MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation49258510.1155/2009/492585492585Research ArticleMRA Parseval Frame Wavelets and Their Multipliers in L2(Rn)GuochangWu1XiaohuiYang2ZhanweiLiu3VampaVictoria1College of InformationHenan University of Finance and EconomicsZhengzhou 450002Chinahnufe.edu.cn2Institute of Applied Mathematics College of Mathematics and Information SciencesHenan UniversityKaifeng 475004Chinahenu.edu.cn3School of Information EngineeringZhengzhou UniversityZhengzhou 450001China200904082009200915122008290620092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We characterize all generalized lowpass filters and multiresolution analysis( MRA) Parseval frame wavelets in L2(Rn) with matrix dilations of the form (Df)(x)=2f(Ax), where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. At first, we study the pseudo-scaling functions, generalized lowpass filters, and multiresolution analysis (MRA) Parseval frame wavelets and give some important characterizations about them. Then, we describe the multiplier classes associated with Parseval frame wavelets in L2(Rn) and give an example to prove our theory.

1. Introduction

Wavelet theory has been studied extensively in both theory and applications since 1980’s (see ). One of the basic advantage of wavelets is that an even can be simultaneously described in the frequency domain as well as in the time domain. This feature permits a multiresolution analysis of data with different behaviors on different scales. The main advantage of wavelets is their time-frequency localization property. Many signals can be efficiently represented by wavelets.

The classical MRA wavelets are probably the most important class of orthonormal wavelets. Because they guarantee the existence of fast implementation algorithm, many of the famous examples often used in applications belong to this class. However, there are useful filters, such as m(ω)=(1/2)(1+e3iω), that do not produce orthonormal basis; nevertheless, they do produce systems that have the reconstruction property as well as many other useful features. It is natural, therefore, to develop a theory involving more general filters that can produce systems having these properties.

A tight wavelet frame is a generalization of an orthonormal wavelet basis by introducing redundancy into a wavelet system . By allowing redundancy in a wavelet system, one has much more freedom in the choice of wavelets. Tight wavelet frames have some desirable features, such as near translation invariant wavelet frame transforms, and it may be easier to recognize patterns in a redundant transform. For advantages and applications of tight wavelet frames, the reader is referred to  and many references therein. Recently, the theory of high dimensional wavelet is widely studied by the people, such as [16, 17].

In , authors discussed wavelet multipliers, scaling function multipliers, and lowpass filter multipliers in L2(Rn). In , authors introduced a class of generalized lowpass filter that allowed them to define and construct the MRA Parseval frame wavelets. This led them to an associated class of generalized scaling functions that were not necessarily obtained from a multiresolution analysis. Also, they generalized notions of the wavelet multipliers in  to the case of wavelet frame and got several properties of the multipliers of Parseval frame wavelets.

In this paper, we characterize all generalized lowpass filter and MRA Parseval frame wavelets (PFWs) in L2(Rn) with matrix dilations of the form (Df)(x)=2f(Ax), where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. Firstly, we study some properties of the generalized wavelets, scaling functions, and filters in L2(Rn). Our result is a generalization of the construction of PFWs from generalized lowpass filters that is introduced in . Though we follow  as a blueprint, it is well known that the situation in higher dimension is so complex that we have to recur to some special matrices to solve problem. Thus, our ways are different from original ones. And we borrow some thoughts and technique in . Then, we give some characterizations of the multiplier classes associated with Parseval frame wavelets in L2(Rn).

Let us now describe the organization of the material that follows. Section 2 is of a preliminary character: it contains various results on matrices belonging to the class En(2) and some facts about a Parseval frame wavelet. In Section 3, we study the pseudoscaling functions, the generalized lowpass filters, and the MRA PFWs and give some important characterizations about them. In Section 4, we describe the multiplier classes associated with Parseval frame wavelets in L2(Rn). At last, we give an example to prove our theory.

2. Preliminaries

Let us now establish some basic notations.

We denote by Tn the n-dimensional torus. By Lp(Tn) we denote the space of all Zn-periodic functions f (i.e., f is 1-periodic in each variable) such that Tn|f(x)|pdx<+. The standard unit cube [-(1/2),1/2)n will be denoted by C. The subsets of Rn invariant under Zn translations and the subsets of Tn are often identified.

