Characterization of Generators for Multiresolution Analyses with Composite Dilations

Abstract

This paper introduces multiresolution analyses with composite dilations (AB-MRAs) and addresses frame multiresolution analyses with composite dilations in the setting of reducing subspaces of 𝐿²(ℝ^𝑛) (AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspace 𝐿²(𝑆)^∨. For a general expansive matrix, we obtain the characterizations for a scaling function to generate an AB-RMRA, and the main theorems generalize the classical results. Finally, some examples are provided to illustrate the general theory.

1. Introduction

As well known, multiresolution analyses (MRAs) play a significant role in the construction of wavelets for 𝐿²(ℝ) [1, 2]. Up to now, different characterizations of the scaling function for an MRA have been presented. It is shown in [1] that a function 𝜑∈𝐿²(ℝ) is a generator for an MRA if and only if (1)∑_𝑘∈ℤ|𝜑(𝜉+𝑘)|²=1, a.e. 𝜉∈[−1/2,1/2]; (2)lim_𝑗→+∞|𝜑(2^−𝑗𝜉)|²=1, a.e. 𝜉∈ℝ;(3)there exists 𝑚₀∈𝐿²([0,1]) such that 𝜑(2𝜉)=𝑚₀(𝜉)𝜑(𝜉), a.e. 𝜉∈ℝ.

If condition (2) is replaced by ℝ=⋃_𝑗∈ℤ2^𝑗supp(𝜑) or another condition that the function 𝐹(𝑥,𝑦)=(1/(𝑦−𝑥))∫^𝑦𝑥|𝜑(𝜔)|²d𝜔 is dyadicaly away from zero at the origin, then the two different characterizations of the scaling functions for MRAs are obtained in [3, 4], respectively.

Similarly, under certain conditions, wavelet with composite dilations can be constructed by AB-MRAs which is the generalized definition of MRAs and permits the existence of fast implementation algorithm [5]. Given an 𝑛×𝑛 invertible matrix 𝑎, 𝑓∈𝐿²(ℝ^𝑛), and 𝑘∈ℤ^𝑛, we define the dilation operator 𝐷 and the shift operator 𝑇_𝑘 on 𝐿²(ℝ^𝑛) by 𝐷_𝑎𝑓(⋅)∶=||det𝑎||^1/2𝑓(𝑎⋅),𝑇_𝑘𝑓(⋅)∶=𝑓(⋅−𝑘).(1.1) The affine system with composite dilations is defined by 𝒜_𝐴𝐵(Ψ)={𝐷_𝑎𝐷_𝑏𝑇_𝑘Ψ∶𝑘∈ℤ^𝑛,𝑏∈𝐵,𝑎∈𝐴} where Ψ={𝜓¹,𝜓²,…,𝜓^𝐿}⊂𝐿²(ℝ^𝑛). By choosing Ψ, 𝐴, and 𝐵 appropriately, we can make 𝒜_𝐴𝐵(Ψ) an orthonormal basis or, more generally, a Parseval frame (PF) for 𝐿²(ℝ^𝑛) [5–7]. In this case, Ψ is called an AB-multiwavelet or a PF  AB-multiwavelet, respectively. Since not all of the AB-multiwavelet come from AB-MRAs, we only focus on the AB-multiwavelet which come from AB-MRAs. For convenience, we denote the operator 𝐷_𝑏𝑇_𝑘 by ℬ.

Before proceeding, we need some conventions. We denote by 𝑇^𝑛=[−1/2,1/2]^𝑛 the 𝑛-dimensional torus. For a Lebesgue measurable set 𝐸 in ℝ^𝑛, we denote by |𝐸| its measure, denote by 𝜒_𝐸 the characteristic function of 𝐸, and define 𝐸^∼∶=𝐸+ℤ^𝑛. An 𝑛×𝑛 matrix 𝐴 is called an expansive matrix if it is an integer matrix with all its eigenvalues greater than 1 in the module. 𝐺 denotes the set of all expansive matrices. We denote by 𝐺𝐿_𝑛(ℤ) the set {𝑎∶𝑎 is an 𝑛×𝑛 integral matrix and |det𝑎|≠0}, by 𝑆𝐿_𝑛(ℤ) the set {𝑎∶𝑎 is an 𝑛×𝑛 integral matrix and |det𝑎|=1}, and by 𝐵 the set of the subgroups of 𝑆𝐿_𝑛(ℤ), respectively. For a Lebesgue measurable function 𝑓, we define its support by supp(𝑓)∶={𝑥∈ℝ^𝑛∶𝑓(𝑥)≠0}.(1.2) The Fourier transform of 𝑓∈𝐿¹(ℝ^𝑛)∩𝐿²(ℝ^𝑛) is defined by 𝑓(𝜉)∶=_ℝ𝑛𝑓(𝑥)𝑒^{−𝑖2𝜋⟨𝜉,𝑥⟩}d𝑥(1.3) on ℝ^𝑛, where ⟨𝜉,𝑥⟩ denotes the inner product in ℝ^𝑛. Let 𝑆 be a Lebesgue nonzero measurable set in ℝ^𝑛. We denote by 𝐿²(𝑆)^∨ the closed subspace of 𝐿²(ℝ^𝑛) of the form 𝐿²(𝑆)^∨∶=𝑓∈𝐿²(ℝ^𝑛)∶supp𝑓⊆𝑆.(1.4)

Definition 1.1 (see [8, 9]). The sequence {𝑥_𝑘,𝑙}_𝑘,𝑙 in a separable Hilbert space 𝐻 is called a semiorthogonal PF for 𝐻 if {𝑥_𝑘,𝑙}_𝑘,𝑙 is a PF for 𝐻 and satisfies ⟨𝑥_𝑘1,𝑙1,𝑥_𝑘2,𝑙2⟩=0 for any 𝑘₁,𝑘₂∈Λ₁, 𝑙₁,𝑙₂∈Λ₂, and 𝑘₁≠𝑘₂, where Λ₁, Λ₂ are two countable index sets. In particular, if {𝑥_𝑘,𝑙}_𝑘,𝑙 is a semiorthogonal PF for span{𝑥_𝑘,𝑙}_𝑘,𝑙, it is called a semiorthogonal sequence.

Definition 1.2 (see [4, 10]). A closed subspace 𝑋 of 𝐿²(ℝ^𝑛) is called a reducing subspace if 𝐷_𝑎𝑋=𝑋 and 𝑇_𝑘𝑋=𝑋 for any 𝑘∈ℤ^𝑛, 𝑎∈𝐺.

The following proposition provides a characterization of reducing subspace.

