is called a normalized tight frame. It is known (cf.[9])that xn is a normalized tight frame for H if and only if x =

∑n∈N〈x, xn〉xn

for all x ∈ H, where the convergence is unconditional in norm.

1

In this article we will investigate a class of normalized tight frames foreither L2(Rd) or certain subspaces of L2(Rd) which are called reducing sub-spaces. The normalized tight frames we will deal with are obtained by apply-ing certain A-dilation and regular translation operators to a single Lebesgueintegrable function. Let us first define the above mentioned operators.

Let A be any d × d real invertible matrix. Then A induces a unitaryoperator DA acting on L2(Rd) defined by

(DAf)(t) : = | detA|1

2f(At),∀f ∈ ÃL2(Rd), t ∈ Rd. (1)

A matrix is called expansive if all its eigenvalues have modulus greater thanone. When a matrix is expansive, it is always invertible. The unitary oper-ator DA corresponding to a real expansive matrix A is called an A-dilationoperator. In an analogous fashion, any vector s in Rd induces a unitarytranslation operator Ts defined by

(Tsf)(t) := f(t− s),∀f ∈ ÃL2(Rd), t ∈ Rd.

In this article we will use only translation operators T` with ` ∈ Zd.

Throughout this article, we will use f or Ff to denote the Fourier trans-form of a function f ∈ L2(Rd). It is defined as

(Ff)(s) : =1

(2π)d/2

∫

Rde−i(st)f(t)dm, (2)

for all f ∈ L2(Rd), where s t denotes the real inner product. Clearly F isa unitary operator. Also, for any bounded linear operator S on L2(Rd), wewill define S := FSF−1. Thus for any d×d real invertible matrix A, we haveDA = D(Aτ )−1 = D−1Aτ = D∗Aτ , where Aτ is the transpose of A. Also, we have

Tλf = ei(λs) · f for any λ ∈ Rd. Furthermore, for any subset X of L2(Rd),we will use X to denote the collection of Fourier transforms of all elementsin X.

We will be interested in families of functions of the form DnAT`ψ : n ∈

Z, ` ∈ Zd for some function ψ ∈ L2(Rd). Such a function is called an A-dilation orthonormal wavelet if the family of functions Dn

AT`ψ : n ∈ Z, ` ∈Zd forms an orthonormal basis for L2(Rd). When the above family forms a(resp. normalized tight) frame for L2(Rd), the function ψ ∈ L2(Rd) is called

2

an A-dilation (resp. normalized tight) frame wavelet. In [4] it was shown thatfor each d ∈ N and each d×d real expansive matrix A, there is a function ψ ∈L2(Rd) such that the set Dn

AT`ψ : n ∈ Z, ` ∈ Zd forms an orthonormal basisfor L2(Rd). In fact, it was proved in [4] that for any real expansive A, thereexists a function ψ ∈ L2(Rd) such that ψ = 1

(2π)d/2χE for some measurable

subset E of Rd and DnAT`ψ : n ∈ Z, ` ∈ Zd forms an orthonormal basis

for L2(Rd). Such set E is called an A-dilation orthonormal wavelet set [3].The corresponding wavelet is called an A-dilation orthonormal MSF wavelet[10]. In this article the notation ψE will be reserved to denote the functiondefined by ψE = 1

(2π)d/2χE. For the sake of simplicity, orthonormal will often

be omitted. Likewise A-dilation will be omitted when it is clear what thematrix A is in the context. Speegle proved in [11] that for any given realexpansive matrix A, the set of all A-dilation MSF wavelets is path-connected.

Theoretically as well as in applications, people are also interested in or-thonormal bases or frames of the form Dn

AT`ψ : n ∈ Z, ` ∈ Zd for subspacesof L2(Rd) such as Hardy space. First let us give the proper definition. Notethat it is consistent with the defintion of frame given at the beginning of thisarticle.

Definition 1 A sequence xn is a normalized tight frame for a closed sub-space X of L2(R) if it is a subset of X and

f =∑

n

〈f, xn〉xn, ∀f ∈ X,

where the convergence is unconditional in norm.

A function ψ is an A-dilation normalized tight frame wavelet for a closedsubspace X if Dn

AT`ψ : n ∈ Z, ` ∈ Zd is a normalized tight frame for X.

We will concentrate on a special class of subspaces, namely the reducingsubspaces. Let A be a real expansive matrix and X be a closed subspace ofL2(Rd). X is called a reducing subspace for DA, T` : ` ∈ Zd if DAX = Xand T`X = X for each ` ∈ Zd. When A is clear in context, we will call suchX just a reducing subspace.