We use the Fourier transform in the form f̂(ω)=Rnf(x)e-2πi<x,ω>dx, where <·,·> denotes the standard inner product in Rn.

For f,gL2(Rn) we denote the function [f,g](ω) as follows: [f,g](ω)=kZnf(ω+k)g(ω+k)¯. In particular, for fL2(Rn), we will write σf(ω):=kZn|f̂(ω+k)|2, which is named as the bracket function of f. For σf(ω)=kZn|f̂(ω+k)|2, we let Ωf be the Zn-translation invariant subset of Rn defined, modulo a null set, by Ωf=suppσf={ωRn:  f̂(ω+k)0,for some  kZn}.

The Lebesgue measure of a set SRn will be denoted by |S|. When measurable sets X and Y are equal up to a set of measure zero, we write XY.

Then we introduce some notations and the existing results about expanding matrices.

Let En(2) denote the set of all expanding matrices A such that |detA|=2. The expanding matrices mean that all eigenvalues have magnitude greater than 1. For AEn(2), we denote by B the transpose of A:B=At. It is obvious that BEn(2).

The following elementary lemma [16,Lemma 2.2] provides us with a convenient description of BZn for an arbitrary AEn(2), and it will be used in Section 3.

Lemma 2.1.

Let BEn(2) be any integer matrix such that |detB|=2. Then the group Zn/BZn is isomorphic to B-1Zn/Znand the order of Zn/BZn is equal to 2. In particular, if αZn/BZn and β=B-1α, then Zn=BZn(BZn+α) and B-1Zn=Zn(Zn+β).

Our standard example that will be frequently used is the quincunx matrix Q=(11-11)En(2). Observe that Q acts on R2 as rotation by π/4 composed with dilation by 2. In the quincunx case, our standard choice will be α=(1,0),β=(1/2,1/2).

In this paper, we will work with two families of unitary operators on L2(Rn). The first one consists of all translation operators Tk:  L2(R)nL2(Rn),kZn, defined by (Tkf)(x)=f(x-k). The second one consists of all integer powers of the dilation operator DA:  L2(Rn)L2(Rn) defined by (Df)(x)=2f(Ax) with AEn(2).

Let us now fix an arbitrary matrix AEn(2). For a function ψL2(Rn), we will consider the affine system Ψ defined by Ψ={ψj,  k(x)ψj,  k(x)=2j/2ψ(Ajx-k),jZ,kZn}.

Let us recall the definition of a Parseval frame and a Parseval frame wavelet.

Definition 2.2.

We say that a countable family {fj},jJ, in a separable Hilbert space H, is a Parseval frame (PF) for H if the equality f2=jJ|<f,fj>|2 is satisfied for all fH.

Definition 2.3.

We say that ψL2(Rn) is a Parseval frame wavelet (briefly: PFW) if the system (2.3) is a Parseval frame for L2(Rn).

Then we recall a result from  that characterizes Parseval frame wavelets associated with more general matrix dilations. We state the special case of that theorem appropriate to the discussion in this paper.

Lemma 2.4 ([<xref ref-type="bibr" rid="B18">20</xref>, Theorem 6.12]).

Let A be an arbitrary matrix in En(2), B=At ,and ψL2(Rn). Then the system (2.3) is a PFW if and only if both the equality         jZ|ψ̂(Bjω)|2=1,a.e. and the equality j=0+ψ̂(Bjω)ψ̂(Bj(ω+Bk+α))¯=0,a.e.,  kZn,αZn/BZn are satisfied.

In the following, we will give some definitions which will be used in this paper. In fact, they are some generalizations of the notations in .

Definition 2.5.

A measurable Zn-periodic function m on Rn is called a generalized filter if it satisfies |m(ω)|2+|m(ω+β)|2=1a.e.ω, where β is defined in Lemma 2.1.

We will denote by F̃ the set of generalized filters and put F̃+={mF̃:m0}. Observe that mF̃|m|F̃+.

Definition 2.6.

A function φL2(Rn) is called a pseudoscaling function if there exists a filter mF̃ such that φ̂(Bω)=m(ω)φ̂(ω)a.e.ω.