Proposition 1.3 (see [4, 10]). A closed subspace 𝑋 of 𝐿²(ℝ^𝑛) is a reducing subspace if and only if 𝑋=𝑓∈𝐿²(ℝ^𝑛)∶supp𝑓⊆𝑆(1.5) for some measurable set 𝑆⊆ℝ^𝑛 with 𝑎𝑆̃=𝑆. So, to be specific, one denotes a reducing subspace by 𝐿²(𝑆)^∨ instead of 𝑋. In particular, 𝐿²(ℝ^𝑛) is a reducing subspace of 𝐿²(ℝ^𝑛).

Definition 1.4 (see [5–7]). Let 𝐵⋉ℤ^𝑛 be a subgroup of the integral affine group 𝑆𝐿_𝑛(ℤ)⋉ℤ^𝑛 (the semidirect product of 𝑆𝐿_𝑛 and ℤ^𝑛). The closed subspace 𝑉 of 𝐿²(ℝ^𝑛) is called a 𝐵⋉ℤ^𝑛 invariant subspace if ℬ𝑉=𝑉 for any (𝑏,𝑘)∈𝐵⋉ℤ^𝑛.

Definition 1.5 (see [2–4]). Let 𝐵 be a countable subset of 𝑆𝐿_𝑛(𝑍) and 𝐴={𝑎^𝑖∶𝑖∈ℤ} where 𝑎∈𝐺𝐿_𝑛(ℤ). We say that a sequence {𝑉_𝑗}_𝑗∈ℤ of closed subspaces of 𝐿²(ℝ^𝑛) is an AB-MRA if the following holds: (1)𝑉₀ is a 𝐵⋉ℤ^𝑛 invariant space; (2)for each 𝑗∈ℤ, 𝑉_𝑗⊂𝑉_𝑗+1, and 𝑉_𝑗=𝐷^𝑗𝑎𝑉₀; (3)⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(ℝ^𝑛); (4)⋂_𝑗∈ℤ𝑉_𝑗={0};(5)there exists 𝜑∈𝑉₀ such that Φ_𝐵={𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛} is a semiorthogonal PF for 𝑉₀.

The space 𝑉₀ is called an AB scaling space, and the function 𝜑 is an AB scaling function for 𝑉₀ or a generator of AB-MRA.

Similarly, we say that a sequence {𝑉_𝑗}_𝑗∈ℤ is an AB-RMRA if it is an AB-MRA on 𝐿²(𝑆)^∨, that is, conditions (1), (2), (4), (5), and (3)′   ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(𝑆)^∨ are satisfied.

The fact that an AB-MRA can induce an AB-RMRA will be demonstrated by the obvious following results.

Proposition 1.6. Let 𝐼 be a countable index set and 𝑃 the orthogonal projection operator from a Hilbert space 𝐻 to its proper subspace 𝐾. If Ψ={𝜓_𝑖∶𝑖∈𝐼} is a Parseval frame on 𝐻, then 𝑃(Ψ)={𝑃(𝜓_𝑖)∶𝑖∈𝐼} is a Parseval frame on 𝐾.

Proposition 1.7. Let 𝑃 be the orthogonal projection operator from a Hilbert space 𝐻 to its reducing subspace 𝐾. Then 𝑃 can commutate with the shift and dilation operators 𝑇_𝑘 and 𝐷_𝑎, respectively.

Theorem 1.8. Suppose that {𝜑;𝑉_𝑗} is an AB-MRA, then {𝜑;𝑉_𝑗} is an AB-RMRA for 𝐿²(𝑆)^∨, where 𝜑∶=𝑃𝜑, 𝑉₀∶=span{𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛}, 𝑉_𝑗∶=span{𝐷_𝑎𝑗𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛}, and 𝑃 is the orthogonal projection operator from 𝐿²(ℝ^𝑛) to 𝐿²(𝑆)^∨.

The rest of this paper is organized as follows. Theorem 1.8 and some properties of an AB-RMRA will be proved in Section 2. In Section 3, the characterization of the generator for an AB-RMRA will be established, which is the main purpose of this paper. Finally, some examples are provided to illustrate the general theory.

2. Preliminaries

In this section, we will firstly prove Theorem 1.8 as follows.

We can easily prove that {𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛} is a Parseval frame sequence by Propositions 1.6 and 1.7. Naturally, {𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛} is a semi-Parseval frame for 𝑉₀. Let 𝜑∈𝑉₀⊂𝐿²(ℝ^𝑛). Thus 𝑃𝜑=𝜑₁. For any 𝑓∈𝑉₀, we have 𝑓=_𝑏_𝑘⟨𝑓,𝐷_𝑏𝑇_𝑘𝜑⟩𝐷_𝑏𝑇_𝑘𝜑=_𝑏_𝑘𝐷_𝑏′⟨𝑓,𝐷_𝑏𝑇_𝑘𝜑₁⟩𝐷_𝑏𝑇_𝑘𝜑₁,𝑇_𝑘^′𝑓=_𝑏_𝑘⟨𝑓,𝐷_𝑏𝑇_𝑘𝜑₁⟩𝜑₁𝑏^𝑏𝑥−𝑏𝑘^−𝑘=_𝑏_𝑘⟨𝑓,𝐷_𝑏𝑇_𝑘𝜑₁⟩𝜑₁(𝑏𝑥−𝑘),(2.1) namely, 𝐷_𝑏^′𝑇_𝑘^′𝑓∈𝑉₀. So 𝑉₀ is a 𝐵⋉ℤ^𝑛 invariant space. On the other hand, 𝑓=_𝑏_𝑘⟨𝑓,𝐷_𝑏𝑇_𝑘𝑃𝜑⟩𝐷_𝑏𝑇_𝑘𝑃𝜑=_𝑏_𝑘=_𝑏⟨𝑃𝑓,𝐷_𝑏𝑇_𝑘𝜑⟩𝑃𝐷_𝑏𝑇_𝑘𝜑_𝑘⟨𝑓,𝐷_𝑏𝑇_𝑘𝜑⟩𝑃_𝑏′_𝑘′=_𝑏⟨𝐷_𝑏𝑇_𝑘𝜑,𝐷_𝑎𝐷_𝑏𝑇_𝑘𝜑⟩𝐷_𝑎𝐷_𝑏𝑇_𝑘𝜑_𝑘𝑐_𝑏,𝑘𝐷_𝑎𝐷_𝑏𝑇_𝑘𝜑∈𝑉₁.(2.2) So 𝑉₀⊂𝑉₁. Notice that 𝑉_𝑖=𝐷_𝑎𝑗𝑉₀. Then 𝑉_𝑖⊂𝑉_𝑖+1. Thus, conditions (1), (2), and (5) in Definition 1.5 have been proved. However, condition (3)′ is the natural consequence of the later Lemma 3.1 in Section 3. Therefore, we complete the proof of Theorem 1.8.