It is still unknown (?) to us whether a closed subspace X with normalizedtight frame of the form Dn

AT`ψ : n ∈ Z, ` ∈ Zd must be a reducing sub-spaces for DA, T` : ` ∈ Zd, though it is necessary that such X must be at

3

least invariant under A-dilation operator. Observe that for any f ∈ ÃL2(Rd),we have (D−1A T`DAf)(t) = f(t − A`). It follows that D−n

A T`DnA = TAn` for

all n ∈ Z, ` ∈ Zd. Thus if all entries of the real expansive matrix A areintegers, then a closed subspace X with normalized tight frame of the formDn

AT`ψ : n ∈ Z, ` ∈ Zd must be also invariant under integral translations,namely X must be a reducing subspace.

In the one dimensional case, Dai and Lu [6] characterized all the reducingsubspaces for DA, T` : ` ∈ Z with A = 2. They proved for any suchsubspace the existence of a subset E of R such that the inverse Fouriertransform of 1√

2πχE (namely ψE ) is an orthonormal wavelet for the subspace.

Such a set is called a subspace wavelet set (with specified real expansive matrixA and subspace). The existence of A-dilation subspace wavelet set for anyreducing subspace for DA, T` : ` ∈ Zd with any d×d real expansive matrixA and any natural number d will be given in another article by the sameauthors. Certain existence results for special cases are found in [5] and [8].

Similarly, a measurable subset E of Rd with finite measure is an A-dilationnormalized tight frame wavelet set for a closed subspace X if ψE is a nor-malized tight frame wavelet for X. Again, A-dilation will be omitted whenit is clear in the context. Normalized tight frame wavelet sets form a muchricher class than the class of wavelet sets [9]. In this paper, for any d × dreal expansive matrix A and any natural number d, we will characterize allthe reducing subspaces for DA, T` : ` ∈ Zd; We will also characterzize allA-dilation normalized tight frame wavelet sets for any given reducing sub-space for DA, T` : ` ∈ Zd; We will prove that for any real expansive matrixA and any reducing subspace for DA, T` : ` ∈ Zd, the family of A-dilationnormalized tight frame wavelet sets is not empty and the corresponding fam-ily of A-dilation normalized tight frame wavelets is path-connected in theL2-norm.

2 Subspace Frame Wavelets

Let d be any fixed natural number. Let A be any fixed d× d real expansivematrix. In this section, we will characterize all the reducing subspaces forDA, T` : ` ∈ Zd and all A-dilation normalized tight frame wavelet sets fora given reducing subspace for DA, T` : ` ∈ Zd; First, let us present several

4

lemmas.

Since A is an expansive matrix, it follows that A is invertible and for anyλ ∈ σ(A−1), |λ| < 1 and thus limk→+∞ λ

−k = 0. This implies that ([7], p.559,Theorem 9) limk→+∞ ‖A

−k‖ = 0. Now let t ∈ Rd with t 6= 0. From the fact

that ‖A−k‖‖Akt‖ ≥ ‖t‖, it follows that ‖Akt‖ ≥ ‖t‖‖A−k‖ . The following Lemma

will be used in the proofs of Lemma 3 and Theorem 3

Lemma 1 Let A be a d×d real expansive matrix. Then limk→+∞ ‖A−k‖ = 0

and limk→+∞ ‖Akt‖ = ∞ for every non-zero element t in Rd.

The proof of Lemma 2 is left to the reader.

Lemma 2 1. Let en : n ∈ N be an orthonormal basis for a separableHilbert space H and let P be an orthogonal projection onto a subspaceH0. Then Pen is a normalized tight frame for H0.

2. Let Hn : n ∈ N be a family of mutually orthogonal subspaces that sumto H. Suppose that for each n ∈ N, xnm : m ∈ N is a normalizedtight frame for Hn. Then xnm : n,m ∈ N is a normalized tight framefor H.

Lemma 3 Let A be a d× d real expansive matrix. Then AnZd : n ∈ Z isdense in Rd.

Proof. Let ε > 0. By Lemma 1 we can choose k ∈ N such that ‖A−k‖ < εd.

Let ej : 1 ≤ j ≤ d be the standard orthonormal basis for Rd. Since A isexpansive hence invertible, the set A−kej : 1 ≤ j ≤ d is also a basis for Rd.Let x ∈ Rd, then x =

∑dj=1 x

(j)A−kej with some x(j) ∈ R for j ∈ 1, 2, .., d.

