Let ℍn be the Heisenberg group. The fundamental manifold of the radial function space for
ℍn can be denoted by [0,+∞)×ℝ, which is just the Laguerre hypergroup. In this paper the
multiresolution analysis on the Laguerre hypergroup 𝕂=[0,+∞)×ℝ 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 [3–6]).

In this paper we will give the definition of acceptable dilations on the Laguerre hypergroup. The multiresolution analysis on the Laguerre hypergroup 𝕂=[0,+∞)×ℝ is also defined. Moreover the properties of Haar wavelet bases for La2(𝕂) are investigated. We will prove the results analogous to those on ℝn in [2], on ℍn in [6], and on ℍ1×ℍ1×⋯×ℍ1 in [7].

Let dma(x,t) be the positive measure defined on 𝕂, for a≥0, by
dma(x,t)=1πΓ(a+1)x2a+1dxdt;
and La2(𝕂) denotes the space of all measurable functions on 𝕂 such that
∥f∥La22=∫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)={12π∫02πf((x2+y2+2xycosθ)1/2,t+s+xysinθ)dθ,ifa=0,aπ∫01∫02πf((x2+y2+2xyρcosθ)1/2,t+s+xyρsinθ)ρ(1-ρ2)a-1dθdρ,ifa>0,
for all (x,t)∈𝕂, f∈La2(𝕂). It is said to be the Fourier transform of a function f∈La2(𝕂) defined as follows:
f̂(λ,m)=∫𝕂φ-λ,m(x,t)f(x,t)dma(x,t),
where φλ,m(x,t)=eiλtℒma(|λ|x2), and the Laguerre function ℒma is defined on ℝ+ by ℒma(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,n∈ℤ} 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, DΓ⊆Γ,

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 f∈La2(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∈ℕ and r≥2, δr is just an acceptable dilation on the Laguerre hypergroup. Now we give the definition of multiresolution analysis on the Laguerre hypergroup.

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

⋂j∈ℤVj={0}, ⋃j∈ℤVj¯=La2(𝕂);

f∈Vj⇔Df∈Vj+1;

f∈V0⇔Tγaf∈V0, for all γ∈Γ;

there exists a scaling function ϕ∈V0 such that {Tγaϕ}γ∈Γ forms an orthonormal basis of V0.

From the above definition it is clear that {DTγaϕ}γ∈Γ is an orthonormal basis of V1. It follows from V0⊆V1 and ϕ∈V0⊆V1 that there exists a sequence {h(γ)}γ∈Γ such that
ϕ=∑γ∈Γh(γ)DTγaϕ.
The solution of (1.7) is often called a refinable function or a scaling function and {h(γ)}γ∈Γ 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∈ℤ and k1>1, |k3|>1.

Proof.

Let (abcd) be the acceptable dilations on 𝕂, where a,b,c,d∈ℝ. From the condition (1), we can obtain
D(xy)=(ax+bycx+dy)∈𝕂,∀(x,t)∈𝕂,
which implies that ax+by≥0 for all x≥0 and y∈ℝ. This yields b=0 and a≥0. From DΓ⊆Γ, we get a,c,d∈ℤ. 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∈ℕ 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.

SupposeVj⊆Vj+1(j∈ℤ) where Vj⊂La2(𝕂) and {Vj}j∈ℤ 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 ⋂j∈ℤVj={0}.

Proof.

Let γ1,γ2∈Γ and γ1≠γ2. 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 TγaχQ≥0 and there exists a constant C>0 such that |TγaχQ|≤C for all a≥0 and γ∈𝕂. This yields Tγ1aχQTγ2aχQ=0, which implies that Tγ1aχQ and Tγ2aχQ cannot be nonzero at the same time.

Let f∈⋂j∈ℤVj. Then f∈V-j for any j∈ℤ which implies that αjf∈V0. Thus there exists a sequence {bj(γ)}γ∈Γ such that αjf=∑γ∈Γbj(γ)TγaχQ. This yields
|αjf|=|∑γ∈Γbj(γ)TγaχQ|≤supγ∈Γ|bj(γ)|∑r∈Γ|TγaχQ|≤∥{bj(γ)}∥l2∑γ∈Γ|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)=C∥αjf∥La2.
Notice that
∥αjf∥La2=(∫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)∥f∥La2.
If we let j tend to infinity, then we can obtain f=0. This implies that ⋂j∈ℤVj={0}. The desired result is thus obtained.

Lemma 3.2.

Suppose Vj⊆Vj+1(j∈ℤ), where Vj⊂La2(𝕂) and {Vj}j∈ℤ 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 ∫𝕂ϕdma≠0, then ⋃j∈ℤVj¯=La2(𝕂).

Proof.

Let P=(x,t)∈𝕂 and a>0. Then we have
TαPaf(y,s)=T(rx,r2t)af(y,s)=aπ∫02π∫01f((r2x2+y2+2rxyr′cosθ)1/2,s+r2t+rxyr′sinθ)r′(1-r′2)a-1dr′dθ=aπ∫02π∫01f(r(x2+yr22+2xyrr′cosθ)1/2,r2(sr2+t+xyrr′sinθ))r′(1-r′2)a-1dr′dθ=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 l∈Γ and a≥0.