Notice that m is not uniquely determined by the pseudo-scaling function φ. Therefore, we shall denote by F̃φ the set of all mF̃ such that m satisfies (2.7) for φ. For example, if φ=0, then, F̃φ=F̃. If φ is a scaling function of an orthonormal MRA wavelet, then F̃φ is a singleton; its only element is the lowpass filter m associated with φ. Notice that for a pseudo-scaling function φ, the function |φ̂| is also a pseudo-scaling function, and if mF̃, then |m|F̃|φ̂|.

Suppose that mF̃+. Since 0m(ω)1,a.e.ω, the function φm̂(ω)=:j=1+m(B-jω) is well defined a.e.; moreover, we have φm̂(Bω)=m(ω)φm̂(ω),a.e.  ω. Following , the function φm̂ defined by (2.9) belongs to L2(Rn), and the function φm̂ is a pseudo-scaling function such that mF̃φm.

Consequently, if mF̃ is an arbitrary generalized filter, then the function φ|m|̂ is a pseudo-scaling function and |m|F̃φ|m|.

Definition 2.7.

For mF̃+, define N0(m)={ωRn:limj+φm̂(B-jω)=0}. We say that mF̃ is a generalized low-pass filter if |N0(|m|)|=0. The set of all generalized low-pass filters is denoted by F̃0.

Then, we will give the definition of MRA PFW.

Definition 2.8.

A PFW ψ is an MRA PFW if there exists a pseudo-scaling function φ and mF̃φ and a unimodular function sL2(Tn) such that ψ̂(Bω)=e2πωis(Bω)m(ω+β)¯φ̂(ω),a.e.    ω.

Let us conclude this introductory section by noting that many of the results that follow can be proved for dilations by expanding integer matrices with arbitrary determinant. Some of these extensions are obtained easily with essentially the same proofs; others require subtler and more involved arguments. But, for the sake of simplicity, we restrict ourselves to the class En(2).

3. MRA Parseval Frame Wavelets

The main purpose of this section is to study the pseudo-scaling functions, the generalized filters, and the MRA PFWs in L2(Rn). We give some important characterizations about them.

In the following we firstly give several lemmas in order to prove our main results.

Lemma 3.1.

Suppose that φ is a pseudo-scaling function and mF̃φ. If limj+|φ̂(B-jω)|=1,a.e.ω, then, |φ̂(ω)|=|j=1+m(B-jω)|,a.e.ω, and |N0(|m|)|=0.

Proof.

By (2.7), we have |φ̂(ω)|=|j=1nm(B-jω)||φ̂(B-jω)|,a.e.ω. Using (2.9), we obtain that |φ̂(ω)|=φ|m|̂ and |N0(|m|)|=0 is clearly satisfied. Thus, the function mF̃ is a generalized low-pass filter.

Lemma 3.2.

If fL1(Rn), then, for a.e.ωRn, limj+|f(Bjω)|=0.

Proof.

Assuming that fL1(Rn) and applying the monotone convergence theorem we obtain RnjN|f(Bjω)|dω=jNRn|f(Bjω)|dω=jN2-jRn|f(ξ)dξ|=f1<+.

It follows that for a.e. ωRn, jN|f(Bjω)| is finite. Therefore, for a.e.    ωRn, limj+|f(Bjω)|=0.

Then, we will give a characterization of the generalized lowpass filter.

Theorem 3.3.

Suppose ψ is an MRA PFW and φ is a pseudo-scaling function satisfying (2.11). Then, m defined by (2.11) is a generalized lowpass filter.

Proof.

Since ψ is an MRA PFW, from (2.4), (2.6), and (2.11), we can obtain 1=jZ|ψ̂(Bjω)|2=jZ|m(Bj-1ω+β)|2|φ̂(Bj-1ω)|2=limn+j=-nn|m(Bj-1ω+β)|2|φ̂(Bj-1ω)|2=limn+j=-nn[1-|m(Bj-1ω)|2]|φ̂(Bj-1ω)|2=limn+{|φ̂(B-n-1ω)|2-|φ̂(Bnω)|2}. Since φL2(Rn), Lemma 3.2 implies limn+|φ̂(Bnω)|2=0 for a.e. ω. This shows that for a.e. ω, limn+|φ̂(B-nω)|=1, thus, by Lemma 3.1, m is a generalized lowpass filter.