Let ` =∑d

j=1[x(j)]ej ∈ Zd, where [x(j)] denotes the largest integer amonge

those that are no greater than x(j). Then

‖A−k`− x‖ = ‖d∑

j=1

(x(j) − [x(j)])A−kej‖ ≤d∑

j=1

‖A−k‖ · ‖ej‖ < ε.

Theorem 1 characterizes the reducing subspaces.

5

Theorem 1 Let A be a d × d real expansive matrix. A closed subspaceX of L2(Rd) is a reducing subspace for DA, T` : ` ∈ Zd if and only ifX = L2(Rd) · χΩ for some Lebesgue measurable subset Ω of Rd with theproperty that Ω = AτΩ.

Proof. Let X be a reducing supspace for DA, T` : ` ∈ Zd and P be theorthogonal projection from L2(Rd) onto X. Then P commutes with bothDnA and T` for all n ∈ Z and ` ∈ Zd. Note that D−nA T`D

nA = TAn`. It follows

that P commutes with TAn` for all n ∈ Z and ` ∈ Zd. By Lemma 3, the setAnZd : n ∈ Z is dense in Rd. So P commutes with Tt for all t ∈ Rd. HenceP commutes with Tt for all t ∈ Rd. If we use Mf to denote the multiplicativeoperator by f(s), then Tt = Me−ts . Me−ts : t ∈ Rd generates a maximalabelian von Neumann algebra A = Mf : f ∈ L∞(Rd). Thus P must besitting in A and P = Mf for some f ∈ L∞(Rd). The relation P 2 = Pimplies that f 2 = f. So f is χΩ for some measurable subset Ω of Rd. ThusX = L2(Rd) · χΩ. Since PDA = DAP, we have P DA = DAP . EquivalentlyMχΩD

−1Aτ = D−1AτMχΩ , hence MχΩDAτ = DAτMχΩ . Let f ∈ L2(Rd). Observe

that

MχΩDAτf(t) = |Aτ |1

2f(Aτ t) · χΩ(t),

DAτMχΩf(t) = |Aτ |1

2f(Aτ t) · χΩ(Aτ t).

This implies that χΩ(Aτ t) = χΩ(t), therefore AτΩ = Ω.The proof of the other direction is simple, we omit it.

Remark.The above proof implies the structure of the commutant of DnA, T`, ` ∈

Zd. We have

DnA, T`, ` ∈ Zd′ = Mf : f ∈ L∞(Rd), f(Atx) = f(x).

Let E be a subset of Rd. Two points x, y ∈ E are said to be 2π-translationequivalent if x− y = 2π` for some ` ∈ Zd. This is an equivalence relation onE. The 2π-translation redundancy index of a point x in E is the cardinalityof its equivalence class. We use E(τ, k) to denote the set of all points in Ewith 2π-translation redundancy index k. For k 6= m, E(τ, k) ∩ E(τ,m) = ∅,so we have E = E(τ,∞) ∪ (

⋃n∈N E(τ, n)).

6

Similarly, two non-zero points x, y ∈ E are said to be A-dilation equiv-alent if y = Akx for some k ∈ Z. This is also an equivalence relation onE. The A-dilation redundancy index of a point x in E is the cardinality inits equivalence class. The set of all points in E with A-dilation redundancyindex k is denoted by E(δA, k). For k 6= m, E(δA, k) ∩ E(δA,m) = ∅. SoE = E(δA,∞) ∪ (

⋃n∈NE(δA, n)).

It is left for the reader to check (cf. [2]) that for a Lebesgue measurableset E of Rd, E(τ, k) and E(δ, k) are measurable for any k ∈ N ∪ ∞.

Theorem 2 Let A be a d × d real expansive matrix. Let X be a reducingsubspace for DA, T` : ` ∈ Zd. Let Ω be the measurable subset Ω of Rd suchthat X = L2(Rd) · χΩ and Ω = AτΩ.

Then a measurable subset E ⊂ Rd is an A-dilation normalized tight framewavelet set for X if and only if (modulo null sets) it satisfies the followingconditions:

1. E = E(δA, 1) and Ω is disjoint union of (Aτ )kE : k ∈ Z,

2. E = E(τ, 1).

Proof. Let E be a Lebesgue measurable set which satisfies conditions (1)and (2). Then E is 2π-translation equivalent to a subset F of [0, 2π)d. Thusthe set G = E ∪ ([0, 2π)d\F ) is 2π-translation equivalent to [0, 2π)d andT`ψG : ` ∈ Zd is an orthogonal basis for L2(G). Let P be the orthogonalprojection from L2(G) onto L2(E). By Lemma 2 (1), P T`ψG : ` ∈ Zd =T`ψE : ` ∈ Zd is a normalized tight frame for L2(E). Therefore for eachk ∈ Z, Dk

AT`ψE : ` ∈ Zd is a normalized tight frame for L2((At)kE). SinceΩ is the disjoint union of (At)kE : k ∈ Z, it follows that L2(Ω) is the(orthogonal) direct sum of L2((At)kE). Thus, by Lemma 2 (2), Dk

AT`ψE :` ∈ Zd, k ∈ Z is a normalized tight frame for L2(Ω), i. e. , E is an A-dilationnormalized tight frame wavelet set for X. Clearly the conclusion is also truewhen E satisfies condition (1) and (2) modulo null sets.

To prove the inverse, we assume that E is an A-dilation normalized tightframe wavelet set for X. We will show that (modulo null sets) the set E hasproperties (1) and (2) in the theorem.

Note that X = L2(Rd) · χΩ contains functions DnAψE, hence it contains

χ(Aτ )nE for any n ∈ Z. This implies that Ω ⊃ ∪n∈Z(Aτ )nE. On the other

7

hand, any function in L2(Rd) ·χΩ = X is in the span of DnAT`ψE : n ∈ Z, ` ∈

Zd, hence it is supported on⋃n∈Z(Aτ )nE. Therefore Ω ⊂ ∪n∈Z(Aτ )nE. In

order to prove that E satisfies (1) we need to show that (modulo null sets) theΩ = ∪n∈Z(Aτ )nE is a disjoint union of (Aτ )nE’s. Namely we need to provethat (modulo null sets) m((Aτ )kE ∩ (Aτ )jE) = 0 for any distinct integersk, j. If this is not true, then there exists an integer j0 > 0 such that m(E ∩(Aτ )j0E) > 0. We can find a subset F of E ∩ (Aτ )j0E with positive measuresuch that elements in the set (Aτ )kF : k ∈ Z are mutually disjoint. We canchoose F in such a way that it is contained in cube Πd

j=1[2mjπ, 2(mj + 1)π)for some integers mj with j ∈ 1, 2, · · · , d. Now we define f by

f = χF .

Since F ⊂ E ⊂ Ω, so f ∈ X. By assumption, ψE is a normalized tight framewavelet for X. Thus f =

∑n∈Z,`∈Zd〈f,Dn

AT`ψE〉DnAT`ψE. It follows that

‖f‖2 =∑

n∈Z,`∈Zd

|〈f,DnAT`ψE〉|

2

≥∑

`∈Zd

|〈χF , T`ψE〉|2 + |〈χF , D

j0A ψE〉|

2

= ‖f‖2 +| detA|2j0

(2π)d‖f‖2

> ‖f‖2.

We reach a contradiction. So (modulo null sets) the set E must satisfy con-dition (1).

In order to prove that (modulo null sets) E = E(τ, 1), it suffices to showthat E ∩ (E + 2π`) is a null set for any ` ∈ Zd\0.

Indeed, otherwise there exists `0 6= ~0 in Zd such that E ∩ (E + 2π`0) haspositive measure. We can choose a subset J of E ∩ (E + 2π`0) with positivemeasure in Πd

j=1[2πmj, 2(mj + 1)π) for some integers mj with j ∈ 1, · · · , d.Now we define g by

g = χJ − χJ−2π`0 .

Since the support of g is contained in E, so g ∈ X. Note that (modulo nullsets) E satisfies condition (1), therefore the elements in the set (Aτ )nE : n ∈

8

Z are mutually disjoint, it follows that for any ` ∈ Zd\0, 〈g, DnAT`ψE〉 = 0.

Also, for any ` ∈ Zd, 〈g, T`ψE〉 = 〈χJ − χJ−2π`0 , T`ψE〉 = 〈χJ , T`ψE〉 −〈χJ−2π`0 , T`ψE〉 = 0, since T`ψE is a 2π-periodic function.

By assumption, ψE is a normalized tight frame wavelet for X, we have

g =∑

n∈Z,`∈Zd

〈g, DnAT`ψE〉D

nAT`ψE = 0.

The contradiction shows that (modulo null sets) E = E(τ, 1).