Some properties of AB-RMRA, which were not discussed in [5–7], will be presented. The first one can be obtained obviously by the definition of AB-RMRA as follows.

Proposition 2.1. Suppose that {𝑉_𝑗}_𝑗∈ℤ is an AB-RMRA. Then (1)for each 𝑗∈ℤ, {𝐷^𝑗𝑎𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛} is a semiorthogonal PF on 𝑉_𝑗;(2)𝑉₀ is a 𝐵⋉ℤ^𝑛 invariant subspace, while 𝑉_𝑗 is a 𝐵⋉𝑎^−𝑗ℤ^𝑛 invariant subspace.

Condition (5) of AB-RMRA can be characterized by the following proposition.

Proposition 2.2. Let 𝜑∈𝐿²(𝑆)^∨. Then Φ_𝐵={𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛} is a semiorthogonal PF sequence if and only if (1)∑_{𝑘∈ℤ𝑛}|𝜑(𝜉+𝑘)|²=𝜒_𝐹(𝜉), a.e., where 𝐹={𝜉∈𝑇^𝑛∩Ω∶𝜑(𝜉+𝑘)≠0,𝑘∈ℤ^𝑛};(2)∑_{𝑘∈ℤ𝑛}𝜑(𝜉+𝑘)𝜑(𝑏̃⁻¹(𝜉+𝑘))=0, a.e. 𝜉∈Ω, for each 𝑏∈𝐵 and 𝑏≠𝐼_𝑛.

Proof. Necessity. For any 𝑓(𝑥)∈span{𝑇_𝑘𝜑∶𝑘∈ℤ^𝑛}, we have 𝑓(𝑥)=_𝑏,𝑘⟨𝑓,ℬ𝜑⟩ℬ𝜑(𝑥)=_𝑘⟨𝑓,𝑇_𝑘𝜑⟩𝑇_𝑘𝜑(𝑥)+_{𝑏≠𝐼𝑛}_𝑘⟨𝑓,ℬ𝜑⟩ℬ𝜑(𝑥)=_𝑘⟨𝑓,𝑇_𝑘𝜑⟩𝑇_𝑘𝜑(𝑥).(2.3) By Theorem  1.6 in [1] and Theorem  7.2.3 in [8], conclusion (1) holds clearly. Using Parseval theorem, we can deduce ⟨𝑇_𝑘𝜑,𝐷_𝑏𝑇_𝑘𝜑⟩==_Ω𝜑(𝑥−𝑘)𝜑(𝑏𝑥−𝑘^)d𝑥_Ω𝜑(𝜉)𝜑(𝑏̃⁻¹𝜉)𝑒^{−2𝜋𝑖(𝑘−}𝑏̃⁻¹_𝑘^′_)⋅𝜉=d𝜉_Ω_𝑙𝜑(𝜉+𝑙)𝜑(𝑏̃⁻¹(𝜉+𝑙))𝑒^{−2𝜋𝑖(𝑘−𝑘1)⋅𝜉}d𝜉,(2.4) where 𝑘₁=𝑏̃⁻¹𝑘^. Note that for any 𝑘,𝑘^∈ℤ^𝑛, 𝑏≠𝑏^∈𝐵, ⟨𝐷_𝑏𝑇_𝑘𝜑,𝐷_𝑏𝑇_𝑘𝜑⟩=0 if and only if for any 𝑏∈𝐵 and 𝑏≠𝐼_𝑛, ⟨𝑇_𝑘𝜑,𝐷_𝑏𝑇_𝑘𝜑⟩=0.(2.5)
Then, we have _𝑙𝜑(𝜉+𝑙)𝜑(𝑏̃⁻¹(𝜉+𝑙))=0,a.e.𝜉∈Ω.(2.6)
Sufficiency. By Theorem  7.2.3 in [8] and conclusion (1), {𝑇_𝑘𝜑∶𝑘∈ℤ^𝑛} is a PF sequence. So is {𝐷_𝑏𝑇_𝑘𝜑∶𝑘∈ℤ^𝑛} for any 𝑏∈𝐵. It follows from (2.4), (2.5), and conclusion (2) that for any 𝑘,𝑘^∈ℤ^𝑛, 𝑏,𝑏^∈𝐵, and 𝑏≠𝑏^, we get ⟨𝐷_𝑏𝑇_𝑘𝜑,𝐷_𝑏^′𝑇_𝑘^′𝜑⟩=0. Thus, for any 𝑓(𝑥)∈span{𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛}, there exists 𝑏∈𝐵 and 𝑓_𝑏(𝑥)∈span{𝐷_𝑏𝑇_𝑘𝜑∶𝑘∈ℤ^𝑛} such that ‖𝑓‖²=_𝑏∈𝐵‖‖𝑓_𝑏‖‖²=_{𝑏∈𝐵𝑘}_∈ℤ𝑛||⟨𝑓_𝑏,𝐷_𝑏𝑇_𝑘𝜑⟩||²=_{𝑏∈𝐵𝑘}_∈ℤ𝑛||⟨𝑓,𝐷_𝑏𝑇_𝑘𝜑⟩||².(2.7) By Theorem  1.6 in [1], the proof of Proposition 2.2 is completed.

Proposition 2.3. Let {𝑉_𝑗}_𝑗∈ℤ be a sequence of closed subspace of 𝐿²(𝑆)^∨, where 𝑉_𝑗∶=span𝐷^𝑗𝑎𝐷_𝑏𝑇_𝑘𝜑∶𝑏∈𝐵,𝑘∈ℤ^𝑛.(2.8) If conditions (1), (2), and (5) of AB-RMRA are satisfied, then one has the following. (1)There exists {𝑐_𝑏,𝑘}∈𝑙²(𝐵×ℤ^𝑛) such that 𝜑(𝑥)=_𝑏_𝑘𝑐_𝑏,𝑘||det𝑎||^1/2𝜑(𝑏𝑎𝑥−𝑘).(2.9)(2)There exists {ℎ_𝑏(𝜉)}_𝑏∈𝐵⊆𝐿^∞(𝑇^𝑛) such that 𝜑(𝜉)=_𝑏ℎ_𝑏(𝑏𝑎)⁻¹𝜉𝜑(𝑏𝑎)⁻¹𝜉,(2.10) for any 𝑏∈𝐵, where ℎ_𝑏(𝜉)=|det𝑎|^−1/2∑_𝑘𝑐_𝑏,𝑘𝑒^{−2𝜋𝑖𝑘⋅𝜉}.