The following theorem provides a way of constructing MRA PFWs from generalized lowpass filters.

In order to obtain main result in this part, we firstly introduce a result in [16,Lemma 2.8].

Lemma 3.4.

Let BMn(Z) be an expanding matrix. Let μ be a Zn-periodic, unimodular function. Then there exists a unimodular function ν that satisfies μ(ω)=ν(Bω)ν(ω)¯,a.e.  ω.

Theorem 3.5.

Suppose that m is a generalized lowpass filter. Then, there exist a pseudo-scaling function φ and an MRA PFW ψ such that they satisfy (2.11).

Proof.

Suppose that m is a generalized lowpass filter. Consider first the signum function μ for m: μ(ω)={m(ω)|m(ω)|,m(ω)0,1,m(ω)0. Clearly, μ is a measurable unimodular function such that, for all ω, m(ω)=μ(ω)|m(ω)|. By Lemma 3.4, there exists a unimodular measurable function ν such that ν(Bω)ν(ω)¯=μ(ω),a.e.ω. Now take the function φ|m|̂ constructed from |m| by (2.9), and put φ̂(ω):=ν(ω)φ|m|̂(ω). Obviously, φL2(Rn). Using (2.9), (3.6), (3.10), and the definition of the signum function μ, we find φ̂(Bω)=ν(Bω)φ|m|̂(Bω)=ν(Bω)|m(ω)|φ|m|̂(ω)=ν(Bω)|m(ω)|ν(ω)¯φ̂(ω)=μ(ω)|m(ω)|φ̂(ω)=m(ω)φ̂(ω).

Thus, φ is a pseudo-scaling function.

For any unimodular function sL2(Tn), we define

ψ̂(Bω)=e2πωis(Bω)m(ω+β)¯φ̂(ω),a.e.ω. We affirm that ψ(x) is an MRA PFW. Lemma 2.4 shows that it is enough to prove that ψ̂ satisfies (2.4) and (2.5).

From the fact that m is a generalized lowpass filter, we can deduce that (3.1) holds. Using Lemma 3.2, we obtain

jZ|ψ̂(Bjω)|2=jZ|m(Bj-1ω+β)|2|φ̂(Bj-1ω)|2=limn+j=-nn|m(Bj-1ω+β)|2|φ̂(Bj-1ω)|2=limn+j=-nn[1-|m(Bj-1ω)|2]  |φ̂(Bj-1ω)|2=limn+{|φ̂(B-n-1ω)|2-|φ̂(Bnω)|2}=limn+|φ̂(B-n-1ω)|2. This computation shows that the first characterizing condition (2.4) in Lemma 2.4 is satisfied precisely when (3.1) holds.

Let us now prove that the function ψ given by above satisfies (2.5).

Let us fix ω and q=Bk+α,kZn,  αZn/BZn, and write

j=0+ψ̂(Bjω)ψ̂(Bj(ω+q))¯=ψ̂(ω)ψ̂((ω+q))¯+j=1+ψ̂(Bjω)ψ̂(Bj(ω+q))¯.

To compute the first term on the right-hand side of (3.14), by the fact e2πiB-1α=e2πiβ=-1 ,and (2.6), we have

ψ̂(ω)ψ̂(ω+q)¯=e2πiB-1ωs(ω)m(B-1ω+β)¯φ̂(B-1ω)×e-2πiB-1(ω+q)s(ω+q)¯m(B-1(ω+q)+β)φ̂(B-1(ω+q))¯=e-2πiB-1q|s(ω)|2m(B-1ω)φ̂(B-1ω)m(B-1ω+k+B-1α)¯φ̂(B-1(ω+q))¯  =e-2πiB-1αφ̂(ω)m(B-1(ω+q))¯φ̂(B-1(ω+q))¯=-φ̂(ω)φ̂(ω+q)¯.