Let φ∈⋃j∈ℤVj. Then there exists a j0∈ℤ such that φ∈Vj0. For any l∈ℤ, let j>-l and j≥j0. Using Vj⊆Vj+1, we immediately obtain φ∈Vj. Then there exists a sequence {aj(γ)}γ∈Γ∈l2(Γ) such that φ=∑γ∈Γaj(γ)αjTγaϕ, which implies
Tαl(P)aφ=∑γ∈Γaj(γ)Tαl(P)aαjTγaϕ=∑γ∈Γaj(γ)αjTαl+j(P)aTγaϕ.
Notice that P∈Γ, l+j>0, and l+j∈ℤ. Thus we can see that αl+j(P)∈Γ and αjTαl+j(P)aTγaϕ∈Vj, which implies Tαl(P)aφ∈Vj⊆⋃j∈ℤVj¯, for all l∈ℤ and P∈Γ.

Let ψ∈⋃j∈ℤVj¯. Then for any ε>0, there exists a φ∈⋃j∈ℤVj such that ∥φ-ψ∥La2<ε. It follows from ∥Tαl(P)aφ-Tαl(P)aψ∥La2=∥Tαl(P)a(φ-ψ)∥La2≤∥φ-ψ∥La2<ε and Tαl(P)aφ∈⋃j∈ℤVj¯ that Tαl(P)aψ∈⋃j∈ℤVj¯, for all l∈ℤ and P∈Γ.

For any g∈𝕂, there must exist an element P∈Γ and l∈ℤ such that |αl(P)-g| is arbitrarily small, which implies that ∥Tαl(P)aψ-Tgaψ∥2<ε for any arbitrarily small ε>0. This yields Tgaψ∈⋃j∈ℤVj¯, for all ψ∈⋃j∈ℤVj¯ and g∈𝕂.

Note ϕ̂(λ,m)=∫𝕂φ-λ,m(x,t)ϕ(x,t)dma and φλ,m(x,t)=eiλtℒma(|λ|x2), ϕ∈La1(𝕂). This shows that ϕ̂(λ,m)→∫𝕂ϕdma,whenλ→0. Since ∫𝕂ϕdma≠0, there exists some ε>0 such that ϕ̂(λ,m)≠0 for all |λ|<ε. Let W=⋃j∈ℤVj¯ and ψ∈W⊥. Then 〈φ,ψ〉=0 for all φ∈W, which implies that for all g∈𝕂,
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+4∫𝕂φ-λ,m(x′r,t′r2)f(x′,t′)dma,φλ,m(xr,tr2)=eiλ(t/r2)ℒma(|λ|(xr)2)=ei(λ/r2)tℒma(|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 φ∈W 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∈ℝ which implies ψ=0. Then ⋃j∈ℤVj¯=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. x∈𝕂, γ1≠γ2 and γ1,γ2∈Γ;

χQ=∑γ∈Γβ(γ)αTγaχQ;

|Q|=1;

Tγ1aTγ2aχQ can be represented by the sequence {TγaχQ}γ∈Γ where γ1,γ2∈Γ.

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 γ1≠γ2 and γ1,γ2∈Γ, which implies
∫𝕂Tγ1aχQTγ2aχQdma=0.
Notice that Tγ1aχQ≥0 and Tγ2aχQ≥0. Thus we can obtain that Tγ1aχQTγ2aχQ=0, almost every x∈𝕂. By (1.7), we know that the second property is satisfied. Because of ∥χQ∥La2=1, we can see that |Q|=1. Let V0∈(MRA(𝕂),Γ,α). Then Tγ2aχQ∈V0. This implies Tγ1aTγ2aχQ∈V0. Therefore, Tγ1aTγ2aχQ can be represented by {TγaχQ}.

To see the converse, let
V0={f∈La2(𝕂):f=∑γ∈Γc(γ)TγaχQ},Vj=αjV0.
Then {Vj}j∈ℤ is a family of closed subspace of La2(𝕂). Let f∈V0. Then
f=∑γ∈Γc(γ)TγaχQ=∑γ∈Γc(γ)Tγa∑γ1∈Γβ(γ1)αTγ1aχQ=∑γ,γ1∈Γc(γ)β(γ1)TγaαTγ1aχQ=∑γ,γ1∈Γc(γ)β(γ1)αTα(γ)aTγ1aχQ.
Since α(γ)∈Γ, we can see that Tα(γ)aTγ1aχQ∈V0, which implies f∈V1. This yields V0⊆V1. Then we can also get Vj⊆Vj+1. Notice that f=∑γ∈Γc(γ)TγaχQ. Thus we can see that Tγ1af=∑γ∈Γc(γ)Tγ1aTγaχQ, for all γ1∈Γ. Because Tγ1aTγ2aχQ can be represented by the sequence {TγaχQ}, thus Tγ1af∈V0.

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

DaubechiesI.GröchenigK.MadychW. R.Multiresolution analysis, Haar bases, and self-similar tilings of ℝnGellerD.Fourier analysis on the Heisenberg group. I. Schwartz spaceLawtonW.Infinite convolution products and refinable distributions on Lie groupsLiuH.PengL.Admissible wavelets associated with the Heisenberg groupLiuH. P.LiuY.WangH. H.Multiresolution analysis, self-similar tilings and Haar wavelets on the Heisenberg groupto appear in Acta Mathematica Scientia Series BXie P. Z.HeJ. X.Multiresolution analysis and Haar wavelets on the product of Heisenberg groupAssalM.Ben AbdallahH.Generalized Besov type spaces on the Laguerre hypergroupNessibiM. M.TrimècheK.Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets