Multiresolution Analysis and Haar Wavelets on the Laguerre Hypergroup

Abstract

Let ℍ^𝑛 be the Heisenberg group. The fundamental manifold of the radial function space for ℍ^𝑛 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 𝐿²_𝑎(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 𝐿²_𝑎(𝕂) are investigated. We will prove the results analogous to those on ℝ^𝑛 in [2], on ℍ^𝑛 in [6], and on ℍ¹×ℍ¹×⋯×ℍ¹ in [7].

Let 𝑑𝑚_𝑎(𝑥,𝑡) be the positive measure defined on 𝕂, for 𝑎≥0, by 𝑑𝑚_𝑎1(𝑥,𝑡)=𝑥𝜋Γ(𝑎+1)^2𝑎+1𝑑𝑥𝑑𝑡;(1.1) and 𝐿²_𝑎(𝕂) denotes the space of all measurable functions on 𝕂 such that ‖𝑓‖²_𝐿2𝑎=_𝕂|𝑓(𝑥,𝑡)|²𝑑𝑚_𝑎(𝑥,𝑡)<∞.(1.2) The generalized translation operator 𝑇^𝑎_(𝑥,𝑡) on 𝕂 is defined by 𝑇^𝑎_(𝑥,𝑡)=1𝑓(𝑦,𝑠)2𝜋₀^2𝜋𝑓((𝑥²+𝑦²+2𝑥𝑦cos𝜃)^1/2𝑎,𝑡+𝑠+𝑥𝑦sin𝜃)𝑑𝜃,if𝑎=0,𝜋¹₀₀^2𝜋𝑓((𝑥²+𝑦²+2𝑥𝑦𝜌cos𝜃)^1/2,𝑡+𝑠+𝑥𝑦𝜌sin𝜃)𝜌(1−𝜌²)^𝑎−1𝑑𝜃𝑑𝜌,if𝑎>0,(1.3) for all (𝑥,𝑡)∈𝕂, 𝑓∈𝐿²_𝑎(𝕂). It is said to be the Fourier transform of a function 𝑓∈𝐿²_𝑎(𝕂) defined as follows: 𝑓(𝜆,𝑚)=_𝕂𝜑_−𝜆,𝑚(𝑥,𝑡)𝑓(𝑥,𝑡)𝑑𝑚_𝑎(𝑥,𝑡),(1.4) where 𝜑_𝜆,𝑚(𝑥,𝑡)=𝑒^𝑖𝜆𝑡ℒ^𝑎_𝑚(|𝜆|𝑥²), and the Laguerre function ℒ^𝑎_𝑚 is defined on ℝ⁺ by ℒ^𝑎_𝑚(𝑥)=𝑒^−𝑥/2(𝐿^𝑎_𝑚(𝑥)/𝐿^𝑎_𝑚(0)), and 𝐿^𝑎_𝑚 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^𝑎_(𝑥,𝑡)𝑓(𝑦,𝑠)𝑔(𝑦,−𝑠)𝑑𝑚_𝑎(𝑦,𝑠),∀(𝑥,𝑡)∈𝕂.(1.5) Further if 𝑓 and 𝑔 are in 𝐿¹(𝕂), then we have 𝑓∗𝑔=𝑓⋅𝑔.(1.6) 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:

(1)𝐷 leaves Γ invariant, that is, 𝐷Γ⊆Γ,(2)all the eigenvalues, 𝜆_𝑖, of 𝐷 satisfy |𝜆_𝑖|>1.

The acceptable dilation 𝐷 on 𝐿²_𝑎(𝕂) is defined by 𝐷𝑓(𝑥,𝑡)=𝑓(𝐷(𝑥,𝑦)), for all 𝑓∈𝐿²_𝑎(𝕂). Let 𝛿_𝑟 (𝑟>0) be the dilation on the Laguerre hypergroup. Hence for all (𝑥,𝑦)∈𝕂, 𝛿_𝑟(𝑥,𝑦)=(𝑟𝑥,𝑟²𝑦). Clearly, for every 𝑟∈ℕ and 𝑟≥2, 𝛿_𝑟 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(𝕂), Γ, 𝐷)). A multiresolution analysis on 𝕂 is an increasing sequence {𝑉_𝑗}_𝑗∈ℤ of closed subspaces of 𝐿²_𝑎(𝕂) satisfying the following conditions:(1)⋂_𝑗∈ℤ𝑉_𝑗={0}, ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²_𝑎(𝕂);(2)𝑓∈𝑉_𝑗⇔𝐷𝑓∈𝑉_𝑗+1;(3)𝑓∈𝑉₀⇔𝑇^𝑎_𝛾𝑓∈𝑉₀, for all 𝛾∈Γ;(4)there exists a scaling function 𝜙∈𝑉₀ such that {𝑇^𝑎_𝛾𝜙}_𝛾∈Γ forms an orthonormal basis of 𝑉₀.