To compute the second term on the right-hand side of (3.14), by (2.6), (2.7), and (2.11), we have

j=1+ψ̂(Bjω)ψ̂(Bj(ω+q))¯=j=1+e2πiBj-1ωs(Bjω)m(Bj-1ω+β)¯φ̂(Bj-1ω)×e-2πiBj-1(ω+q)s(Bj(ω+q))¯m(Bj-1(ω+q)+β)φ̂(Bj-1(ω+q))¯=j=1+e-2πiBj-1q|m(Bj-1ω+β)|2φ̂(Bj-1ω)φ̂(Bj-1(ω+q))¯=j=1+(1-|m(Bj-1ω)|2)φ̂(Bj-1ω)φ̂(Bj-1(ω+q))¯  =j=1+{φ̂(Bj-1ω)φ̂(Bj-1(ω+q))¯-φ̂(Bjω)φ̂(Bj(ω+q))¯}  =φ̂(ω)φ̂(ω+q)¯-limp+φ̂(Bpω)φ̂(Bp(ω+q))¯=φ̂(ω)φ̂(ω+q)¯.

This shows that the expression in (3.14) is equal to 0. Hence, ψ̂ satisfies (2.5), and ψ is a PFW.

4. The Multiplier Classes Associated with PFWs

In this section, we will describe the multiplier classes associated with PFWs in L2(Rn). We firstly give their definitions.

Definition 4.1.

A PFW multiplier ν is a function such that ψ̃=(ψ̂ν)̆ is a PFW whenever ψ is a PFW.

An MRA PFW multiplier is a function ν such that ψ̃=(ψ̂ν)̆is an MRA PFW whenever ψ is an MRA PFW.

A pseudo-scaling function multiplier is a function ν such that φ̃=(φ̂ν)̆is a pseudo-scaling function associated with an MRA PFW whenever φ has the same property.

A generalized lowpass filter multiplier is a function ν such that m̃=mν is a generalized lowpass filter whenever m is a generalized lowpass filter.

At first, we will obtain a property of PFW multiplier.

Property 4.2.

If a measurable function ν is a PFW multiplier, then, ν is unimodular.

Proof.

Let ν be a PFW multiplier; that is, there exist PFWs ψ and ψ̃ such that ψ̃̂=ψ̂ν.

Let ψL2(Rn) be any wavelet satisfying |ψ̂(ω)|>0 for a.e. ω. By assumption, for every l1, we easily deduce that (ψ̂νl)̆ is a PFW and, thus, satisfies (2.4):

jZ|ν(Bjω)|2l|ψ̂(Bjω)|2=1,a.e.ωRn.

In particular, for a.e. ω and every lN,

|ν(ω)|l|ψ̂(ω)|1. This is only possible if |ν(ω)|1 a.e. since |ψ̂(ω)| almost never vanishes. Using (2.4) for PFWs ψ and ψ̃ such that ψ̃̂=ψ̂ν, we easily obtain jZ|ψ̂(Bjω)|2(1-|ν(Bjω)|2)=1,a.e.ωRn. which is only possible if all terms vanish. Thus, ν(ω)=1 a.e.

The next theorem gives a characterization of PFW multiplier in L2(Rn).

Theorem 4.3.

If a measurable function ν is unimodular and ν(Bω)ν(ω)¯ is Zn-periodic, then, this measurable functionν is a PFW multiplier.

Proof.

PFW is characterized as an element of L2(Rn) satisfying (2.4) and (2.5) in Lemma 2.4.

Let ψ be a PFW, so that (2.4) and (2.5) hold. Let us assume that ν is unimodular and ν(Bω)ν(ω)¯ is a Zn-periodic function, necessarily unimodular. Let ψ̃̂=ψ̂ν For the function ψ̃, (2.4) obviously holds from the fact that ν is a unimodular function and |ψ̂ν|=|ψ̂|. In the following, for the function ψ̃, let us consider (2.5). Let kZn,  αZn/BZn, and let j>0. Then

ψ̃̂(Bjω)ψ̃̂(Bj(ω+Bk+α))¯=ψ̂(Bjω)ψ̂(Bj(ω+Bk+α))¯ν(Bjω)ν(Bj(ω+Bk+α))¯.

If j1, according to Lemma 3.4, there exists a unimodular function μ that satisfies

μ(ω)=ν(Bω)ν(ω)¯,a.e.ω.