Proof. By conditions (2) and (5) of AB-RMRA and the fact that 𝜑∈𝑉₀⊂𝑉₁, we obtain 𝜑(𝑥)=_𝑏,𝑘⟨𝜑,𝐷_𝑎ℬ𝜑⟩𝐷_𝑎ℬ𝜑(𝑥)=_𝑏𝑘_∈ℤ𝑛𝑐_𝑏,𝑘||det𝑎||^1/2𝜑(𝑏𝑎𝑥−𝑘),(2.11) where 𝑐_𝑏,𝑘=⟨𝜑,𝐷_𝑎ℬ𝜑⟩ and {𝑐_𝑏,𝑘}∈𝑙²(𝐵×ℤ^𝑛). Therefore (2.9) holds. Taking Fourier transform on both sides of (2.9), we obtain (2.10), where for any 𝑏∈𝐵, ℎ_𝑏(𝜉)=|det𝑎|^−1/2∑_𝑘𝑐_𝑏,𝑘𝑒^{−2𝜋𝑖𝑘⋅𝜉}. In what follows, we will only prove {ℎ_𝑏(𝜉)}_𝑏∈𝐵⊆𝐿^∞(𝑇^𝑛). Indeed, for 𝑎∈𝐺𝐿_𝑛(ℤ),𝑎ℤ̃^𝑛 is a subgroup of ℤ^𝑛 and the quotient group ℤ^𝑛/𝑎ℤ̃^𝑛 has order 𝑀=|det𝑎|. Thus, we can choose a complete set of representatives of ℤ^𝑛/𝑎ℤ̃^𝑛, that is, the set {𝛼₀,𝛼₁,…,𝛼_𝑀−1} so that each 𝑘∈ℤ^𝑛 can be uniquely expressed in the form 𝑘=𝑎𝑘̃^+𝛼_𝑖 with 𝑘^∈ℤ^𝑛, 0≤𝑖≤𝑀−1. For simplicity, we denote (𝑏𝑎)⁻¹ and (𝑏^𝑎)⁻¹ by 𝑏^∗ and 𝑏^∗1, respectively. Then we have _𝑘||𝜑(𝜉+𝑘)||²=_{𝑘∈ℤ𝑛}|||||_𝑏ℎ_𝑏𝑏^∗(𝜉+𝑘)𝜑𝑏^∗(𝜉+𝑘)|||||²=_{𝑘∈ℤ𝑛}_𝑏,𝑏1=ℎ_𝑏𝑏^∗(𝜉+𝑘)ℎ_𝑏1[𝑏^∗1(𝜉+𝑘)]𝜑𝑏^∗(𝜉+𝑘)𝜑[𝑏^∗1(𝜉+𝑘)]_𝑀−1_𝑖=0_{𝑘′,𝑏,𝑏1}×ℎ_𝑏𝑏^∗𝜉+𝛼_𝑖ℎ_𝑏1[𝑏^∗1(𝜉+𝛼_𝑖)]𝜑𝑏^∗𝜉+𝛼_𝑖+𝑏̃⁻¹𝑘^𝜑[𝑏^∗1(𝜉+𝛼_𝑖)+𝑏₁₋₁𝑘^](2.12)=^𝑀−1_𝑖=0_𝑏,𝑏1×ℎ_𝑏𝑏^∗𝜉+𝛼_𝑖ℎ_𝑏1[𝑏^∗1(𝜉+𝛼_𝑖)]_{𝑘′∈ℤ𝑛}=_𝑀−1𝜑𝑏^∗𝜉+𝛼_𝑖+𝑏̃⁻¹𝑘^𝜑[𝑏^∗1(𝜉+𝛼_𝑖)+𝑏₁₋₁𝑘^]_𝑖=0_𝑏||ℎ_𝑏𝑏^∗𝜉+𝛼_𝑖||²_𝑘_{′∈ℤ𝑛}|||𝜑𝑏_∗𝜉+𝛼_𝑖+𝑏̃⁻¹𝑘^|||²,(2.13) where (2.12) is obtained by the periodicity of function sequence {ℎ_𝑏(𝜉)}_𝑏 and (2.13) is proved by conclusion (2) in Proposition 2.2. In addition, using Proposition 2.2 again and (2.13) above, for any 𝜉∈𝐹, we get ∑_{𝑘∈ℤ𝑛}|𝜑(𝜉+𝑘)|²=∑^{𝑀−1𝑖=0}∑_𝑏|ℎ_𝑏[𝑏^∗𝜉+𝑏^∗𝛼_𝑖)]|²=1. Then, for 𝜉∈ℝ^𝑛, |ℎ_𝑏(𝜉)|²≤1, so ℎ_𝑏(𝜉)∈𝐿^∞(𝑇^𝑛). Therefore, the proof of Proposition 2.3 is completed.

3. Characterization of the Generator for an AB-RMRA

In this section, we will characterize the scaling function of AB-RMRA which will determine a multiresolution structure and AB-wavelets and the obtained results can be easily extended to the whole space 𝐿²(ℝ^𝑛).

Lemma 3.1. Let {𝑉_𝑗}_𝑗∈ℤ be a sequence of closed subspaces of 𝐿²(𝑆)^∨ and defined by (2.8). Assume that conditions (1), (2), and (5) in an AB-RMRA are satisfied. Then the following results are equivalent: (1)⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(𝑆)^∨; (2)lim_𝑗→+∞∑_𝑏|𝜑[(𝑏𝑎_𝑗)⁻¹𝜉]|²=1, a.e. 𝜉∈ℝ^𝑛.