From the above definition it is clear that {𝐷𝑇^𝑎_𝛾𝜙}_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)𝐷𝑇^𝑎_𝛾𝜙.(1.7) 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:

(1)they must be a automorphism of Laguerre hypergroup;(2)they must leave Γ invariant;(3)the modulus of their eigenvalues must be more than 1.

Theorem 2.1. The acceptable dilations on 𝕂 must be the form 𝑘𝐷=¹0𝑘₂𝑘₃,(2.1) where 𝑘₁, 𝑘₂, 𝑘₃∈ℤ and 𝑘₁>1, |𝑘₃|>1.

Proof. Let (^𝑎𝑏_𝑐𝑑) be the acceptable dilations on 𝕂, where 𝑎,𝑏,𝑐,𝑑∈ℝ. From the condition (1), we can obtain 𝐷𝑥𝑦=𝑎𝑥+𝑏𝑦𝑐𝑥+𝑑𝑦∈𝕂,∀(𝑥,𝑡)∈𝕂,(2.2) 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 𝛿_𝑟, where 𝑟∈ℕ and 𝑟>1. For simplicity we denote it by 𝛿_𝑟=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. Suppose𝑉_𝑗⊆𝑉_𝑗+1 (𝑗∈ℤ) where 𝑉_𝑗⊂𝐿²_𝑎(𝕂) and {𝑉_𝑗}_𝑗∈ℤ satisfies (2) and (4) of the definition of multiresolution analysis on the Laguerre hypergroup. The characteristic function 𝜒_𝑄 of the set 𝑄 is a scaling function of multiresolution analysis. Then ⋂_𝑗∈ℤ𝑉_𝑗={0}.

Proof. Let 𝛾₁,𝛾₂∈Γ and 𝛾₁≠𝛾₂. By using the property (4) of the Definition 1.1 we can obtain ⟨𝑇^𝑎_𝛾1𝜒_𝑄,𝑇^𝑎_𝛾2𝜒_𝑄⟩=0,(3.1) which implies that ∫_𝕂𝑇^𝑎_𝛾1𝜒_𝑄𝑇^𝑎_𝛾2𝜒_𝑄𝑑𝑚_𝑎=0. From (1.3), we know that 𝑇^𝑎_𝛾𝜒_𝑄≥0 and there exists a constant 𝐶>0 such that |𝑇^𝑎_𝛾𝜒_𝑄|≤𝐶 for all 𝑎≥0 and 𝛾∈𝕂. This yields 𝑇^𝑎_𝛾1𝜒_𝑄𝑇^𝑎_𝛾2𝜒_𝑄=0, which implies that 𝑇^𝑎_𝛾1𝜒_𝑄 and 𝑇^𝑎_𝛾2𝜒_𝑄 cannot be nonzero at the same time.
Let 𝑓∈⋂_𝑗∈ℤ𝑉_𝑗. Then 𝑓∈𝑉_−𝑗 for any 𝑗∈ℤ which implies that 𝛼^𝑗𝑓∈𝑉₀. Thus there exists a sequence {𝑏_𝑗(𝛾)}_𝛾∈Γ such that 𝛼^𝑗𝑓=∑_𝛾∈Γ𝑏_𝑗(𝛾)𝑇^𝑎_𝛾𝜒_𝑄. This yields |𝛼^𝑗𝑓|=|_𝛾∈Γ𝑏_𝑗(𝛾)𝑇^𝑎_𝛾𝜒_𝑄|≤sup_𝛾∈Γ|𝑏_𝑗(𝛾)|_𝑟∈Γ|𝑇^𝑎_𝛾𝜒_𝑄|≤‖{𝑏_𝑗(𝛾)}‖_{𝑙2𝛾∈Γ}|𝑇^𝑎_𝛾𝜒_𝑄|,(3.2) which implies that |𝛼^𝑗𝑓|≤𝐶‖{𝑏_𝑗(𝛾)}‖_𝑙2. Then we can see that |𝑓(𝑃)|=|𝛼^𝑗(𝛼^−𝑗𝑓(𝑃))|=|𝛼^𝑗𝑓(𝛼^−𝑗𝑃)|≤𝐶‖{𝑏_𝑗(𝛾)}‖_𝑙2(Γ)=𝐶‖𝛼^𝑗𝑓‖_𝐿2𝑎.(3.3) Notice that ‖𝛼^𝑗𝑓‖_𝐿2𝑎=(_𝐾|𝛼^𝑗𝑓(𝑃)|²𝑑𝑚_𝑎)^1/2=(_𝐾|𝑓(𝛼^𝑗𝑃)|²𝑑𝑚_𝑎)^1/2=(^+∞_−∞0^+∞|𝑓(𝑟^𝑗𝑥,𝑟^2𝑗𝑦)|²𝑥^2𝑎+1𝜋Γ(𝑎+1)𝑑𝑥𝑑𝑦)^1/2=𝑟^{−(𝑎+2𝑗)}‖𝑓‖_𝐿2𝑎.(3.4) If we let 𝑗 tend to infinity, then we can obtain 𝑓=0. This implies that ⋂_𝑗∈ℤ𝑉_𝑗={0}. The desired result is thus obtained.