3 Path Connectivity

In this section we will prove that for a fixed d × d real expansive matrixA, in a reducing subspace for DA, T` : ` ∈ Zd, the set of all A-dilationnormalized tight frame wavelets is not empty and that the family of A-dilation normalized tight frame wavelets ψE induced by measurable sets Ethrough ψE = 1

(2π)d/2χE is path-connected in the L2-norm.

Theorem 3 Let A be a d× d real expansive matrix A and X be a reducingsubspace for DA, T` : ` ∈ Zd. Then there exists an A-dilation normalizedtight frame wavelet set for X.

Furthermore, for any given ε with 0 < ε < 1, there exists ε1 > 0 such thatthere is an A-dilation normalized tight frame wavelet set for X contained inthe ring B(0, ε)\B(0, ε1).

Proof. Let us use C to denote the set (AτB(0, 1))\B(0, 1). Since A isexpansive, Aτ is also expansive. Thus for any x ∈ Rd\0, by Lemma 1, wehave limk→+∞ ‖(A

τ )−kx‖ = 0 and limk→+∞ ‖(Aτ )kx‖ = ∞. So there is an

integer n such that (Aτ )nx /∈ B(0, 1) and (Aτ )n−1x ∈ B(0, 1). It follows that(Aτ )nx ∈ C, hence x ∈ (Aτ )−nC. Therefore, we have

Rd\0 =⋃

n∈Z

(Aτ )nC.

Now for any ε with 0 < ε < 1, since C is a bounded set and ‖(Aτ )−k‖ → 0,there is an integer k0 such that (Aτ )k0C ⊂ B(0, ε). Note that 0 is an exterior

9

point of C, hence it is also an exterior point of (Aτ )k0C. So there is an ε1 > 0such that

E := (Aτ )k0C ⊂ B(0, ε)\B(0, ε1) ⊂ [−π, π)d.

It is clear that E = E(τ, 1). Now define F := E(δA, 1). We see that F =F (τ, 1), F = F (δA, 1) and Rd\0 =

⋃n∈Z(Aτ )nF.

Since X is a reducing subspace for DA, T` : ` ∈ Zd, by Theorem 1, thereis a mearsurable subset Ω of Rd such that AτΩ = Ω, and X = L2(Rd) · χΩ.If we define W := F ∪ Ω, then W is an A-dilation normalized tight framewavelet set for X. Indeed, it is clear that W = W (τ, 1) = W (δA, 1) andthat (Aτ )nW : n ∈ Z is a disjoint family of subsets in Ω. Also from thedefinition of W and the facts that Rd\0 =

⋃n∈Z(Aτ )nF and that AτΩ = Ω,

it follows that Ω\0 =⋃n∈Z(Aτ )nW.

Theorem 4 Let A be a d × d real expansive matrix and X be a reducingsubspace for DA, T` : ` ∈ Zd. The set of all A-dilation normalized tightframe wavelet sets for X is path connected in the norm topology.

Proof. First of all, by Theorem 1, there exists a measurable subset Ωof Rd such that X = L2(Rd) · χΩ and Ω = AτΩ. Let E be any A-dilationnormalized tight frame wavelet set for X, by Theorem 2, (modulo null sets)the following holds: E = E(δA, 1), Ω is disjoint union of (Aτ )kE : k ∈ Zand E = E(τ, 1). By the same argument used in the proof of Theorem 3,there exist ε1, ε2 with 0 < ε1 < ε2 < 1 and two subsets G1, G2 of Ω withG1 ⊂ B(O, ε2)\B(O, ε1) and G2 ⊂ B(O, 1)\B(O, ε2), such that G1 and G2are both A-dilation normalized tight frame wavelet sets for X.

Let Fn = (Aτ )−n(G1)∩E. Then Fn’s are mutually disjoint, and ∪n∈ZFn =E. Also (Aτ )n(Fn) ⊂ G1 for each integer n and ∪n∈Z(Aτ )n(Fn) = G1. Foreach integer n, since Fn is bounded, there exist real numbers rn and qn with0 ≤ rn < qn, such that Fn ⊂ B(O, qn)\B(O, rn). Now for each s such that0 ≤ s ≤ 1, we define F s

n = B(O, rn+(qn−rn)s)∩Fn and F s = ∪n∈ZFsn. It can

be checked that F 0 = ∅ and F 1 = E. For i ∈ 1, 2 and any subset K of Rd,we will use the notation gi(K) to denote the set ∪n∈Z(Aτ )n((Aτ )−n(Gi)∩K) =Gi ∩ (∪n∈Z(Aτ )nK). Finally, we define Ks := (E\F s) ∩ τ−1(g1(F

