MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation97590110.1155/2009/975901975901Research ArticleMultiresolution Analysis and Haar Wavelets on the Laguerre HypergroupXiePeizhuHeJianxunSeyranianAlexander P.Department of MathematicsSchool of Mathematics and Information SciencesGuangzhou UniversityGuangzhou 510006Chinagzhu.edu.cn200916042009200926092008070420092009Copyright © 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.

Let n be the Heisenberg group. The fundamental manifold of the radial function space for n can be denoted by [0,+)×, which is just the Laguerre hypergroup. In this paper the multiresolution analysis on the Laguerre hypergroup 𝕂=[0,+)× is defined. Moreover the properties of Haar wavelet bases for La2(𝕂) are investigated.

1. Introduction

In the past decade research on the multiresolution analysis has made considerable progress due to its wide applications. For the basic theory of multiresolution we refer readers to the work in [1, 2]. Recently, we find that a lot of authors try to extend the theory of wavelets on the Euclidean space to nilpotent Lie groups (see ).

In this paper we will give the definition of acceptable dilations on the Laguerre hypergroup. The multiresolution analysis on the Laguerre hypergroup 𝕂=[0,+)× is also defined. Moreover the properties of Haar wavelet bases for La2(𝕂) are investigated. We will prove the results analogous to those on n in , on n in , and on 1×1××1 in .

Let dma(x,t) be the positive measure defined on 𝕂, for a0, by dma(x,t)=1πΓ(a+1)x2a+1dxdt; and La2(𝕂) denotes the space of all measurable functions on 𝕂 such that fLa22=𝕂|f(x,t)|2dma(x,t)<. The generalized translation operator T(x,t)a on 𝕂 is defined by T(x,t)af(y,s)={12π02πf((x2+y2+2xycosθ)1/2,  t+s+xysinθ)dθ,if  a=0,aπ0102πf((x2+y2+2xyρcosθ)1/2,t+s+xyρsinθ)ρ(1-ρ2)a-1dθdρ,if  a>0, for all (x,t)𝕂, fLa2(𝕂). It is said to be the Fourier transform of a function fLa2(𝕂) defined as follows: f̂(λ,m)=𝕂φ-λ,m(x,t)f(x,t)dma(x,t), where φλ,m(x,t)=eiλtma(|λ|x2), and the Laguerre function ma is defined on + by ma(x)=e-x/2(Lma(x)/Lma(0)), and Lma is the Laguerre polynomial of degree m and order a. We know that for a pair of functions f and g, the generalized convolution product on the Laguerre hypergroup is defined by f*g(x,t)=𝕂T(x,t)af(y,s)g(y,-s)dma(y,s),    (x,t)𝕂. Further if f and g are in L1(𝕂), then we have f*g^=f̂·ĝ. The functional analysis and Fourier analysis on 𝕂 and its dual have been extensively studied in [8, 9].

Let Γ={(m,n):m,n} be a discrete subspace of 𝕂. An automorphism D is said to be an acceptable dilation for Γ if it satisfies the following properties:

D leaves Γ invariant, that is, DΓΓ,

all the eigenvalues, λi, of D satisfy |λi|>1.

The acceptable dilation D on La2(𝕂) is defined by Df(x,t)=f(D(x,y)), for all   fLa2(𝕂). Let δr  (r>0) be the dilation on the Laguerre hypergroup. Hence for all (x,y)𝕂, δr(x,y)=(rx,r2y). Clearly, for every r and r2, δr is just an acceptable dilation on the Laguerre hypergroup. Now we give the definition of multiresolution analysis on the Laguerre hypergroup.

Definition 1.1 ((MRA(<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M61"><mml:mrow><mml:mi>𝕂</mml:mi></mml:mrow></mml:math></inline-formula>), <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M62"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M63"><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula>)).

A multiresolution analysis on 𝕂 is an increasing sequence {Vj}j of closed subspaces of La2(𝕂) satisfying the following conditions:

jVj={0}, jVj¯=La2(𝕂);

fVjDfVj+1;

fV0TγafV0, for all γΓ;

there exists a scaling function ϕV0 such that {Tγaϕ}γΓ forms an orthonormal basis of V0.

From the above definition it is clear that {DTγaϕ}γΓ is an orthonormal basis of V1. It follows from V0V1 and ϕV0V1 that there exists a sequence {h(γ)}γΓ such that ϕ=γΓh(γ)DTγaϕ. The solution of (1.7) is often called a refinable function or a scaling function and {h(γ)}γΓ is called a refinement sequence.

2. Acceptable Dilations on the Laguerre Hypergroup

In this section we will investigate the acceptable dilations on the Laguerre hypergroup. From the previous argument, we know that the acceptable dilations on the Laguerre hypergroup must satisfy three conditions:

they must be a automorphism of Laguerre hypergroup;

they must leave Γ invariant;

the modulus of their eigenvalues must be more than 1.

Theorem 2.1.