Lemma 3.2. Suppose 𝑉_𝑗⊆𝑉_𝑗+1(𝑗∈ℤ), where 𝑉_𝑗⊂𝐿²_𝑎(𝕂) and {𝑉_𝑗}_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 𝐿¹_𝑎(𝕂) and ∫_𝕂𝜙𝑑𝑚_𝑎≠0, then ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²_𝑎(𝕂).

Proof. Let 𝑃=(𝑥,𝑡)∈𝕂 and 𝑎>0. Then we have 𝑇^𝑎_𝛼𝑃𝑓(𝑦,𝑠)=𝑇^𝑎_{(𝑟𝑥,𝑟2𝑡)}=𝑎𝑓(𝑦,𝑠)𝜋₀^2𝜋1₀𝑓((𝑟²𝑥²+𝑦²+2𝑟𝑥𝑦𝑟^′cos𝜃)^1/2,𝑠+𝑟²𝑡+𝑟𝑥𝑦𝑟^′sin𝜃)𝑟′(1−𝑟^′2)^𝑎−1𝑑𝑟^′=𝑎𝑑𝜃𝜋₀^2𝜋1₀𝑓(𝑟(𝑥²+𝑦𝑟²²𝑦+2𝑥𝑟𝑟^′cos𝜃)^1/2,𝑟²(𝑠𝑟²𝑦+𝑡+𝑥𝑟𝑟^′sin𝜃))𝑟^′(1−𝑟^′2)^𝑎−1𝑑𝑟^′𝑑𝜃=𝑇^𝑎_(𝑥,𝑡)𝛿_𝑟𝑦𝑓(𝑟,𝑠𝑟²)=𝛼⁻¹𝑇^𝑎_(𝑥,𝑡)𝛼𝑓,(3.5) which implies 𝑇^𝑎_𝛼𝑃=𝛼⁻¹𝑇^𝑎_𝑃𝛼. For 𝑎=0, we can get the same result. It is easy to see that 𝑇^𝑎_𝛼𝑙𝑃=𝛼^−𝑙𝑇^𝑎_𝑃𝛼^𝑙, for all 𝑙∈Γ and 𝑎≥0.
Let 𝜑∈⋃_𝑗∈ℤ𝑉_𝑗. Then there exists a 𝑗₀∈ℤ such that 𝜑∈𝑉_𝑗0. For any 𝑙∈ℤ, let 𝑗>−𝑙 and 𝑗≥𝑗₀. Using 𝑉_𝑗⊆𝑉_𝑗+1, we immediately obtain 𝜑∈𝑉_𝑗. Then there exists a sequence {𝑎_𝑗(𝛾)}_𝛾∈Γ∈𝑙²(Γ) such that 𝜑=∑_𝛾∈Γ𝑎_𝑗(𝛾)𝛼^𝑗𝑇^𝑎_𝛾𝜙, which implies 𝑇^𝑎_{𝛼𝑙(𝑃)}𝜑=_𝛾∈Γ𝑎_𝑗(𝛾)𝑇^𝑎_{𝛼𝑙(𝑃)}𝛼^𝑗𝑇^𝑎_𝛾𝜙=_𝛾∈Γ𝑎_𝑗(𝛾)𝛼^𝑗𝑇^𝑎_{𝛼𝑙+𝑗(𝑃)}𝑇^𝑎_𝛾𝜙.(3.6) Notice that 𝑃∈Γ, 𝑙+𝑗>0, and 𝑙+𝑗∈ℤ. Thus we can see that 𝛼^𝑙+𝑗(𝑃)∈Γ and 𝛼^𝑗𝑇^𝑎_{𝛼𝑙+𝑗(𝑃)}𝑇^𝑎_𝛾𝜙∈𝑉_𝑗, which implies 𝑇^𝑎_{𝛼𝑙(𝑃)}𝜑∈𝑉_𝑗⊆⋃_𝑗∈ℤ𝑉_𝑗, for all 𝑙∈ℤ and 𝑃∈Γ.
Let 𝜓∈⋃_𝑗∈ℤ𝑉_𝑗. Then for any 𝜀>0, there exists a 𝜑∈⋃_𝑗∈ℤ𝑉_𝑗 such that ‖𝜑−𝜓‖_𝐿2𝑎<𝜀. It follows from ‖𝑇^𝑎_{𝛼𝑙(𝑃)}𝜑−𝑇^𝑎_{𝛼𝑙(𝑃)}𝜓‖_𝐿2𝑎=‖𝑇^𝑎_{𝛼𝑙(𝑃)}(𝜑−𝜓)‖_𝐿2𝑎≤‖𝜑−𝜓‖_𝐿2𝑎<𝜀 and 𝑇^𝑎_{𝛼𝑙(𝑃)}𝜑∈⋃_𝑗∈ℤ𝑉_𝑗 that 𝑇^𝑎_{𝛼𝑙(𝑃)}𝜓∈⋃_𝑗∈ℤ𝑉_𝑗, for all 𝑙∈ℤ and 𝑃∈Γ.
