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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 850850, 13 pages
Research Article

Characterization of Generators for Multiresolution Analyses with Composite Dilations

1School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China
2Basic Department, Henan Quality and Engineering Vocational College, Pingdingshan 467000, China
3School of Mathematics and Information Sciences, Henan University, Kaifeng 475001, China

Received 20 July 2010; Revised 12 December 2010; Accepted 2 February 2011

Academic Editor: RomanΒ Simon Hilscher

Copyright Β© 2011 Yuan Zhu et al. This 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.


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(ℝ𝑛) (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 πœ‘βˆˆπΏ2(ℝ) is a generator for an MRA if and only if (1)βˆ‘π‘˜βˆˆβ„€|ξπœ‘(πœ‰+π‘˜)|2=1, a.e. πœ‰βˆˆ[βˆ’1/2,1/2]; (2)lim𝑗→+∞|ξπœ‘(2βˆ’π‘—πœ‰)|2=1, a.e. πœ‰βˆˆβ„;(3)there exists π‘š0∈𝐿2([0,1]) such that ξπœ‘(2πœ‰)=π‘š0(πœ‰)ξπœ‘(πœ‰), a.e. πœ‰βˆˆβ„.

If condition (2) is replaced by ⋃ℝ=π‘—βˆˆβ„€2𝑗supp(ξπœ‘) or another condition that the function ∫𝐹(π‘₯,𝑦)=(1/(π‘¦βˆ’π‘₯))𝑦π‘₯|ξπœ‘(πœ”)|2dπœ” 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 π‘Ž, π‘“βˆˆπΏ2(ℝ𝑛), and π‘˜βˆˆβ„€π‘›, we define the dilation operator 𝐷 and the shift operator π‘‡π‘˜ on 𝐿2(ℝ𝑛) by π·π‘Ž||||𝑓(β‹…)∢=detπ‘Ž1/2𝑓(π‘Žβ‹…),π‘‡π‘˜π‘“(β‹…)∢=𝑓(β‹…βˆ’π‘˜).(1.1) The affine system with composite dilations is defined by π’œπ΄π΅(Ξ¨)={π·π‘Žπ·π‘π‘‡π‘˜Ξ¨βˆΆπ‘˜βˆˆβ„€π‘›,π‘βˆˆπ΅,π‘Žβˆˆπ΄} where Ξ¨={πœ“1,πœ“2,…,πœ“πΏ}βŠ‚πΏ2(ℝ𝑛). By choosing Ξ¨, 𝐴, and 𝐡 appropriately, we can make π’œπ΄π΅(Ξ¨) an orthonormal basis or, more generally, a Parseval frame (PF) for 𝐿2(ℝ𝑛) [5–7]. In this case, Ξ¨ 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 π·π‘π‘‡π‘˜ by ℬ.

Before proceeding, we need some conventions. We denote by 𝑇𝑛=[βˆ’1/2,1/2]𝑛 the 𝑛-dimensional torus. For a Lebesgue measurable set 𝐸 in ℝ𝑛, we denote by |𝐸| its measure, denote by πœ’πΈ the characteristic function of 𝐸, and define 𝐸∼∢=𝐸+℀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 𝐺𝐿𝑛(β„€) the set {π‘ŽβˆΆπ‘Ž is an 𝑛×𝑛 integral matrix and |detπ‘Ž|β‰ 0}, by 𝑆𝐿𝑛(β„€) the set {π‘ŽβˆΆπ‘Ž is an 𝑛×𝑛 integral matrix and |detπ‘Ž|=1}, and by 𝐡 the set of the subgroups of 𝑆𝐿𝑛(β„€), respectively. For a Lebesgue measurable function 𝑓, we define its support by supp(𝑓)∢={π‘₯βˆˆβ„π‘›βˆΆπ‘“(π‘₯)β‰ 0}.(1.2) The Fourier transform of π‘“βˆˆπΏ1(ℝ𝑛)∩𝐿2(ℝ𝑛) is defined by ξξ€œπ‘“(πœ‰)∢=ℝ𝑛𝑓(π‘₯)π‘’βˆ’π‘–2πœ‹βŸ¨πœ‰,π‘₯⟩dπ‘₯(1.3) on ℝ𝑛, where βŸ¨πœ‰,π‘₯⟩ denotes the inner product in ℝ𝑛. Let 𝑆 be a Lebesgue nonzero measurable set in ℝ𝑛. We denote by 𝐿2(𝑆)∨ the closed subspace of 𝐿2(ℝ𝑛) of the form 𝐿2(𝑆)βˆ¨ξ‚†βˆΆ=π‘“βˆˆπΏ2(ℝ𝑛𝑓.)∢suppβŠ†π‘†(1.4)

Definition 1.1 (see [8, 9]). The sequence {π‘₯π‘˜,𝑙}π‘˜,𝑙 in a separable Hilbert space 𝐻 is called a semiorthogonal PF for 𝐻 if {π‘₯π‘˜,𝑙}π‘˜,𝑙 is a PF for 𝐻 and satisfies ⟨π‘₯π‘˜1,𝑙1,π‘₯π‘˜2,𝑙2⟩=0 for any π‘˜1,π‘˜2βˆˆΞ›1, 𝑙1,𝑙2βˆˆΞ›2, and π‘˜1β‰ π‘˜2, where Ξ›1, Ξ›2 are two countable index sets. In particular, if {π‘₯π‘˜,𝑙}π‘˜,𝑙 is a semiorthogonal PF for span{π‘₯π‘˜,𝑙}π‘˜,𝑙, it is called a semiorthogonal sequence.

Definition 1.2 (see [4, 10]). A closed subspace 𝑋 of 𝐿2(ℝ𝑛) is called a reducing subspace if π·π‘Žπ‘‹=𝑋 and π‘‡π‘˜π‘‹=𝑋 for any π‘˜βˆˆβ„€π‘›, π‘ŽβˆˆπΊ.