The acceptable dilations on 𝕂 must be the form D=(k10k2k3), where k1, k2, k3 and k1>1, |k3|>1.

Proof.

Let (abcd) be the acceptable dilations on 𝕂, where a,b,c,d. From the condition (1), we can obtain D(xy)=(ax+bycx+dy)𝕂,  (x,t)𝕂, which implies that ax+by0 for all x0 and y. This yields b=0 and a0. From DΓΓ, we get a,c,d. By using the condition (3) we can obtain that a>1 and |d|>1. This concludes the proof of the theorem.

3. Multiresolution Analysis on the Laguerre Hypergroup

In this section, we only consider the dilation δr, where r and r>1. For simplicity we denote it by δr=α. In order to obtain the main theorem, we need to give some lemmas to characterize the properties of the multiresolution analysis on 𝕂.

Lemma 3.1.

SupposeVjVj+1  (j) where VjLa2(𝕂) and {Vj}j satisfies (2) and (4) of the definition of multiresolution analysis on the Laguerre hypergroup. The characteristic function χQ of the set Q is a scaling function of multiresolution analysis. Then jVj={0}.

Proof.

Let γ1,γ2Γ and γ1γ2. By using the property (4) of the Definition 1.1 we can obtain Tγ1aχQ,Tγ2aχQ=0, which implies that 𝕂Tγ1aχQTγ2aχQdma=0. From (1.3), we know that TγaχQ0 and there exists a constant C>0 such that |TγaχQ|C for all a0 and γ𝕂. This yields Tγ1aχQTγ2aχQ=0, which implies that Tγ1aχQ and Tγ2aχQ cannot be nonzero at the same time.

Let fjVj. Then fV-j for any j which implies that αjfV0. Thus there exists a sequence {bj(γ)}γΓ such that αjf=γΓbj(γ)TγaχQ. This yields |αjf|=|γΓbj(γ)TγaχQ|supγΓ|bj(γ)|rΓ|TγaχQ|{bj(γ)}l2γΓ|TγaχQ|, which implies that |αjf|C{bj(γ)}l2. Then we can see that |f(P)|=|αj(α-jf(P))|=|αjf(α-jP)|C{bj(γ)}l2(Γ)=CαjfLa2. Notice that αjfLa2=(K|αjf(P)|2dma)1/2=(K|f(αjP)|2dma)1/2=(-+0+|f(rjx,r2jy)|2x2a+1πΓ(a+1)dxdy)1/2=r-(a+2j)fLa2. If we let j tend to infinity, then we can obtain f=0. This implies that jVj={0}. The desired result is thus obtained.

Lemma 3.2.

Suppose VjVj+1    (j), where VjLa2(𝕂) and {Vj}j satisfies (2), (3), and (4) of the definition of multiresolution analysis on the Laguerre hypergroup. If the scaling function ϕ in (4) is in La1(𝕂) and 𝕂ϕdma0, then jVj¯=La2(𝕂).

Proof.

Let P=(x,t)𝕂 and a>0. Then we have TαPaf(y,s)=T(rx,r2t)af(y,s)=aπ02π01f((r2x2+y2+2rxyrcosθ)1/2,s+r2t+rxyrsinθ)r(1-r2)a-1drdθ=aπ02π01f(r(x2+yr22+2xyrrcosθ)1/2,r2(sr2+t+xyrrsinθ))r(1-r2)a-1drdθ=T(x,t)aδrf(yr,sr2)=α-1T(x,t)aαf, which implies TαPa=α-1TPaα. For a=0, we can get the same result. It is easy to see that TαlPa=α-lTPaαl, for all lΓ and a0.

Let φjVj. Then there exists a j0 such that φVj0. For any l, let j>-l and jj0. Using VjVj+1, we immediately obtain φVj. Then there exists a sequence {aj(γ)}γΓl2(Γ) such that φ=γΓaj(γ)αjTγaϕ, which implies Tαl(P)aφ=γΓaj(γ)Tαl(P)aαjTγaϕ=γΓaj(γ)αjTαl+j(P)aTγaϕ. Notice that PΓ, l+j>0, and l+j. Thus we can see that αl+j(P)Γ and αjTαl+j(P)aTγaϕVj, which implies Tαl(P)aφVjjVj¯, for all l and PΓ.

Let ψjVj¯. Then for any ε>0, there exists a φjVj such that φ-ψLa2<ε. It follows from Tαl(P)aφ-Tαl(P)aψLa2=Tαl(P)a(φ-ψ)La2φ-ψLa2<ε and Tαl(P)aφjVj¯ that Tαl(P)aψjVj¯, for all l and PΓ.

For any g𝕂, there must exist an element PΓ and l such that |αl(P)-g| is arbitrarily small, which implies that Tαl(P)aψ-Tgaψ2<ε for any arbitrarily small ε>0. This yields TgaψjVj¯, for all ψjVj¯ and g𝕂.