Proof. Theorems  1.7 and  5.2 in [1] imply that for any 𝑓∈𝐿²(ℝ^𝑛), lim_𝑗→+∞‖𝑃_𝑗𝑓‖²=‖𝑓‖² is equivalent to ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(ℝ^𝑛). Thus, for any 𝑓∈𝐿²(𝑆)^∨, lim_𝑗→+∞‖𝑃_𝑗𝑓‖²=‖𝑓‖² is equivalent to ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(𝑆)^∨. Hence, we have to prove that lim_𝑗→+∞‖𝑃_𝑗𝑓‖²=‖𝑓‖² is equivalent to lim_𝑗→+∞∑_𝑏|𝜑[(𝑏𝑎_𝑗)⁻¹𝜉]|²=1, a.e. 𝜉∈Ω. First, we prove (1)⇒(2) For any 𝑓∈𝐿²(𝑆)^∨, 𝑓=𝑃_𝑗𝑓+𝑄_𝑗𝑓, where 𝑄_𝑗∶𝐿²(ℝ)→(𝑉_𝑗)^⟂ is the orthogonal projection operator. Set 𝑓(𝜉)=𝜒_{𝑇𝑛∩𝑆}(𝜉). Then, when 𝑗 is large enough, we have ‖‖𝑃_𝑗𝑓‖‖²=_𝑏_𝑘||𝑓,𝐷^𝑗𝑎𝐷_𝑏𝑇_𝑘𝜑||²=_𝑏_𝑘|||||_ℝ𝑛||det𝑎_𝑗||^−1/2𝑓(𝜉)𝜑𝑏𝑎_𝑗⁻¹𝜉𝑒^{2𝜋𝑖𝑘⋅}(𝑏𝑎_𝑗)⁻¹_𝜉d𝜉|||||²=||det𝑎_𝑗||_𝑏_𝑘|||||(𝑏𝑎_𝑗)⁻¹_𝑇^𝑛𝜑(𝜉)𝑒^{2𝜋𝑖𝑘⋅𝜉}|||||²=||det𝑎_𝑗d𝜉||_𝑏_𝑇𝑛||||𝜒(𝑏𝑎_𝑗)⁻¹_𝑇^𝑛=(𝜉)𝜑(𝜉)||||²d𝜉_𝑏_𝑇𝑛||||𝜑𝑏𝑎_𝑗=⁻¹(𝜉)||||²d𝜉_𝑇𝑛_𝑏||||𝜑𝑏𝑎_𝑗⁻¹(𝜉)||||²d𝜉.(3.1) Before proving the equivalence, we need to prove two assertions as follows:(i)∑_𝑏|𝜑[(𝑏𝑎_𝑗)⁻¹𝜉]|²∈𝐿¹(𝑇^𝑛); (ii)lim_𝑗→+∞∑_𝑏|𝜑[(𝑏𝑎_𝑗)⁻¹𝜉]|² makes sense.Since _𝑇𝑛_𝑏||||𝜑𝑏𝑎_𝑗⁻¹𝜉||||²d𝜉=_𝑏_𝑇𝑛||||𝜑𝑏𝑎_𝑗=||det𝑎_𝑗⁻¹𝜉||||²d𝜉||_𝑏(𝑏𝑎_𝑗)⁻¹_𝑇^𝑛=||det𝑎_𝑗||𝜑(𝜉)||²d𝜉||_⋃𝑏(𝑏𝑎_𝑗)⁻¹_𝑇^𝑛≤||det𝑎_𝑗||𝜑(𝜉)||²d𝜉||_ℝ𝑛||𝜑(𝜉)||²d𝜉<∞,(3.2) it follows that (i) holds.
For (ii), we will only prove that {∑_𝑏|𝜑[(𝑏𝑎_𝑗)⁻¹𝜉]|²}_𝑗∈ℤ is a monotonic bounded sequence when 𝜉(∈𝑆) is fixed. Indeed, by the orthogonality, for each 𝑏≠𝑏^∈𝐵, we have supp(𝜑(𝑏̃⁻¹𝜉))∩supp(𝜑(𝑏^−1𝜉))=∅. In addition, we deduce from (2.10) that, for any 𝑏∈𝐵, 𝜑(𝑏𝑎_𝑗)⁻¹𝜉=_{𝑏′∈𝐵}ℎ_𝑏^′(𝑏^=_{𝑏′∈𝐵}𝑎)⁻¹(𝑏𝑎_𝑗)⁻¹𝜉𝜑(𝑏^𝑎)⁻¹(𝑏𝑎_𝑗)⁻¹𝜉ℎ_𝑏^′𝑏^−1𝑏̃⁻¹𝑎_𝑗+1−1𝜉𝜑𝑏^−1𝑏̃⁻¹𝑎_𝑗+1−1𝜉.(3.3)
Set 𝑏^∗=𝑏̃⁻¹𝑎_𝑗+1−1. Then by the orthogonality and Proposition 2.3, we obtain _𝑏_∈𝐵||||𝜑𝑏𝑎_𝑗⁻¹𝜉||||²=_𝑏∈𝐵|||||_{𝑏′∈𝐵}ℎ_𝑏^′𝑏^−1𝑏^∗𝜉𝜑𝑏^−1𝑏^∗𝜉|||||²=_{𝑏∈𝐵𝑏}_1∈𝐵=_𝑏∈𝐵ℎ_𝑏1𝑏¹⁻¹𝑏^∗𝜉𝜑𝑏₁₋₁𝑏^∗𝜉_𝑏2∈𝐵ℎ_𝑏2𝑏²⁻¹𝑏^∗𝜉𝜑𝑏₂₋₁𝑏^∗𝜉||||ℎ_𝑏𝑏𝑎^𝑗+1⁻¹𝜉𝜑𝑏𝑎_𝑗+1⁻¹𝜉||||²≤_𝑏∈𝐵||||𝜑𝑏𝑎_𝑗+1⁻¹𝜉||||².