s)) and

Es := g1(Fs) ∪ ((E\F s)\Ks) ∪ g2(K

s),

10

To finish the proof, it suffices to show that χEs is a continuous pathof A-dilation normalized tight frame wavelet sets connecting χE and χG1 .For this, we need to show several things. First, we need to show that forany ε > 0 and 0 ≤ s ≤ 1, there exists δ > 0, such that if |s1 − s| ≤ δ,then ‖χEs1 − χEs‖2 < ε. Secondly, we need to show that for each s with0 ≤ s ≤ 1, Es is an A-dilation normalized tight frame wavelet set for X aswell. We leave this to our reader (apply Theorem 2). Thirdly, we need toverify that E0 = E and E1 = G1. This is immediate from the definition ofEs.

Note that

‖χEs1 − χEs‖2 ≤ ‖χg1(F s1 ) − χg1(F s)‖2

+ ‖χ(E\F s1 )\Ks1 − χ(E\F s)\Ks‖2 + ‖χg2(Ks1 ) − χg2(Ks)‖2.

Now let ε > 0. Since ∪n∈Z(Aτ )n(Fn) = G1, there exists a natural numberN , so that m(∪|n|≥N (Aτ )n(Fn)) < ε/6. On the other hand, χF s

nis continuous

for each fixed n and so is χ(Aτ )n(F sn). Thus, for some δ1 > 0, whenever

|s1 − s| < δ1, we will have

‖χ∪|n|<N (Aτ )n(F

s1n ) − χ∪|n|<N (A

τ )n(F sn)‖

2 ≤∑

|n|<N‖χ(Aτ )n(F

s1n ) − χ(Aτ )n(F s

n)‖2 <

ε

6.

This shows that ‖χg1(F s1 )− χg1(F s)‖2 < ε

3. With similar argument applied to

‖χ(E\F s1 )\Ks1 − χ(E\F s)\Ks‖2 and‖χg2(Ks1 ) − χg2(Ks)‖

2, eventually we find some δ > 0, such that whenever|s1 − s| < δ, we always have ‖χEs1 − χEs‖2 < ε.

Remark. In any given reducing subspace X for DA, T` : ` ∈ Zd, an A-dilation normalized tight frame wavelet set E for X is an A-dilation waveletset if and only if E has measure 2π. Since we can not control the measuresof the normalized tight frame wavelet sets in our proof, we are not ableto recapture D. Speegle’s connectivity result in the case when X is L2(Rd)itself. On the other hand, our approach does provide a more elementary wayof proving the path connectivity property for a larger class of basis functions.

References

[1]

11

[2] X. Dai, Y. Diao and Q. Gu, Normalized Tight Frame Wavelet Sets,Proc. Amer. Math. Soc., to appear.

[3] X. Dai and D. Larson, Wandering vectors for unitary systems andorthogonal wavelets, Memoirs Amer. Math. Soc., 134(1998), N0. 640.

[4] X. Dai, D. Larson and D. Speegle, Wavelet sets in Rn, J. Fourier Anal.Appl., 3(1997), 451-456.

[5] X. Dai, D. Larson and D. Speegle, Wavelet sets in Rn II, Contemp.Math., 216 (1998), 15-40.

[6] X. Dai and S. Lu, Wavelets in subspaces, Mich. J. Math., 43(1996),81-89.

961-1005.

[7] N. Dunford and J. Schwartz, Linear Operators Part I, Wiley-Interscience. 1958.

[8] Q. Gu and D. Han, On multiresolution analysis (MRA) wavelets in Rd,J. Fourier Anal. Appl., to appear.

[9] D. Han and D. Larson, Bases, Frames and Group representations,Memoirs. Amer. Math. Soc., to appear.

[10] E. Hernandez, G. Weiss, A first course on wavelets, CRC Press, BocaRaton, (1996).

[11] D. Speegle, The s-elementary wavelets are path-connected, Proc. Amer.Math. Soc., 127(1999), 223-233.

X.DAI and Y.DIAO

Department of Mathematics

University of North Carolina at Charlotte

Charlotte, NC 28223 USA

Q.GU

Department of Mathematics

Beijing University

12

Beijing, China

D.HAN

Department of Mathematics and Statistics

McMaster University

Hamilton, ON L8S 4K1 Canada Canada

13