Note ϕ̂(λ,m)=𝕂φ-λ,m(x,t)ϕ(x,t)dma and φλ,m(x,t)=eiλtma(|λ|x2), ϕLa1(𝕂). This shows that ϕ̂(λ,m)𝕂ϕdma,  when      λ0. Since 𝕂ϕdma0, there exists some ε>0 such that ϕ̂(λ,m)0 for all |λ|<ε. Let W=jVj¯ and ψW. Then φ,ψ=0 for all φW, which implies that for all g𝕂, 0=Tgaφ,ψ=KTgaφ(x,y)ψ(x,y)dma(x,y)=KTgaφ(x,y)ψ̃(x,-y)dma(x,y)=φ*ψ̃(g), where ψ̃(x,y)=ψ(x,-y). Then φ*ψ̃̂(λ,m)=φ̂(λ,m)ψ̃̂(λ,m)=0. Notice that αf̂(λ,m)=𝕂φ-λ,m(x,t)f(rx,r2t)dma=1r2a+4𝕂φ-λ,m(xr,tr2)f(x,t)dma,φλ,m(xr,tr2)=eiλ(t/r2)ma(|λ|(xr)2)=ei(λ/r2)tma(|1r2λ|x2)=φλ/r2,m(x,t). Thus we can see that αf̂(λ,m)=(1/r2a+4)f̂(λ/r2,m), which implies αjf̂(λ,m)=(1/rj(2a+4))f̂(λ/r2j,m). Let φ=rj(2a+4)αjϕ. Then φW and φ̂=ϕ̂(λ/r2j,m). This yields ϕ̂(λr2j,m)ψ̃̂(λ,m)=0. Taking into account the fact that ϕ̂(λ/r2j,m)0 when |λ|<r2jε, we see ψ̃̂(λ,m)=0 when |λ|<r2jε. Let j tend to infinity, then ψ̃̂=0 for all λ which implies ψ=0. Then jVj¯=La2(𝕂). We complete the proof of this theorem.

Theorem 3.3.

Suppose ϕ=χQ is a scaling function for a multiresolution analysis associated with (Γ,α), where χQ is the characteristic function of a measurable set Q. Then Q satisfies the following properties:

Tγ1aχQTγ2aχQ=0, for a.e. x𝕂, γ1γ2 and γ1,γ2Γ;

χQ=γΓβ(γ)αTγaχQ;

|Q|=1;

Tγ1aTγ2aχQ can be represented by the sequence {TγaχQ}γΓ where γ1,γ2Γ.

Conversely, the characteristic function of a bounded measurable set Q that satisfies properties (1), (2), (3), and (4) is the scaling function of a multiresolution analysis associated with (Γ,α).

Proof.

Suppose ϕ=χQ is a scaling function for a multiresolution analysis associated with (Γ,α). Then Tγ1aχQ,Tγ2aχQ=0 for all γ1γ2 and γ1,γ2Γ, which implies 𝕂Tγ1aχQTγ2aχQdma=0. Notice that Tγ1aχQ0 and Tγ2aχQ0. Thus we can obtain that Tγ1aχQTγ2aχQ=0, almost every x𝕂. By (1.7), we know that the second property is satisfied. Because of χQLa2=1, we can see that |Q|=1. Let V0(MRA(𝕂),Γ,α). Then Tγ2aχQV0. This implies Tγ1aTγ2aχQV0. Therefore, Tγ1aTγ2aχQ can be represented by {TγaχQ}.

To see the converse, let V0={fLa2(𝕂):f=γΓc(γ)TγaχQ},  Vj=αjV0. Then {Vj}j is a family of closed subspace of La2(𝕂). Let fV0. Then f=γΓc(γ)TγaχQ=γΓc(γ)Tγaγ1Γβ(γ1)αTγ1aχQ=γ,γ1Γc(γ)β(γ1)TγaαTγ1aχQ=γ,γ1Γc(γ)β(γ1)αTα(γ)aTγ1aχQ. Since α(γ)Γ, we can see that Tα(γ)aTγ1aχQV0, which implies fV1. This yields V0V1. Then we can also get VjVj+1. Notice that f=γΓc(γ)TγaχQ. Thus we can see that Tγ1af=γΓc(γ)Tγ1aTγaχQ, for all γ1Γ. Because Tγ1aTγ2aχQ can be represented by the sequence {TγaχQ}, thus Tγ1afV0.

In order to show that {Vj}j is a multiresolution analysis associated with (Γ,α), it suffices to show that jVj=0 and jVj¯=La2(𝕂). Further, it follows easily from Lemmas 3.1 and 3.2 that jVj=0,jVj¯=La2(𝕂). Our result is proved.

In this paper orthonormal Haar wavelet bases for La2(𝕂) are not constructed. But we believe that orthonormal Haar wavelet bases for La2(𝕂) can be constructed just as that in [2, 6, 7]. The details will appear elsewhere.

Acknowledgments

The second author is supported by the National Natural Science Foundation of China (no. 10671041) and the Doctoral Program Foundation of the Ministry of Education of China (no. 200810780002). The authors would be grateful to the referee for his/her invaluable suggestions.

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