(3.4) Hence, {∑_𝑏∈𝐵|𝜑((𝑏𝑎_𝑗)⁻¹𝜉)|²}_𝑗∈ℤ is a monotonic sequence when 𝜉 is fixed. On the other hand, by the property of 𝐵, we deduce that 𝐷_𝑏𝑇_𝑘𝜑(𝑥)=𝜑(𝑏𝑥−𝑘)=𝜑(𝑏(𝑥−𝑏⁻¹𝑘))=𝜑(𝑏(𝑥−𝑘^))=𝑇_𝑘𝐷_𝑏𝜑(𝑥). Note that {𝐷_𝑏𝑇_𝑘𝜑∶𝑘∈ℤ^𝑛} is a PF for span{𝐷_𝑏𝑇_𝑘𝜑∶𝑘∈ℤ^𝑛} for any 𝑏∈𝐵. Then {𝑇_𝑘^′𝐷_𝑏𝜑∶𝑘^∈ℤ^𝑛} is a PF on span{𝐷_𝑏𝑇_𝑘𝜑∶𝑘∈ℤ^𝑛}, which implies that there exists a set 𝑀_𝑏 such that ∑_𝑘|𝜑(𝑏̃⁻¹𝜉+𝑘)|²=𝜒_𝑀𝑏(𝜉). By the orthogonality, ∑_𝑏∑_𝑘|𝜑(𝑏̃⁻¹𝜉+𝑘)|²=𝜒_{⋃𝑏𝑀𝑏}(𝜉), consequently ⋃_𝑏𝑀_𝑏=𝐹. Hence, ∑_𝑏|𝜑(𝑏̃⁻¹𝜉)|²≤1 holds for 𝜉∈𝑆. So lim_𝑗→+∞∑_𝑏∈𝐵|𝜑((𝑏𝑎_𝑗)⁻¹𝜉)|² exists. Now we have proved the two assertions. By the Lebesgue dominant convergence theorem, we get lim_𝑗→+∞‖𝑃_𝑗𝑓‖²=∫_𝑇𝑛lim_𝑗→+∞∑_𝑏|𝜑[(𝑏𝑎_𝑗)⁻¹𝜉]|²d𝜉=‖𝑓‖²=|𝑇^𝑛|=1. Thus, lim_𝑗→+∞∑_𝑏∈𝐵|𝜑[(𝑏𝑎_𝑗)⁻¹𝜉]|²=1, a.e. 𝜉∈ℝ^𝑛.
Next, we prove (2)⇒(1). Let 𝐷 be the class of all functions 𝑓∈𝐿²(ℝ^𝑛) such that 𝑓∈𝐿^∞(ℝ^𝑛) and 𝑓 is compactly supported in ℝ^𝑛⧵{0}. If we can show that lim_𝑗→+∞‖𝑃_𝑗𝑓‖²=‖𝑓‖² for all 𝑓∈𝐷, then, by Lemma  1.10 in [1], the proof is finished. Indeed, denoting (𝑏𝑎_𝑗)⁻¹ by 𝑏^∗, we have ‖‖𝑃_𝑗𝑓‖‖²=_𝑏_𝑘|||𝑓,𝐷_𝑎𝑗𝐷_𝑏𝑇_𝑘𝜑|||²=_𝑏_𝑘||||_ℝ𝑛||det𝑎_𝑗||^−1/2𝑓(𝜉)𝜑𝑏^∗𝜉𝑒^{2𝜋𝑖𝑘⋅𝑏∗𝜉}d𝜉||||²=_𝑏_𝑘|||||_𝑚_{𝑎𝑗𝑇𝑛}||det𝑎_𝑗||^−1/2𝑓𝜉+𝑎_𝑗𝑚𝜑𝑏^∗𝜉+𝑎_𝑗𝑚𝑒^{2𝜋𝑖𝑘⋅𝑏∗𝜉}d𝜉|||||²=_𝑏_𝑘|||||_{𝑎𝑗𝑇𝑛}||det𝑎_𝑗||^−1/2_𝑚𝑓𝜉+𝑎_𝑗𝑚𝜑𝑏^∗𝜉+𝑎_𝑗𝑚𝑒^{2𝜋𝑖𝑘⋅𝑏∗𝜉}d𝜉|||||²=_𝑏_{𝑎𝑗𝑇𝑛}|||||_𝑚𝑓𝜉+𝑎_𝑗𝑚𝜑𝑏^∗=𝜉+𝑎^𝑇𝑗𝑚|||||²d𝜉_𝑏_{𝑎𝑗𝑇𝑛}_𝑚=𝑓𝜉+𝑎_𝑗𝑚𝜑[𝑏^∗(𝜉+𝑎_𝑗𝑚)]_𝑛𝑓(𝜉+𝑎_𝑗𝑛)𝜑𝑏^∗𝜉+𝑎_𝑗𝑛d𝜉_𝑏_𝑚_{𝑎𝑗𝑇𝑛+𝑚}_𝑝=𝑓(𝜂)𝑓(𝜂+𝑎_𝑗𝑝)𝜑𝑏^∗𝜂+𝑏̃𝑝𝜑(𝑏^∗𝜂)d𝜂_𝑏_ℝ𝑛_𝑝=𝑓(𝜂)𝑓(𝜂+𝑎_𝑗𝑝)𝜑𝑏^∗𝜂+𝑏𝑝̃𝜑(𝑏^∗𝜂)d𝜂_ℝ𝑛||𝑓(𝜂)||²_𝑏|||𝜑𝑏̃⁻¹𝑎_𝑗𝜂|||²d𝜂+𝑅_𝑓,(3.5) where 𝑅_𝑓=∑_𝑏∑_𝑝≠0∫_ℝ𝑛𝑓(𝜂)𝑓(𝜂+𝑎_𝑗𝑝)𝜑(𝑏^∗𝜂+𝑏𝑝̃)𝜑(𝑏^∗𝜂)𝑑𝜂. Since 𝑓 has compact support, when 𝑗 is large enough, supp(𝑓(𝜂))∩supp(𝑓(𝜂+𝑎_𝑗𝑝))=∅, consequently, 𝑅_𝑓=0. Thus, taking 𝑗→+∞ in (3.5), we obtain lim_𝑗→+∞‖‖𝑃_𝑗𝑓‖‖²=lim_𝑗→+∞_𝑏_𝑘|||𝑓,𝐷_𝑎𝑗𝐷_𝑏𝑇_𝑘𝜑|||²=lim_𝑗→+∞_𝑏_ℝ𝑛||𝑓(𝜂)||²_𝑏|||𝜑𝑏̃⁻¹=𝑎_𝑗−1𝜂|||²d𝜂_ℝ𝑛||𝑓(𝜂)||²lim_𝑗→+∞_𝑏|||𝜑𝑏̃⁻¹=𝑎_𝑗−1𝜂|||²d𝜂_ℝ𝑛||𝑓(𝜂)||²d𝜂=‖𝑓‖².(3.6)