By Zn-periodicity and unimodularity of μ, we have

ν(Bjω)ν(Bj(ω+Bk+α))¯=μ(Bj-1ω)μ(Bj-1(ω+Bk+α))¯ν(Bj-1ω)¯ν(Bj-1(ω+Bk+α))=μ(Bj-1ω)μ(Bj-1(ω+Bk+α))¯ν(Bj-1ω)ν(Bj-1(ω+Bk+α))¯=μ(Bj-1ω)μ(Bj-1(ω))¯ν(Bj-1ω)ν(Bj-1(ω+Bk+α))¯=ν(Bj-1ω)ν(Bj-1(ω+Bk+α))¯. Then, we can repeat the above argument until we obtain ν(Bj-1ω)ν(Bj-1(ω+Bk+α))¯=ν(ω)ν(ω+Bk+α)¯,for  j2.

By the equalities (4.4), (4.7), and summing over j0, we obtain

j=0+ψ̃̂(Bjω)ψ̃̂(Bj(ω+Bk+α))¯=ν(ω)ν(ω+Bk+α)¯j=0+j=0+ψ̂(Bjω)ψ̂(Bj(ω+Bk+α))¯.

Since the above equation of the right-hand side is 0 by (2.5) in Lemma 2.4, we conclude that ψ̃ also satisfies (2.5).

Hence, from Lemma 2.4, ψ̃ is a PFW, thus, ν is a PFW multiplier.

Remark 4.4.

It is proved in [19, Theorem 3.2] that if a measurable function ν is a PFW multiplier, ν(2ω)ν(ω)¯ is Z-periodic. However, it still is a unsolved question whether this conclusion holds in L2(Rn). We will discuss it in the future. The next result give a characterization of the MRA PFW multipliers. In order to obtain this result, we firstly introduce a result in [16,Lemma 2.8].

Lemma 4.5.

Let BMn(Z) be an expanding matrix such that |detB|=2. Let αZn/BZn and β=B-1α. Suppose that s is a Zn- periodic, unimodular function on Rn. Then there exists a Zn-periodic, unimodular function t on Rn such that s(Bω)=t(ω)t(ω+β)t(Bω)¯.

Theorem 4.6.

A measurable function ν is an MRA PFW multiplier if and only if ν is unimodular, and ν(Bω)ν(ω)¯ is Zn-periodic.

Proof.

(if) Let ν be unimodular, and let s(ω)=ν(Bω)ν(ω)¯ be Zn-periodic, necessarily unimodular. We now use Lemma 4.5 to obtain a unimodular, Zn-periodic function t such that s(Bω)=t(ω)t(ω+β)t(Bω)¯.

Let

μ(ω)=ν(ω)t(B-1ω)t(B-1ω+β). Then μ is unimodular, and μ(Bω)μ(ω)¯=ν(Bω)ν(ω)¯t(ω)t(ω+β)t(B-1ω)t(B-1ω+β)¯=s(ω)(t(ω)t(B-1ω)t(B-1ω+β)¯)t(ω+β)=s(ω)s(ω)¯t(ω+β)=t(ω+β) is a Zn-periodic function. In the above computation, we have used (4.9) and (4.10) as well as the unimodularity and the periodicity of t.

Let ψ be an MRA PFW, and let φ be an associated pseudo-scaling function with mF̃φ such that (2.11) holds. Let

m̃(ω)=m(ω)t(ω+β),φ̃̂(ω)=φ̂(ω)μ(ω). Then, we have φ̃̂(ω)=φ̂(Bω)μ(Bω)=m(ω)φ̂(ω)t(ω+β)μ(ω)=m̃(ω)φ̃̂(ω), where the second equation is deduced by (4.13) and the definition of pseudo-scaling function.

It is obvious that

|m̃(ω)|2+|m̃(ω+β)|2=1a.e.ωRn. Thus, we obtain m̃(ω)F̃φ̃.