For any 𝑔∈𝕂, there must exist an element 𝑃∈Γ and 𝑙∈ℤ such that |𝛼^𝑙(𝑃)−𝑔| is arbitrarily small, which implies that ‖𝑇^𝑎_{𝛼𝑙(𝑃)}𝜓−𝑇^𝑎_𝑔𝜓‖₂<𝜀 for any arbitrarily small 𝜀>0. This yields 𝑇^𝑎_𝑔𝜓∈⋃_𝑗∈ℤ𝑉_𝑗, for all 𝜓∈⋃_𝑗∈ℤ𝑉_𝑗 and 𝑔∈𝕂.
Note 𝜙(𝜆,𝑚)=∫_𝕂𝜑_−𝜆,𝑚(𝑥,𝑡)𝜙(𝑥,𝑡)𝑑𝑚_𝑎 and 𝜑_𝜆,𝑚(𝑥,𝑡)=𝑒^𝑖𝜆𝑡ℒ^𝑎_𝑚(|𝜆|𝑥²), 𝜙∈𝐿¹_𝑎(𝕂). This shows that 𝜙(𝜆,𝑚)→∫_𝕂𝜙𝑑𝑚_𝑎,𝑤ℎ𝑒𝑛𝜆→0. Since ∫_𝕂𝜙𝑑𝑚_𝑎≠0, there exists some 𝜀>0 such that 𝜙(𝜆,𝑚)≠0 for all |𝜆|<𝜀. Let 𝑊=⋃_𝑗∈ℤ𝑉_𝑗 and 𝜓∈𝑊^⟂. Then ⟨𝜑,𝜓⟩=0 for all 𝜑∈𝑊, which implies that for all 𝑔∈𝕂, 0=⟨𝑇^𝑎_𝑔𝜑,𝜓⟩=_𝐾𝑇^𝑎_𝑔𝜑(𝑥,𝑦)𝜓(𝑥,𝑦)𝑑𝑚_𝑎(𝑥,𝑦)=_𝐾𝑇^𝑎_𝑔𝜑(𝑥,𝑦)𝜓(𝑥,−𝑦)𝑑𝑚_𝑎(𝑥,𝑦)=𝜑∗𝜓(𝑔),(3.7) where 𝜓(𝑥,𝑦)=𝜓(𝑥,−𝑦). Then 𝜑∗𝜓(𝜆,𝑚)=𝜑(𝜆,𝑚)𝜓(𝜆,𝑚)=0. Notice that 𝛼𝑓(𝜆,𝑚)=_𝕂𝜑_−𝜆,𝑚(𝑥,𝑡)𝑓(𝑟𝑥,𝑟²𝑡)𝑑𝑚_𝑎=1𝑟^2𝑎+4_𝕂𝜑_−𝜆,𝑚(𝑥^′𝑟,𝑡^′𝑟²)𝑓(𝑥^′,𝑡^′)𝑑𝑚_𝑎,𝜑_𝜆,𝑚(𝑥𝑟,𝑡𝑟²)=𝑒^{𝑖𝜆(𝑡/𝑟2)}ℒ^𝑎_𝑚𝑥(|𝜆|(𝑟)²)=𝑒^{𝑖(𝜆/𝑟2)𝑡}ℒ^𝑎_𝑚1(|𝑟²𝜆|𝑥²)=𝜑_{𝜆/𝑟2,𝑚}(𝑥,𝑡).(3.8) Thus we can see that 𝛼𝑓(𝜆,𝑚)=(1/𝑟^2𝑎+4)𝑓(𝜆/𝑟²,𝑚), which implies 𝛼^𝑗𝑓(𝜆,𝑚)=(1/𝑟^{𝑗(2𝑎+4)})𝑓(𝜆/𝑟^2𝑗,𝑚). Let 𝜑=𝑟^{𝑗(2𝑎+4)}𝛼^𝑗𝜙. Then 𝜑∈𝑊 and 𝜑=𝜙(𝜆/𝑟^2𝑗,𝑚). This yields 𝜆𝜙(𝑟^2𝑗,𝑚)𝜓(𝜆,𝑚)=0.(3.9) Taking into account the fact that 𝜙(𝜆/𝑟^2𝑗,𝑚)≠0 when |𝜆|<𝑟^2𝑗𝜀, we see 𝜓(𝜆,𝑚)=0 when |𝜆|<𝑟^2𝑗𝜀. Let 𝑗 tend to infinity, then 𝜓=0 for all 𝜆∈ℝ which implies 𝜓=0. Then ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²_𝑎(𝕂). We complete the proof of this theorem.