Lemma 3.2. Let 𝑎∈𝐺, 𝜑∈𝐿²(ℝ^𝑛) satisfy (2.10), and let {𝑉_𝑗}_𝑗∈ℤ be defined by (2.8). Then _𝑗∈ℤ𝑉_𝑗=𝐿²(𝑆)^∨,(3.7) where 𝑆=⋃_𝑗∈ℤ⋃_𝑏∈𝐵𝑎_𝑗̃𝑏̃supp(𝜑).

Proof. By the definition of {𝑉_𝑗}_𝑗∈ℤ, we have 𝐷_𝑎𝑉_𝑗=𝑉_𝑗+1 for any 𝑗∈ℤ. It follows that 𝐷_𝑎(⋃_𝑗∈ℤ𝑉_𝑗)=⋃_𝑗∈ℤ𝑉_𝑗. Note that 𝐷_𝑎 is a unitary operator. Hence 𝐷_𝑎(⋃_𝑗∈ℤ𝑉_𝑗)=𝐷_𝑎(⋃_𝑗∈ℤ𝑉_𝑗)=⋃_𝑗∈ℤ𝑉_𝑗. It is obvious to see that 𝑉_𝑗={𝑓∶𝑓=∑_𝑏𝐹_𝑏[(𝑏𝑎_𝑗)⁻¹𝜉]𝜑[(𝑏𝑎_𝑗)⁻¹𝜉]}, where {𝐹_𝑏}_𝑏∈𝐿²(𝑇^𝑛). Then, for any 𝑓∈𝑉₀, we have 𝑓(𝜉)=∑_𝑏𝐹_𝑏(𝑏̃⁻¹𝜉)𝜑(𝑏̃⁻¹𝜉), 𝐹_𝑏∈𝐿²(𝑇^𝑛). Notice that for any 𝑏∈𝐵, 𝜑(𝑏̃⁻¹𝜉)=∑_𝑏ℎ_𝑏(𝑏̃⁻¹𝑎̃⁻¹𝜉)𝜑(𝑏̃⁻¹𝑎̃⁻¹𝜉). Thus 𝑓(𝜉)=_𝑏1𝐹_𝑏1𝑏^1−𝑇𝜉_𝑏2ℎ_𝑏2=_𝑏1𝑏²⁻¹𝑎̃⁻¹𝜉𝜑𝑏₂₋₁𝑎̃⁻¹𝜉_𝑏2𝐹_𝑏1𝑏¹⁻¹𝜉ℎ_𝑏2𝑏²⁻¹𝑎̃⁻¹𝜉𝜑𝑏₂₋₁𝑎̃⁻¹𝜉.(3.8) Put 𝐻_𝑏2(𝑏̃⁻¹𝑎̃⁻¹𝜉)=∑_𝑏1𝐹_𝑏1(𝑏₁₋₁𝜉)𝑀_𝑏2(𝑏₂₋₁𝑎̃⁻¹𝜉). Then, we obtain 𝑓(𝜉)=∑_𝑏2𝐻_𝑏2(𝑏₂₋₁𝑎̃⁻¹𝜉)𝜑(𝑏₂₋₁𝑎̃⁻¹𝜉). Recalling that 𝐹_𝑏(𝜉)∈𝐿²(𝑇^𝑛) and ℎ_𝑏(𝜉)∈𝐿^∞(𝑇^𝑛), we have 𝑀_𝑏2(𝜉)∈𝐿²(𝑇^𝑛), so 𝑓(𝑥)∈𝑉₁. Thus, for any 𝑓(𝑥), we get 𝑉₀⊂𝑉₁, so 𝑉_𝑗⊂𝑉_𝑗+1. Therefore, for any 𝑓(𝑥)∈⋃_𝑗∈ℤ𝑉_𝑗, we can choose 𝑗_𝑓>0 such that 𝑓(𝑥)∈𝑉_𝑗𝑓, that is, 𝑓(𝑥)=∑_𝑏∑_𝑘𝑐_𝑏,𝑘𝜑(𝑏𝑎_𝑗𝑓𝑥−𝑘). Hence, for any 𝑚∈ℤ^𝑛, we have 𝑇_𝑚𝑓(𝑥)=_𝑏_𝑘=𝑐_𝑏,𝑘𝜑𝑏𝑎_𝑗𝑓(𝑥−𝑚)−𝑘_𝑏_𝑘=𝑐_𝑏,𝑘𝜑𝑏𝑎_𝑗𝑓𝑥−𝑏𝑎_𝑗𝑓𝑚−𝑘_𝑏_𝑘𝑐_𝑏,𝑘𝜑𝑏𝑎_𝑗𝑓𝑥−𝑘,(3.9) which implies 𝑇_𝑚(⋃_𝑗∈ℤ𝑉_𝑗)⊆⋃_𝑗∈ℤ𝑉_𝑗. Thus, 𝑇_𝑚(⋃_𝑗∈ℤ𝑉_𝑗)=⋃_𝑗∈ℤ𝑉_𝑗 holds. Note that 𝑇_𝑘 is also a unitary operator. Hence, 𝑇_𝑚(⋃_𝑗∈ℤ𝑉_𝑗)=𝑇_𝑚(⋃_𝑗∈ℤ𝑉_𝑗)=⋃_𝑗∈ℤ𝑉_𝑗. By Proposition 1.3, ⋃_𝑗∈ℤ𝑉_𝑗 is a reducing subspace. We notate ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(𝑆)^∨. Next we have to prove 𝑆=⋃_𝑗∈ℤ⋃_𝑏𝑎_𝑗𝑏̃supp(𝜑). By 𝜑(𝑏𝑎_𝑗⋅)∈𝑉_𝑗, we have supp(𝜑((𝑏𝑎_𝑗)⁻¹𝜉))⊂𝑆. Obviously, we will only prove that 𝑆⧵⋃_𝑗∈ℤ⋃_𝑏𝑎_𝑗𝑏̃supp(𝜑) is a zero measurable set. If 𝑆⧵⋃_𝑗∈ℤ⋃_𝑏𝑎_𝑗𝑏̃supp(𝜑) is a set with nonzero measure, then a contradiction is led. In fact, choosing a set 𝑀⊂𝑆⧵⋃_𝑗∈ℤ⋃_𝑏𝑎_𝑗𝑏̃supp(𝜑) with 0<|𝑀|<+∞ and using Plancherel theorem, we have ‖‖𝑃_𝑗𝑓‖‖²=𝑃_𝑗𝑓,𝑃_𝑗𝑓=𝑓,𝑃_𝑗𝑓=𝑓,𝑓^𝑗,(3.10) for any 𝑓∈𝐿²(ℝ^𝑛), where 𝑓^𝑗∶=𝑃_𝑗𝑓. In particular, setting 𝑓(𝜉)=𝜒_𝑀(𝜉) and taking 𝑗→+∞ in (3.10), we get ‖𝑃_𝑗𝑓‖²=⟨𝑓,𝑓^𝑗⟩=0, but ‖𝑓‖²=|𝑀|>0, and this is a contradiction. Therefore, we complete the proof of Lemma 3.2.

By Lemmas 3.1 and 3.2 and Theorems  1.7 and  5.2 in [1], we characterize the density condition of AB-RMRA as follows.

Theorem 3.3. Let 𝑎∈𝐺 and {𝑉_𝑗}_𝑗∈ℤ be defined by (2.8). If conditions (1), (2), and (5) of AB-RMRA are satisfied, then the following results are equivalent: (1)⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(𝑆)^∨; (2)lim_𝑗→+∞‖𝑃_𝑗𝑓‖²=‖𝑓‖², where 𝑃_𝑗∶𝐿²(ℝ^𝑛)→𝑉_𝑗 denotes an orthogonal projection operator;(3)lim_𝑗→+∞∑_𝑏∈𝐵|𝜑((𝑏𝑎_𝑗)⁻¹𝜉)|²=1, a.e. 𝜉∈𝑆;(4)𝑆=⋃_𝑗∈ℤ⋃_𝑏𝑎_𝑗̃𝑏̃supp𝜑.

It is well known that ⋂_𝑗∈ℤ𝑉_𝑗={0} can be deduced by the other conditions of MRA. Similarly, the condition ⋂_𝑗∈ℤ𝑉_𝑗={0} of AB-RMRA can be also deduced by the others. Thus, using Proposition 2.2 and Theorem 3.3, we can get the main theorem as follows.

Theorem 3.4. Let 𝑎∈𝐺, and let 𝐵 be a subgroup of 𝑆𝐿_𝑛(ℤ) with 𝑎𝐵𝑎⁻¹⊆𝐵. Suppose that 𝜑∈𝐿²(𝑆)^∨ and {𝑉_𝑗}_𝑗∈ℤ is defined by (2.8). Then 𝜑 is a generator of AB-RMRA if and only if (1)there exists {ℎ_𝑏(𝜉)}_𝑏∈𝐵⊂𝐿^∞(𝑇^𝑛) such that 𝜑(𝜉)=_𝑏𝑀_𝑏𝑏̃⁻¹𝑎̃⁻¹𝜉𝜑𝑏̃⁻¹𝑎̃⁻¹𝜉;(3.11)(2)∑_𝑘|𝜑(𝜉+𝑘)|²=𝜒_𝐹(𝜉), a.e., where 𝐹={𝜉∈𝑇^𝑛∩𝑆∶𝜑(𝜉+𝑘)≠0,𝑘∈ℤ^𝑛};(3)for any 𝑏∈𝐵 and 𝑏≠𝐼_𝑛, ∑_𝑘𝜑(𝜉+𝑘)𝜑(𝑏̃⁻¹(𝜉+𝑘))=0, a.e. 𝜉∈𝑆;(4)𝑆=⋃_𝑗∈ℤ⋃_𝑏𝑎_𝑗̃𝑏̃supp(𝜑).

Remarks 1. (1) Condition (4) in Theorem 3.4 can be replaced by any one of conditions in Theorem 3.3.
(2) If 𝑆=ℝ^𝑛 in Theorem 3.4, then we obtain the characterization of generator of AB-MRA.
(3) If 𝑛=1, 𝑎=(2), and 𝐵=(1) in Theorem 3.4, then we obtain the characterization of {𝑉_𝑗}_𝑗∈ℤ as a generator of MRA on 𝐿²(ℝ).

Corollary 3.5. Let 𝑎∈𝐺, and let 𝐵 be a subgroup of 𝑆𝐿_𝑛(ℤ) with 𝑎𝐵𝑎⁻¹⊆𝐵. Suppose that 𝐸 is a bounded nonzero measurable set satisfying 𝑎̃⁻¹𝐸⊂𝐸 with 𝑆=⋃_𝑗∈ℤ⋃_𝑏𝑎_𝑗̃𝑏𝐸̃. Define 𝜑(𝜉)=𝜒_𝐸(𝜉) and 𝐹={𝜉∈𝑇^𝑛∩Ω∶𝜑(𝜉+𝑘)≠0,𝑘∈ℤ^𝑛}. Then 𝜑 generates an AB-RMRA if and only if (1)𝐹=⋃_{𝑘∈ℤ𝑛}(𝐸+𝑘) with |𝐸∩(𝐸+𝑘)|=0, |𝑏^𝑇𝐸∩𝑏^𝑇𝐸|=0(𝑏≠𝑏^);(2)there exists 𝑏∈𝐵 such that (𝑏𝑎)⁻¹𝐸⊂𝐸 and [(𝑏𝑎)⁻¹𝐸]^∼∩𝐸=(𝑏𝑎)⁻¹𝐸;(3)𝑇_𝑘𝜒(𝑏𝑎)⁻¹_𝐸(𝜉)=𝜒(𝑏𝑎)⁻¹_𝐸(𝜉) on (𝑏𝑎)⁻¹𝐸∩[(𝑏𝑎)⁻¹𝐸+𝑘]≠∅ for 𝑘∈ℤ^𝑛⧵{0}.

4. Some Examples

Three examples are provided to illustrate the general theory in this section.

Example 4.1. Let 𝜑(𝜉)=𝜒_𝐼(𝜉), where 𝐼=𝐼⁺∪𝐼⁻, 𝐼⁻={𝜉∈ℝ²∣−𝜉∈𝐼⁺}, and 𝐼⁺ is a triangle region with vertices (0,0),(𝛼,0),(𝛼,𝛼). Let 𝑎=²⁰⁰², 𝐵={^1𝑖01∶𝑖∈ℤ}, 𝑆₀={(𝜉₁,𝜉₂)∈ℝ²∶0≤𝜉₁≤𝛼,𝜉₂∈ℝ}, 𝑆_𝑗=(𝑎^𝑗)𝑆₀. Define 𝑉_𝑗=𝐿²(𝑆_𝑗)^∨. Then by Corollary 3.5, we get the following:(1)when 𝛼≤1, 𝜑(𝑥) is a generator for an AB-RMRA;(2)when 𝛼>1, 𝜑(𝑥) is not a generator.

Example 4.2. Given 𝑘∈ℤ, let 𝐸=[0,𝛽₀]²∪(⋃^{−1𝑗=−𝑘}[2^𝑗𝛼,2^𝑗𝛽]²), where 0<𝛽₀≤2^−𝑘𝛼, 𝛼<𝛽≤min{2𝛼,2^𝑘+1𝛽₀,2}. If 𝜑=𝜒_𝐸(⋅), then 𝜑 generates an AB-RMRA, where 𝑆={(𝑥,𝑦)∶𝑥≥0,𝑦≥0}, 𝑎=⁰²²⁰.

Example 4.3. Let 𝐸=[0,1/4]²∪[3/8,3/8+𝜀]², where 0<𝜀≤1/16. Assume that 𝜑=𝜒_𝐸(⋅). Then 𝜑 generates an AB-RMRA, where 𝑆={(𝑥,𝑦)∶𝑥≥0,𝑦≥0}, 𝑎=²⁰⁰².

Acknowledgments

The paper was supported by the National Natural Science Foundation of China (no. 61071189) and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province of China (no. 084100510012).