Let

ψ̃̂(ω)=ψ̂(ω)ν(ω). Since ψ is a PFW, we can apply Lemma 2.4 to deduce that ψ̃ is a PFW. It follows, that ψ̃ is an MRA PFW, since e2πiωs(Bω)m̃(ω+β)¯φ̃̂(ω)=e2πiωs(Bω)m(ω+β)t(ω+2β)¯φ̃̂(ω)μ(ω)=ψ̂(Bω)t(ω+2β)¯μ(ω)=ψ̂(Bω)t(ω+2β)¯μ(Bω)t(ω+β)¯=ν(Bω)ψ̂(Bω)=ψ̃̂(Bω), where we have used (4.13) and the definition of μ.

(only if) The Haar wavelet is, in particular, an MRA TFW; we can then proceed as in the proof of Property 4.2 to show the unimodularity of ν. The proof that ν(Bω)ν(ω)¯ is Zn-periodic is similar to Theorem 2 in , and we omit it.

Thus, we complete the proof.

The next theorem characterizes the class of the generalized lowpass filter multipliers. Note that the generalized lowpass filters are the functions mF̃φ for some pseudo-scaling function φ satisfying Definition 2.5, that is, mF̃φ and |N0(m)|=0. We will use this fact in the following result.

Theorem 4.7.

A measurable function ν is a generalized lowpass filter multiplier if and only if ν is unimodular and Zn-periodic.

Proof.

(if) Suppose that ν is unimodular and Zn-periodic, and m(ω) is a generalized lowpass filter.

Let

m̃(ω)=m(ω)ν(ω). Then, from the definition of the generalized lowpass filter, we have mF̃φ and |N0(m)|=0. Therefore, according to the function ν being unimodular and Zn-periodic, it is obvious that these properties are also true for m̃(ω).

(only if) We proceed similarly in the proof of the two previous theorems.

Let ψL2(Rn) be any wavelet satisfying |ψ̂(ω)|>0 for a.e. ω, and m the corresponding lowpass filter. Then, it follows that |m(ω)|>0 a.e. ωRn.

Let

m̃(ω)=m(ω)ν(ω). By the assumption that the measurable function ν is a generalized lowpass filter multiplier, we have m̃F̃. In particular, m̃ is Zn-periodic, and so ν is also Zn-periodic.

Applying ν repetitively, we obtain

|ν(ω)|l|m(ω)|1,fora.e.ω,l1.

This implies that |ν(ω)|1 a.e. Unimodularity follows, since both m and m̃ are in F̃ and, thus, satisfy

|m̃(ω)|2+|m̃(ω+β)|2=1=|m(ω)|2+|m(ω+β)|2.

In a conclusion, ν is unimodular and Zn-periodic.

The above results provide description of PFW, MRA PFW, and generalized lowpass filter multipliers. These classes are identical with the respective multiplier classes of wavelets. This fact is basically a consequence of the fact that all of these multiplier operations necessarily preserve the L2(Rn) norm of the PFW ψ(ω).

Let us introduce a notation for the next theorem. For a measurable function ν, let E={ω:ν(ω)0} and μ(ω)=(ν(Bω))/(ν(ω)) on E.

The following result shows that the situation for pseudo-scaling function multipliers is completely different from others.

Theorem 4.8.

ν is a pseudo-scaling function multiplier if and only if

|ν(Bω)||ν(ω)| a.e. and limj|ν(Bjω)|=1 a.e.;

μ(ω) extends to a Zn-periodic function;

If ξ,ηE, and ξ-η is an odd multiple of β, then |μ(ω)|=|ν(ω)|=1.

Proof.

(if) Let φ be a pseudo-scaling function satisfying (2.6), and suppose that ν satisfies (1)–(3).

Let

φ̃̂(ω)=φ̂(ω)ν(ω).

Using condition (1), we see that φ̃ satisfies

limj|φ̃̂(Bjω)|=limj|ν(Bjω)|limj|φ̂(Bjω)|=1.

Let us now examine the scale equationφ̃̂(Bω)=m̃(ω)φ̃̂(ω) from (4.15).

We claim that there exists m̃F̃ such that

ν(Bω)m(ω)=ν(ω)m̃(ω).

If ω̅E, then, by (1), Bω̅E and, thus, (4.25) is satisfied automatically.

If ωE, then (4.25) is equivalent to

m̃(ω)=ν(Bω)ν(ω)m(ω).

The condition (2) implies that m̃(ω) defined on E by (4.26) is Zn-periodic on E. We will now define a Zn-periodic extension of m̃(ω) to Rn satisfying

|m̃(ω)|2+|m̃(ω+β)|2=1.

Let us consider [-1,1]n. Define m̃(ω) on (part of)[-1,1]n by the following: if ω+kE for some kZn, then

m̃(ω)=m̃(ω+k). The definition is consistent with the Zn periodicity of m on E. Let ξ,ηE, and ξ-η=β. Then one of the following conditions must hold:

k,lZn, such that ξ+kE and η+lE;

kZn such that ξ+kE and for any lZn, η+l¯E;

For any k,lZn, such that ξ+k¯E and η+l¯E.

In (a) case, (ξ+k)-(η+l) is an odd multiple of β , so by (3), 1=|μ(ξ+k)|=|μ(η+l)|. Thus, we have

|m̃(ξ)|=|m̃(ξ+k)|=|m(ξ+k)|=|m(ξ)|. Similarly, we can deduce |m̃(η)|=|m(η)|. Since mF, this implies |m̃(ξ)|2+|m̃(η)|2=1. We have either ξ+β=η or η+β=ξ, and so (4.27) holds.

In (b) case we extend the definition of m̃(ω) to the set of all such η[-1,1]n by

m̃(η)=1-|m̃(ξ)|2, and so we get (4.27).

In (c) case, we let m̃(ζ)=1 if ζB-1[-1,1]n, and m̃(ζ)=0 if ζ[-1,1]nB-1[-1,1]n, whenever ζ=ξ or η.

We have thus extended m̃(ω) to the entire interval [-1,1]n so that (4.27) holds if ξ and η are in [-1,1]n. We now extend m̃(ω) to Rn by Zn-periodicity.

Clearly, m̃F̃, and (4.15) is satisfied.

(only if) Let φ be the Shannon scaling function in L2(Rn); that is, φ̂(ω)=χ[-1,1)n(ω). Then φ̃̂(ω)=φ̂(ω)ν(ω)=ν(ω),ω[-1,1)n. By (2.6) for φ̃̂(ω), we have limj|ν(Bjω)|=1.

This establishes (2), since the right-hand side is Zn-periodic, The remain proof is similar to ones in , and so we omit them. Then, we will provide an example to prove our results.

Example 4.9.

Let A be the quincunx matrix Q=(1-111)En(2), then we get B=Qt=(11-11)En(2). Furthermore, we have B-1CC and BjCBj+1C,  for all  jN, where C is the standard unit square in R2.

Let us define

φ̂(ω)={12,ωB-1CB-2C,1,ωB-2C,0,ω  ̅  B-1C,m(ω)={0,ωB-1CB-2C,12,ωB-2CB-3C,1,ωB-3C.

Now we extend m(ω) to C such that the equality |m(ω)|2+|m(ω+β)|2=1 is satisfied for all ωC, where we take β=(1/2,1/2) and extend m(ω) to R2 by Z2-periodicity.

From the definitions of the functions φ and m, we easily deduce that φ is a pseudo-scaling function, and m is a generalized lowpass filter.

Finally, we define

ψ̂(Bω)=e2πiωm(ω+β)¯φ̂(ω). Therefore, by Theorem 3.5, we know that we get an MRA Parseval frame wavelet.

However, from , we know that this MRA Parseval frame wavelet is not a Parseval frame wavelet associated to MRA, which does not permit fast algorithm.

5. Conclusion

In this paper, we characterize all generalized lowpass filter and MRA Parseval frame wavelets in L2(Rn) with matrix dilations of the form (Df)(x)=2f(Ax), where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. We study some properties of the multipliers of generalized wavelets, scaling functions, and filters in L2(Rn). Our result is a generalization of the construction of PFWs from generalized low-pass filters that is introduced in . However, our ways are different from original ones. Then, we describe the multiplier classes associated with Parseval frame wavelets in L2(Rn) and give an example to prove our theory.

Acknowledgments

The authors would like to express their gratitude to the referee for his (or her) valuable comments and suggestions that lead to a significant improvement of their manuscript. This work was supported by the National Natural Science Foundation of China (no. 60802061) and the Natural Science Foundation for the Education Department of Henan Province of China (no. 2008B510001).

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