Theorem 3.3. Suppose 𝜙=𝜒_𝑄 is a scaling function for a multiresolution analysis associated with (Γ,𝛼), where 𝜒_𝑄 is the characteristic function of a measurable set 𝑄. Then 𝑄 satisfies the following properties:(1)𝑇^𝑎_𝛾1𝜒_𝑄𝑇^𝑎_𝛾2𝜒_𝑄=0, for a.e. 𝑥∈𝕂, 𝛾₁≠𝛾₂ and 𝛾₁,𝛾₂∈Γ;(2)𝜒_𝑄=∑_𝛾∈Γ𝛽(𝛾)𝛼𝑇^𝑎_𝛾𝜒_𝑄;(3)|𝑄|=1;(4)𝑇^𝑎_𝛾1𝑇^𝑎_𝛾2𝜒_𝑄 can be represented by the sequence {𝑇^𝑎_𝛾𝜒_𝑄}_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=𝜒_𝑄 is a scaling function for a multiresolution analysis associated with (Γ,𝛼). Then ⟨𝑇^𝑎_𝛾1𝜒_𝑄,𝑇^𝑎_𝛾2𝜒_𝑄⟩=0 for all 𝛾₁≠𝛾₂ and 𝛾₁,𝛾₂∈Γ, which implies _𝕂𝑇^𝑎_𝛾1𝜒_𝑄𝑇^𝑎_𝛾2𝜒_𝑄𝑑𝑚_𝑎=0.(3.10) Notice that 𝑇^𝑎_𝛾1𝜒_𝑄≥0 and 𝑇^𝑎_𝛾2𝜒_𝑄≥0. Thus we can obtain that 𝑇^𝑎_𝛾1𝜒_𝑄𝑇^𝑎_𝛾2𝜒_𝑄=0, almost every 𝑥∈𝕂. By (1.7), we know that the second property is satisfied. Because of ‖𝜒_𝑄‖_𝐿2𝑎=1, we can see that |𝑄|=1. Let 𝑉₀∈(MRA(𝕂),Γ,𝛼). Then 𝑇^𝑎_𝛾2𝜒_𝑄∈𝑉₀. This implies 𝑇^𝑎_𝛾1𝑇^𝑎_𝛾2𝜒_𝑄∈𝑉₀. Therefore, 𝑇^𝑎_𝛾1𝑇^𝑎_𝛾2𝜒_𝑄 can be represented by {𝑇^𝑎_𝛾𝜒_𝑄}.
To see the converse, let 𝑉₀={𝑓∈𝐿²_𝑎(𝕂)∶𝑓=_𝛾∈Γ𝑐(𝛾)𝑇^𝑎_𝛾𝜒_𝑄},𝑉_𝑗=𝛼^𝑗𝑉₀.(3.11) Then {𝑉_𝑗}_𝑗∈ℤ is a family of closed subspace of 𝐿²_𝑎(𝕂). Let 𝑓∈𝑉₀. Then 𝑓=_𝛾∈Γ𝑐(𝛾)𝑇^𝑎_𝛾𝜒_𝑄=_𝛾∈Γ𝑐(𝛾)𝑇^𝑎_{𝛾𝛾1∈Γ}𝛽(𝛾₁)𝛼𝑇^𝑎_𝛾1𝜒_𝑄=_{𝛾,𝛾1∈Γ}𝑐(𝛾)𝛽(𝛾₁)𝑇^𝑎_𝛾𝛼𝑇^𝑎_𝛾1𝜒_𝑄=_{𝛾,𝛾1∈Γ}𝑐(𝛾)𝛽(𝛾₁)𝛼𝑇^𝑎_𝛼(𝛾)𝑇^𝑎_𝛾1𝜒_𝑄.(3.12) Since 𝛼(𝛾)∈Γ, we can see that 𝑇^𝑎_𝛼(𝛾)𝑇^𝑎_𝛾1𝜒_𝑄∈𝑉₀, which implies 𝑓∈𝑉₁. This yields 𝑉₀⊆𝑉₁. Then we can also get 𝑉_𝑗⊆𝑉_𝑗+1. Notice that 𝑓=∑_𝛾∈Γ𝑐(𝛾)𝑇^𝑎_𝛾𝜒_𝑄. Thus we can see that 𝑇^𝑎_𝛾1𝑓=∑_𝛾∈Γ𝑐(𝛾)𝑇^𝑎_𝛾1𝑇^𝑎_𝛾𝜒_𝑄, for all 𝛾₁∈Γ. Because 𝑇^𝑎_𝛾1𝑇^𝑎_𝛾2𝜒_𝑄 can be represented by the sequence {𝑇^𝑎_𝛾𝜒_𝑄}, thus 𝑇^𝑎_𝛾1𝑓∈𝑉₀.
In order to show that {𝑉_𝑗}_𝑗∈ℤ is a multiresolution analysis associated with (Γ,𝛼), it suffices to show that ⋂_𝑗∈ℤ𝑉_𝑗=0 and ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²_𝑎(𝕂). Further, it follows easily from Lemmas 3.1 and 3.2 that _𝑗∈ℤ𝑉_𝑗=0,_𝑗∈ℤ𝑉_𝑗=𝐿²_𝑎(𝕂).(3.13) Our result is proved.

In this paper orthonormal Haar wavelet bases for 𝐿²_𝑎(𝕂) are not constructed. But we believe that orthonormal Haar wavelet bases for 𝐿²_𝑎(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.