The following proposition provides a characterization of reducing subspace.

Proposition 1.3 (see [4, 10]). A closed subspace 𝑋 of 𝐿2(ℝ𝑛) is a reducing subspace if and only if 𝑋=π‘“βˆˆπΏ2(ℝ𝑛𝑓)∢suppβŠ†π‘†(1.5) for some measurable set π‘†βŠ†β„π‘› with Μƒπ‘Žπ‘†=𝑆. So, to be specific, one denotes a reducing subspace by 𝐿2(𝑆)∨ instead of 𝑋. In particular, 𝐿2(ℝ𝑛) is a reducing subspace of 𝐿2(ℝ𝑛).

Definition 1.4 (see [5–7]). Let 𝐡⋉℀𝑛 be a subgroup of the integral affine group 𝑆𝐿𝑛(β„€)⋉℀𝑛 (the semidirect product of 𝑆𝐿𝑛 and ℀𝑛). The closed subspace 𝑉 of 𝐿2(ℝ𝑛) is called a 𝐡⋉℀𝑛 invariant subspace if ℬ𝑉=𝑉 for any (𝑏,π‘˜)βˆˆπ΅β‹‰β„€π‘›.

Definition 1.5 (see [2–4]). Let 𝐡 be a countable subset of 𝑆𝐿𝑛(𝑍) and 𝐴={π‘Žπ‘–βˆΆπ‘–βˆˆβ„€} where π‘ŽβˆˆπΊπΏπ‘›(β„€). We say that a sequence {𝑉𝑗}π‘—βˆˆβ„€ of closed subspaces of 𝐿2(ℝ𝑛) is an AB-MRA if the following holds: (1)𝑉0 is a 𝐡⋉℀𝑛 invariant space; (2)for each π‘—βˆˆβ„€, π‘‰π‘—βŠ‚π‘‰π‘—+1, and 𝑉𝑗=π·π‘—π‘Žπ‘‰0; (3)β‹ƒπ‘—βˆˆβ„€π‘‰π‘—=𝐿2(ℝ𝑛); (4)β‹‚π‘—βˆˆβ„€π‘‰π‘—={0};(5)there exists πœ‘βˆˆπ‘‰0 such that Φ𝐡={π·π‘π‘‡π‘˜πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›} is a semiorthogonal PF for 𝑉0.

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

Similarly, we say that a sequence {𝑉𝑗}π‘—βˆˆβ„€ is an AB-RMRA if it is an AB-MRA on 𝐿2(𝑆)∨, that is, conditions (1), (2), (4), (5), and (3)β€² β€‰β€‰β‹ƒπ‘—βˆˆβ„€π‘‰π‘—=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 Ξ¨={πœ“π‘–βˆΆπ‘–βˆˆπΌ} is a Parseval frame on 𝐻, then 𝑃(Ξ¨)={𝑃(πœ“π‘–)βˆΆπ‘–βˆˆπΌ} 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 π‘‡π‘˜ and π·π‘Ž, respectively.

Theorem 1.8. Suppose that {πœ‘;𝑉𝑗} is an AB-MRA, then 𝑉{ξ‚πœ‘;𝑗} is an AB-RMRA for 𝐿2(𝑆)∨, where ξ‚πœ‘βˆΆ=π‘ƒπœ‘, 𝑉0∢=span{π·π‘π‘‡π‘˜ξ‚πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›}, ξ‚π‘‰π‘—βˆΆ=span{π·π‘Žπ‘—π·π‘π‘‡π‘˜ξ‚πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›}, and 𝑃 is the orthogonal projection operator from 𝐿2(ℝ𝑛) 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 {π·π‘π‘‡π‘˜ξ‚πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›} is a Parseval frame sequence by Propositions 1.6 and 1.7. Naturally, {π·π‘π‘‡π‘˜ξ‚πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›} is a semi-Parseval frame for 𝑉0. Let πœ‘βˆˆπ‘‰0βŠ‚πΏ2(ℝ𝑛). Thus π‘ƒπœ‘=πœ‘1. For any ξ‚π‘‰π‘“βˆˆ0, we have 𝑓=π‘ξ“π‘˜βŸ¨π‘“,π·π‘π‘‡π‘˜ξ‚πœ‘βŸ©π·π‘π‘‡π‘˜ξ“ξ‚πœ‘=π‘ξ“π‘˜βŸ¨π‘“,π·π‘π‘‡π‘˜πœ‘1βŸ©π·π‘π‘‡π‘˜πœ‘1,π·π‘β€²π‘‡π‘˜β€²ξ“π‘“=π‘ξ“π‘˜βŸ¨π‘“,π·π‘π‘‡π‘˜πœ‘1βŸ©πœ‘1ξ€·π‘ξ…žπ‘π‘₯βˆ’π‘π‘˜ξ…žξ€Έ=ξ“βˆ’π‘˜π‘ξ“π‘˜βŸ¨π‘“,π·π‘π‘‡π‘˜πœ‘1βŸ©πœ‘1(𝑏π‘₯βˆ’π‘˜),(2.1) namely, π·π‘β€²π‘‡π‘˜β€²ξ‚π‘‰π‘“βˆˆ0. So 𝑉0 is a 𝐡⋉℀𝑛 invariant space. On the other hand, 𝑓=π‘ξ“π‘˜βŸ¨π‘“,π·π‘π‘‡π‘˜π‘ƒπœ‘βŸ©π·π‘π‘‡π‘˜ξ“π‘ƒπœ‘=π‘ξ“π‘˜βŸ¨π‘ƒπ‘“,π·π‘π‘‡π‘˜πœ‘βŸ©π‘ƒπ·π‘π‘‡π‘˜πœ‘=ξ“π‘ξ“π‘˜βŸ¨π‘“,π·π‘π‘‡π‘˜ξ“πœ‘βŸ©π‘ƒπ‘β€²ξ“π‘˜β€²βŸ¨π·π‘π‘‡π‘˜πœ‘,π·π‘Žπ·π‘π‘‡π‘˜πœ‘βŸ©π·π‘Žπ·π‘ξ…žπ‘‡π‘˜ξ…žπœ‘=ξ“π‘ξ“π‘˜π‘π‘,π‘˜π·π‘Žπ·π‘π‘‡π‘˜ξ‚π‘‰ξ‚πœ‘βˆˆ1.(2.2) So 𝑉0βŠ‚ξ‚π‘‰1. Notice that 𝑉𝑖=π·π‘Žπ‘—ξ‚π‘‰0. Then ξ‚π‘‰π‘–βŠ‚ξ‚π‘‰π‘–+1. Thus, conditions (1), (2), and (5) in Definition 1.5 have been proved. However, condition (3)β€² 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 {𝑉𝑗}π‘—βˆˆβ„€ is an AB-RMRA. Then (1)for each π‘—βˆˆβ„€, {π·π‘—π‘Žπ·π‘π‘‡π‘˜πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›} is a semiorthogonal PF on 𝑉𝑗;(2)𝑉0 is a 𝐡⋉℀𝑛 invariant subspace, while 𝑉𝑗 is a π΅β‹‰π‘Žβˆ’π‘—β„€π‘› invariant subspace.

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

Proposition 2.2. Let πœ‘βˆˆπΏ2(𝑆)∨. Then Φ𝐡={π·π‘π‘‡π‘˜πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›} is a semiorthogonal PF sequence if and only if (1)βˆ‘π‘˜βˆˆβ„€π‘›|ξπœ‘(πœ‰+π‘˜)|2=πœ’πΉ(πœ‰), a.e., where 𝐹={πœ‰βˆˆπ‘‡π‘›βˆ©Ξ©βˆΆξπœ‘(πœ‰+π‘˜)β‰ 0,π‘˜βˆˆβ„€π‘›};(2)βˆ‘π‘˜βˆˆβ„€π‘›ξπœ‘(πœ‰+π‘˜)Μƒπ‘ξπœ‘(βˆ’1(πœ‰+π‘˜))=0, a.e. πœ‰βˆˆΞ©, for each π‘βˆˆπ΅ and 𝑏≠𝐼𝑛.

Proof. Necessity. For any 𝑓(π‘₯)∈span{π‘‡π‘˜πœ‘βˆΆπ‘˜βˆˆβ„€π‘›}, we have 𝑓(π‘₯)=𝑏,π‘˜ξ“βŸ¨π‘“,β„¬πœ‘βŸ©β„¬πœ‘(π‘₯)=π‘˜βŸ¨π‘“,π‘‡π‘˜πœ‘βŸ©π‘‡π‘˜ξ“πœ‘(π‘₯)+π‘β‰ πΌπ‘›ξ“π‘˜ξ“βŸ¨π‘“,β„¬πœ‘βŸ©β„¬πœ‘(π‘₯)=π‘˜βŸ¨π‘“,π‘‡π‘˜πœ‘βŸ©π‘‡π‘˜πœ‘(π‘₯).(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 βŸ¨π‘‡π‘˜πœ‘,π·π‘π‘‡π‘˜ξ…žξ€œπœ‘βŸ©=Ξ©πœ‘(π‘₯βˆ’π‘˜)πœ‘(𝑏π‘₯βˆ’π‘˜ξ…ž=ξ€œ)dπ‘₯Ξ©ξπœ‘(πœ‰)Μƒπ‘ξπœ‘(βˆ’1πœ‰)π‘’βˆ’2πœ‹π‘–(π‘˜βˆ’Μƒπ‘βˆ’1π‘˜β€²)β‹…πœ‰=ξ€œdπœ‰Ξ©ξ“π‘™ξπœ‘(πœ‰+𝑙)Μƒπ‘ξπœ‘(βˆ’1(πœ‰+𝑙))π‘’βˆ’2πœ‹π‘–(π‘˜βˆ’π‘˜1)β‹…πœ‰dπœ‰,(2.4) where π‘˜1=Μƒπ‘βˆ’1π‘˜ξ…ž. Note that for any π‘˜,π‘˜ξ…žβˆˆβ„€π‘›, π‘β‰ π‘ξ…žβˆˆπ΅, βŸ¨π·π‘π‘‡π‘˜πœ‘,π·π‘ξ…žπ‘‡π‘˜ξ…žπœ‘βŸ©=0 if and only if for any π‘βˆˆπ΅ and 𝑏≠𝐼𝑛, βŸ¨π‘‡π‘˜πœ‘,π·π‘π‘‡π‘˜ξ…žπœ‘βŸ©=0.(2.5)
Then, we have ξ“π‘™ξπœ‘(πœ‰+𝑙)Μƒπ‘ξπœ‘(βˆ’1(πœ‰+𝑙))=0,a.e.πœ‰βˆˆΞ©.(2.6)
Sufficiency. By Theorem  7.2.3 in [8] and conclusion (1), {π‘‡π‘˜πœ‘βˆΆπ‘˜βˆˆβ„€π‘›} is a PF sequence. So is {π·π‘π‘‡π‘˜πœ‘βˆΆπ‘˜βˆˆβ„€π‘›} for any π‘βˆˆπ΅. It follows from (2.4), (2.5), and conclusion (2) that for any π‘˜,π‘˜ξ…žβˆˆβ„€π‘›, 𝑏,π‘ξ…žβˆˆπ΅, and π‘β‰ π‘ξ…ž, we get βŸ¨π·π‘π‘‡π‘˜πœ‘,π·π‘β€²π‘‡π‘˜β€²πœ‘βŸ©=0. Thus, for any 𝑓(π‘₯)∈span{π·π‘π‘‡π‘˜πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›}, there exists π‘βˆˆπ΅ and 𝑓𝑏(π‘₯)∈span{π·π‘π‘‡π‘˜πœ‘βˆΆπ‘˜βˆˆβ„€π‘›} such that ‖𝑓‖2=ξ“π‘βˆˆπ΅β€–β€–π‘“π‘β€–β€–2=ξ“π‘βˆˆπ΅ξ“π‘˜βˆˆβ„€π‘›||βŸ¨π‘“π‘,π·π‘π‘‡π‘˜||πœ‘βŸ©2=ξ“π‘βˆˆπ΅ξ“π‘˜βˆˆβ„€π‘›||βŸ¨π‘“,π·π‘π‘‡π‘˜||πœ‘βŸ©2.(2.7) By Theorem  1.6 in [1], the proof of Proposition 2.2 is completed.

Proposition 2.3. Let {𝑉𝑗}π‘—βˆˆβ„€ be a sequence of closed subspace of 𝐿2(𝑆)∨, where π‘‰π‘—βˆΆ=𝐷spanπ‘—π‘Žπ·π‘π‘‡π‘˜πœ‘βˆΆπ‘βˆˆπ΅,π‘˜βˆˆβ„€π‘›ξ€Ύ.(2.8) If conditions (1), (2), and (5) of AB-RMRA are satisfied, then one has the following. (1)There exists {𝑐𝑏,π‘˜}βˆˆπ‘™2(𝐡×℀𝑛) such that ξ“πœ‘(π‘₯)=π‘ξ“π‘˜π‘π‘,π‘˜||||detπ‘Ž1/2πœ‘(π‘π‘Žπ‘₯βˆ’π‘˜).(2.9)(2)There exists {β„Žπ‘(πœ‰)}π‘βˆˆπ΅βŠ†πΏβˆž(𝑇𝑛) such that ξ“ξπœ‘(πœ‰)=π‘β„Žπ‘ξ‚ƒξƒ€(π‘π‘Ž)βˆ’1πœ‰ξ‚„ξ‚ƒξƒ€ξπœ‘(π‘π‘Ž)βˆ’1πœ‰ξ‚„,(2.10) for any π‘βˆˆπ΅, where β„Žπ‘(πœ‰)=|detπ‘Ž|βˆ’1/2βˆ‘π‘˜π‘π‘,π‘˜π‘’βˆ’2πœ‹π‘–π‘˜β‹…πœ‰.

Proof. By conditions (2) and (5) of AB-RMRA and the fact that πœ‘βˆˆπ‘‰0βŠ‚π‘‰1, we obtain ξ“πœ‘(π‘₯)=𝑏,π‘˜βŸ¨πœ‘,π·π‘Žβ„¬πœ‘βŸ©π·π‘Žξ“β„¬πœ‘(π‘₯)=π‘ξ“π‘˜βˆˆβ„€π‘›π‘π‘,π‘˜||||detπ‘Ž1/2πœ‘(π‘π‘Žπ‘₯βˆ’π‘˜),(2.11) where 𝑐𝑏,π‘˜=βŸ¨πœ‘,π·π‘Žβ„¬πœ‘βŸ© and {𝑐𝑏,π‘˜}βˆˆπ‘™2(𝐡×℀𝑛). Therefore (2.9) holds. Taking Fourier transform on both sides of (2.9), we obtain (2.10), where for any π‘βˆˆπ΅, β„Žπ‘(πœ‰)=|detπ‘Ž|βˆ’1/2βˆ‘π‘˜π‘π‘,π‘˜π‘’βˆ’2πœ‹π‘–π‘˜β‹…πœ‰. In what follows, we will only prove {β„Žπ‘(πœ‰)}π‘βˆˆπ΅βŠ†πΏβˆž(𝑇𝑛). Indeed, for π‘ŽβˆˆπΊπΏπ‘›(β„€),Μƒπ‘Žβ„€π‘› is a subgroup of ℀𝑛 and the quotient group ℀𝑛/Μƒπ‘Žβ„€π‘› has order 𝑀=|detπ‘Ž|. Thus, we can choose a complete set of representatives of ℀𝑛/Μƒπ‘Žβ„€π‘›, that is, the set {𝛼0,𝛼1,…,π›Όπ‘€βˆ’1} so that each π‘˜βˆˆβ„€π‘› can be uniquely expressed in the form π‘˜=Μƒπ‘Žπ‘˜ξ…ž+𝛼𝑖 with π‘˜ξ…žβˆˆβ„€π‘›, 0β‰€π‘–β‰€π‘€βˆ’1. For simplicity, we denote (π‘π‘Ž)βˆ’1 and (π‘ξ…žπ‘Ž)βˆ’1 by π‘βˆ— and π‘βˆ—1, respectively. Then we have ξ“π‘˜||||ξπœ‘(πœ‰+π‘˜)2=ξ“π‘˜βˆˆβ„€π‘›|||||ξ“π‘β„Žπ‘ξ€Ίπ‘βˆ—ξ€»ξ€Ίπ‘(πœ‰+π‘˜)ξπœ‘βˆ—ξ€»|||||(πœ‰+π‘˜)2=ξ“π‘˜βˆˆβ„€π‘›ξ“π‘,𝑏1β„Žπ‘ξ€Ίπ‘βˆ—ξ€»(πœ‰+π‘˜)β„Žπ‘1[π‘βˆ—1𝑏(πœ‰+π‘˜)]ξπœ‘βˆ—ξ€»(πœ‰+π‘˜)ξπœ‘[π‘βˆ—1=(πœ‰+π‘˜)]π‘€βˆ’1𝑖=0ξ“π‘˜β€²,𝑏,𝑏1β„Žπ‘ξ€Ίπ‘βˆ—ξ€·πœ‰+π›Όπ‘–ξ€Έξ€»β„Žπ‘1[π‘βˆ—1(πœ‰+𝛼𝑖𝑏)]ξπœ‘βˆ—ξ€·πœ‰+𝛼𝑖+Μƒπ‘βˆ’1π‘˜ξ…žξ‚„Γ—ξπœ‘[π‘βˆ—1(πœ‰+𝛼𝑖𝑏)+1βˆ’1π‘˜ξ…ž](2.12)=π‘€βˆ’1𝑖=0𝑏,𝑏1β„Žπ‘ξ€Ίπ‘βˆ—ξ€·πœ‰+π›Όπ‘–ξ€Έξ€»β„Žπ‘1[π‘βˆ—1(πœ‰+𝛼𝑖×)]π‘˜β€²βˆˆβ„€π‘›ξ‚ƒπ‘ξπœ‘βˆ—ξ€·πœ‰+𝛼𝑖+Μƒπ‘βˆ’1π‘˜ξ…žξ‚„ξπœ‘[π‘βˆ—1(πœ‰+𝛼𝑖𝑏)+1βˆ’1π‘˜ξ…ž]=π‘€βˆ’1𝑖=0𝑏||β„Žπ‘ξ€Ίπ‘βˆ—ξ€·πœ‰+𝛼𝑖||ξ€Έξ€»2ξ“π‘˜β€²βˆˆβ„€π‘›|||ξ‚ƒπ‘ξπœ‘βˆ—ξ€·πœ‰+𝛼𝑖+Μƒπ‘βˆ’1π‘˜ξ…žξ‚„|||2,(2.13) where (2.12) is obtained by the periodicity of function sequence {β„Žπ‘(πœ‰)}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 πœ‰βˆˆπΉ, we get βˆ‘π‘˜βˆˆβ„€π‘›|ξπœ‘(πœ‰+π‘˜)|2=βˆ‘π‘€βˆ’1𝑖=0βˆ‘π‘|β„Žπ‘[π‘βˆ—πœ‰+π‘βˆ—π›Όπ‘–)]|2=1. Then, for πœ‰βˆˆβ„π‘›, |β„Žπ‘(πœ‰)|2≀1, so β„Žπ‘(πœ‰)∈𝐿∞(𝑇𝑛). 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(ℝ𝑛).

Lemma 3.1. Let {𝑉𝑗}π‘—βˆˆβ„€ 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)β‹ƒπ‘—βˆˆβ„€π‘‰π‘—=𝐿2(𝑆)∨; (2)lim𝑗→+βˆžβˆ‘π‘ξƒ΄|ξπœ‘[(π‘π‘Žπ‘—)βˆ’1πœ‰]|2=1, a.e. πœ‰βˆˆβ„π‘›.

Proof. Theorems  1.7 and  5.2 in [1] imply that for any π‘“βˆˆπΏ2(ℝ𝑛), lim𝑗→+βˆžβ€–π‘ƒπ‘—π‘“β€–2=‖𝑓‖2 is equivalent to β‹ƒπ‘—βˆˆβ„€π‘‰π‘—=𝐿2(ℝ𝑛). Thus, for any π‘“βˆˆπΏ2(𝑆)∨, lim𝑗→+βˆžβ€–π‘ƒπ‘—π‘“β€–2=‖𝑓‖2 is equivalent to β‹ƒπ‘—βˆˆβ„€π‘‰π‘—=𝐿2(𝑆)∨. Hence, we have to prove that lim𝑗→+βˆžβ€–π‘ƒπ‘—π‘“β€–2=‖𝑓‖2 is equivalent to lim𝑗→+βˆžβˆ‘π‘ξƒ΄|ξπœ‘[(π‘π‘Žπ‘—)βˆ’1πœ‰]|2=1, a.e. πœ‰βˆˆΞ©. First, we prove (1)β‡’(2) For any π‘“βˆˆπΏ2(𝑆)∨, 𝑓=𝑃𝑗𝑓+𝑄𝑗𝑓, where π‘„π‘—βˆΆπΏ2(ℝ)β†’(𝑉𝑗)βŸ‚ is the orthogonal projection operator. Set 𝑓(πœ‰)=πœ’π‘‡π‘›βˆ©π‘†(πœ‰). Then, when 𝑗 is large enough, we have ‖‖𝑃𝑗𝑓‖‖2=ξ“π‘ξ“π‘˜||𝑓,π·π‘—π‘Žπ·π‘π‘‡π‘˜πœ‘ξ¬||2=ξ“π‘ξ“π‘˜|||||ξ€œβ„π‘›||detπ‘Žπ‘—||βˆ’1/2𝑓(πœ‰)ξ‚Έξƒ΄ξ€·ξπœ‘π‘π‘Žπ‘—ξ€Έβˆ’1πœ‰ξ‚Ήπ‘’2πœ‹π‘–π‘˜β‹…ξƒ΄(π‘π‘Žπ‘—)βˆ’1πœ‰|||||dπœ‰2=||detπ‘Žπ‘—||ξ“π‘ξ“π‘˜|||||ξ€œξƒ΄(π‘π‘Žπ‘—)βˆ’1π‘‡π‘›ξπœ‘(πœ‰)𝑒2πœ‹π‘–π‘˜β‹…πœ‰|||||2=||dπœ‰detπ‘Žπ‘—||ξ“π‘ξ€œπ‘‡π‘›||||πœ’ξƒ΄(π‘π‘Žπ‘—)βˆ’1𝑇𝑛(πœ‰)||||ξπœ‘(πœ‰)2=dπœ‰π‘ξ€œπ‘‡π‘›||||ξ‚Έξƒ΄ξ€·ξπœ‘π‘π‘Žπ‘—ξ€Έβˆ’1ξ‚Ή||||(πœ‰)2=ξ€œdπœ‰π‘‡π‘›ξ“π‘||||ξ‚Έξƒ΄ξ€·ξπœ‘π‘π‘Žπ‘—ξ€Έβˆ’1ξ‚Ή||||(πœ‰)2dπœ‰.(3.1) Before proving the equivalence, we need to prove two assertions as follows:(i)βˆ‘π‘ξƒ΄|ξπœ‘[(π‘π‘Žπ‘—)βˆ’1πœ‰]|2∈𝐿1(𝑇𝑛); (ii)lim𝑗→+βˆžβˆ‘π‘ξƒ΄|ξπœ‘[(π‘π‘Žπ‘—)βˆ’1πœ‰]|2 makes sense.Since ξ€œπ‘‡π‘›ξ“π‘||||ξ‚Έξƒ΄ξ€·ξπœ‘π‘π‘Žπ‘—ξ€Έβˆ’1πœ‰ξ‚Ή||||2dπœ‰=π‘ξ€œπ‘‡π‘›||||ξ‚Έξƒ΄ξ€·ξπœ‘π‘π‘Žπ‘—ξ€Έβˆ’1πœ‰ξ‚Ή||||2=||dπœ‰detπ‘Žπ‘—||ξ“π‘ξ€œξƒ΄(π‘π‘Žπ‘—)βˆ’1𝑇𝑛||||ξπœ‘(πœ‰)2=||dπœ‰detπ‘Žπ‘—||ξ€œβ‹ƒπ‘ξƒ΄(π‘π‘Žπ‘—)βˆ’1𝑇𝑛||||ξπœ‘(πœ‰)2≀||dπœ‰detπ‘Žπ‘—||ξ€œβ„π‘›||||ξπœ‘(πœ‰)2dπœ‰<∞,(3.2) it follows that (i) holds.
For (ii), we will only prove that {βˆ‘π‘ξƒ΄|ξπœ‘[(π‘π‘Žπ‘—)βˆ’1πœ‰]|2}π‘—βˆˆβ„€ is a monotonic bounded sequence when πœ‰(βˆˆπ‘†) is fixed. Indeed, by the orthogonality, for each π‘β‰ π‘ξ…žβˆˆπ΅, we have ̃𝑏supp(ξπœ‘(βˆ’1ξ‚π‘πœ‰))∩supp(ξπœ‘(ξ…žβˆ’1πœ‰))=βˆ…. In addition, we deduce from (2.10) that, for any π‘βˆˆπ΅, ξ‚ƒξƒ΄ξπœ‘(π‘π‘Žπ‘—)βˆ’1πœ‰ξ‚„=ξ“π‘β€²βˆˆπ΅β„Žπ‘β€²ξ‚Έξƒ΄(π‘ξ…žπ‘Ž)βˆ’1(π‘π‘Žπ‘—)βˆ’1πœ‰ξ‚Ήξ‚Έξƒ΄ξπœ‘(π‘ξ…žπ‘Ž)βˆ’1(π‘π‘Žπ‘—)βˆ’1πœ‰ξ‚Ή=ξ“π‘β€²βˆˆπ΅β„Žπ‘β€²ξ‚΅ξ‚π‘ξ…žβˆ’1Μƒπ‘βˆ’1ξƒ€π‘Žπ‘—+1βˆ’1πœ‰ξ‚Άξ‚΅ξ‚π‘ξπœ‘ξ…žβˆ’1Μƒπ‘βˆ’1ξƒ€π‘Žπ‘—+1βˆ’1πœ‰ξ‚Ά.(3.3)
Set π‘βˆ—=Μƒπ‘βˆ’1ξƒ€π‘Žπ‘—+1βˆ’1. Then by the orthogonality and Proposition 2.3, we obtain ξ“π‘βˆˆπ΅||||ξ‚Έξƒ΄ξ€·ξπœ‘π‘π‘Žπ‘—ξ€Έβˆ’1πœ‰ξ‚Ή||||2=ξ“π‘βˆˆπ΅|||||ξ“π‘β€²βˆˆπ΅β„Žπ‘β€²ξ‚΅ξ‚π‘ξ…žβˆ’1π‘βˆ—πœ‰ξ‚Άξ‚΅ξ‚π‘ξπœ‘ξ…žβˆ’1π‘βˆ—πœ‰ξ‚Ά|||||2=ξ“π‘βˆˆπ΅ξ“π‘1βˆˆπ΅β„Žπ‘1𝑏1βˆ’1π‘βˆ—πœ‰ξ‚ξ‚€ξ‚π‘ξπœ‘1βˆ’1π‘βˆ—πœ‰ξ‚ξ“π‘2βˆˆπ΅β„Žπ‘2𝑏2βˆ’1π‘βˆ—πœ‰ξ‚ξ‚€ξ‚π‘ξπœ‘2βˆ’1π‘βˆ—πœ‰ξ‚=ξ“π‘βˆˆπ΅||||β„Žπ‘ξ‚΅ξ„Ÿξ€·π‘π‘Žπ‘—+1ξ€Έβˆ’1πœ‰ξ‚Άξ‚΅ξ„Ÿξ€·ξπœ‘π‘π‘Žπ‘—+1ξ€Έβˆ’1πœ‰ξ‚Ά||||2β‰€ξ“π‘βˆˆπ΅||||ξ‚΅ξ„Ÿξ€·ξπœ‘π‘π‘Žπ‘—+1ξ€Έβˆ’1πœ‰ξ‚Ά||||2.(3.4) Hence, {βˆ‘π‘βˆˆπ΅ξƒ΄|ξπœ‘((π‘π‘Žπ‘—)βˆ’1πœ‰)|2}π‘—βˆˆβ„€ is a monotonic sequence when πœ‰ is fixed. On the other hand, by the property of 𝐡, we deduce that π·π‘π‘‡π‘˜πœ‘(π‘₯)=πœ‘(𝑏π‘₯βˆ’π‘˜)=πœ‘(𝑏(π‘₯βˆ’π‘βˆ’1π‘˜))=πœ‘(𝑏(π‘₯βˆ’π‘˜ξ…ž))=π‘‡π‘˜ξ…žπ·π‘πœ‘(π‘₯). Note that {π·π‘π‘‡π‘˜πœ‘βˆΆπ‘˜βˆˆβ„€π‘›} is a PF for span{π·π‘π‘‡π‘˜πœ‘βˆΆπ‘˜βˆˆβ„€π‘›} for any π‘βˆˆπ΅. Then {π‘‡π‘˜β€²π·π‘πœ‘βˆΆπ‘˜ξ…žβˆˆβ„€π‘›} is a PF on span{π·π‘π‘‡π‘˜πœ‘βˆΆπ‘˜βˆˆβ„€π‘›}, which implies that there exists a set 𝑀𝑏 such that βˆ‘π‘˜Μƒπ‘|ξπœ‘(βˆ’1πœ‰+π‘˜)|2=πœ’π‘€π‘(πœ‰). By the orthogonality, βˆ‘π‘βˆ‘π‘˜Μƒπ‘|ξπœ‘(βˆ’1πœ‰+π‘˜)|2=πœ’β‹ƒπ‘π‘€π‘(πœ‰), consequently ⋃𝑏𝑀𝑏=𝐹. Hence, βˆ‘π‘Μƒπ‘|ξπœ‘(βˆ’1πœ‰)|2≀1 holds for πœ‰βˆˆπ‘†. So lim𝑗→+βˆžβˆ‘π‘βˆˆπ΅ξƒ΄|ξπœ‘((π‘π‘Žπ‘—)βˆ’1πœ‰)|2 exists. Now we have proved the two assertions. By the Lebesgue dominant convergence theorem, we get lim𝑗→+βˆžβ€–π‘ƒπ‘—π‘“β€–2=βˆ«π‘‡π‘›lim𝑗→+βˆžβˆ‘π‘ξƒ΄|ξπœ‘[(π‘π‘Žπ‘—)βˆ’1πœ‰]|2dπœ‰=‖𝑓‖2=|𝑇𝑛|=1. Thus, lim𝑗→+βˆžβˆ‘π‘βˆˆπ΅ξƒ΄|ξπœ‘[(π‘π‘Žπ‘—)βˆ’1πœ‰]|2=1, a.e. πœ‰βˆˆβ„π‘›.
Next, we prove (2)β‡’(1). Let 𝐷 be the class of all functions π‘“βˆˆπΏ2(ℝ𝑛) such that ξπ‘“βˆˆπΏβˆž(ℝ𝑛) and 𝑓 is compactly supported in ℝ𝑛⧡{0}. If we can show that lim𝑗→+βˆžβ€–π‘ƒπ‘—π‘“β€–2=‖𝑓‖2 for all π‘“βˆˆπ·, then, by Lemma  1.10 in [1], the proof is finished. Indeed, denoting (π‘π‘Žπ‘—)βˆ’1 by π‘βˆ—, we have ‖‖𝑃𝑗𝑓‖‖2=ξ“π‘ξ“π‘˜|||𝑓,π·π‘Žπ‘—π·π‘π‘‡π‘˜πœ‘ξ‚­|||2=ξ“π‘ξ“π‘˜||||ξ€œβ„π‘›||detπ‘Žπ‘—||βˆ’1/2𝑓(πœ‰)ξ€·π‘ξπœ‘βˆ—πœ‰ξ€Έπ‘’2πœ‹π‘–π‘˜β‹…π‘βˆ—πœ‰||||dπœ‰2=ξ“π‘ξ“π‘˜|||||ξ“π‘šξ€œξ‚π‘Žπ‘—π‘‡π‘›||detπ‘Žπ‘—||βˆ’1/2ξπ‘“ξ€·πœ‰+ξ‚π‘Žπ‘—π‘šξ€Έξ€Ίπ‘ξπœ‘βˆ—ξ€·πœ‰+ξ‚π‘Žπ‘—π‘šπ‘’ξ€Έξ€»2πœ‹π‘–π‘˜β‹…π‘βˆ—πœ‰|||||dπœ‰2=ξ“π‘ξ“π‘˜|||||ξ€œξ‚π‘Žπ‘—π‘‡π‘›||detπ‘Žπ‘—||βˆ’1/2ξ“π‘šξπ‘“ξ€·πœ‰+ξ‚π‘Žπ‘—π‘šξ€Έξ€Ίπ‘ξπœ‘βˆ—ξ€·πœ‰+ξ‚π‘Žπ‘—π‘šπ‘’ξ€Έξ€»2πœ‹π‘–π‘˜β‹…π‘βˆ—πœ‰|||||dπœ‰2=ξ“π‘ξ€œξ‚π‘Žπ‘—π‘‡π‘›|||||ξ“π‘šξπ‘“ξ€·πœ‰+ξ‚π‘Žπ‘—π‘šξ€Έξ‚ƒπ‘ξπœ‘βˆ—ξ‚€πœ‰+π‘Žπ‘‡π‘—π‘š|||||2=dπœ‰π‘ξ€œξ‚π‘Žπ‘—π‘‡π‘›ξ“π‘šξπ‘“ξ€·πœ‰+ξ‚π‘Žπ‘—π‘šξ€Έξπœ‘[π‘βˆ—(πœ‰+ξ‚π‘Žπ‘—ξ“π‘š)]𝑛𝑓(πœ‰+ξ‚π‘Žπ‘—ξ€Ίπ‘π‘›)ξπœ‘βˆ—ξ€·πœ‰+ξ‚π‘Žπ‘—π‘›=dπœ‰π‘ξ“π‘šξ€œξ‚π‘Žπ‘—π‘‡π‘›+π‘šξ“π‘ξπ‘“(πœ‚)𝑓(πœ‚+ξ‚π‘Žπ‘—ξ€Ίπ‘π‘)ξπœ‘βˆ—Μƒξ€»πœ‚+π‘π‘ξπœ‘(π‘βˆ—=ξ“πœ‚)dπœ‚π‘ξ€œβ„π‘›ξ“π‘ξπ‘“(πœ‚)𝑓(πœ‚+ξ‚π‘Žπ‘—ξ€·π‘π‘)ξπœ‘βˆ—Μƒξ€Έπœ‚+π‘π‘ξπœ‘(π‘βˆ—=ξ€œπœ‚)dπœ‚β„π‘›||||𝑓(πœ‚)2𝑏|||ξ‚€Μƒπ‘ξπœ‘βˆ’1ξ‚π‘Žπ‘—πœ‚ξ‚|||2dπœ‚+𝑅𝑓,(3.5) where 𝑅𝑓=βˆ‘π‘βˆ‘π‘β‰ 0βˆ«β„π‘›ξπ‘“(πœ‚)𝑓(πœ‚+ξ‚π‘Žπ‘—π‘)ξπœ‘(π‘βˆ—Μƒπœ‚+𝑏𝑝)ξπœ‘(π‘βˆ—πœ‚)π‘‘πœ‚. Since 𝑓 has compact support, when 𝑗 is large enough, supp(𝑓(πœ‚))∩supp(𝑓(πœ‚+ξ‚π‘Žπ‘—π‘))=βˆ…, consequently, 𝑅𝑓=0. Thus, taking 𝑗→+∞ in (3.5), we obtain lim𝑗→+βˆžβ€–β€–π‘ƒπ‘—π‘“β€–β€–2=lim𝑗→+βˆžξ“π‘ξ“π‘˜|||𝑓,π·π‘Žπ‘—π·π‘π‘‡π‘˜πœ‘ξ‚­|||2=lim𝑗→+βˆžξ“π‘ξ€œβ„π‘›||||𝑓(πœ‚)2𝑏|||ξ‚€Μƒπ‘ξπœ‘βˆ’1ξ‚π‘Žπ‘—βˆ’1πœ‚ξ‚|||2=ξ€œdπœ‚β„π‘›||||𝑓(πœ‚)2lim𝑗→+βˆžξ“π‘|||ξ‚€Μƒπ‘ξπœ‘βˆ’1ξ‚π‘Žπ‘—βˆ’1πœ‚ξ‚|||2=ξ€œdπœ‚β„π‘›||𝑓||(πœ‚)2dπœ‚=‖𝑓‖2.(3.6)

Lemma 3.2. Let π‘ŽβˆˆπΊ, πœ‘βˆˆπΏ2(ℝ𝑛) satisfy (2.10), and let {𝑉𝑗}π‘—βˆˆβ„€ be defined by (2.8). Then ξšπ‘—βˆˆβ„€π‘‰π‘—=𝐿2(𝑆)∨,(3.7) where ⋃𝑆=π‘—βˆˆβ„€β‹ƒπ‘βˆˆπ΅Μƒπ‘Žπ‘—Μƒπ‘supp(ξπœ‘).

Proof. By the definition of {𝑉𝑗}π‘—βˆˆβ„€, we have π·π‘Žπ‘‰π‘—=𝑉𝑗+1 for any π‘—βˆˆβ„€. It follows that π·π‘Ž(β‹ƒπ‘—βˆˆβ„€π‘‰π‘—β‹ƒ)=π‘—βˆˆβ„€π‘‰π‘—. Note that π·π‘Ž is a unitary operator. Hence π·π‘Ž(β‹ƒπ‘—βˆˆβ„€π‘‰π‘—)=π·π‘Ž(β‹ƒπ‘—βˆˆβ„€π‘‰π‘—)=β‹ƒπ‘—βˆˆβ„€π‘‰π‘—. It is obvious to see that π‘‰π‘—ξβˆ‘={π‘“βˆΆπ‘“=𝑏𝐹𝑏[(π‘π‘Žπ‘—)βˆ’1ξƒ΄πœ‰]ξπœ‘[(π‘π‘Žπ‘—)βˆ’1πœ‰]}, where {𝐹𝑏}π‘βˆˆπΏ2(𝑇𝑛). Then, for any π‘“βˆˆπ‘‰0, we have ξβˆ‘π‘“(πœ‰)=𝑏𝐹𝑏(Μƒπ‘βˆ’1Μƒπ‘πœ‰)ξπœ‘(βˆ’1πœ‰), πΉπ‘βˆˆπΏ2(𝑇𝑛). Notice that for any π‘βˆˆπ΅, Μƒπ‘ξπœ‘(βˆ’1βˆ‘πœ‰)=π‘β„Žπ‘(Μƒπ‘βˆ’1Μƒπ‘Žβˆ’1Μƒπ‘πœ‰)ξπœ‘(βˆ’1Μƒπ‘Žβˆ’1πœ‰). Thus 𝑓(πœ‰)=𝑏1𝐹𝑏1𝑏1βˆ’π‘‡πœ‰ξ€Έξ“π‘2β„Žπ‘2𝑏2βˆ’1Μƒπ‘Žβˆ’1πœ‰ξ‚ξ‚€ξ‚π‘ξπœ‘2βˆ’1Μƒπ‘Žβˆ’1πœ‰ξ‚=𝑏1𝑏2𝐹𝑏1𝑏1βˆ’1πœ‰ξ‚β„Žπ‘2𝑏2βˆ’1Μƒπ‘Žβˆ’1πœ‰ξ‚ξ‚€ξ‚π‘ξπœ‘2βˆ’1Μƒπ‘Žβˆ’1πœ‰ξ‚.(3.8) Put 𝐻𝑏2(Μƒπ‘βˆ’1Μƒπ‘Žβˆ’1βˆ‘πœ‰)=𝑏1𝐹𝑏1(𝑏1βˆ’1πœ‰)𝑀𝑏2(𝑏2βˆ’1Μƒπ‘Žβˆ’1πœ‰). Then, we obtain ξβˆ‘π‘“(πœ‰)=𝑏2𝐻𝑏2(𝑏2βˆ’1Μƒπ‘Žβˆ’1ξ‚π‘πœ‰)ξπœ‘(2βˆ’1Μƒπ‘Žβˆ’1πœ‰). Recalling that 𝐹𝑏(πœ‰)∈𝐿2(𝑇𝑛) and β„Žπ‘(πœ‰)∈𝐿∞(𝑇𝑛), we have 𝑀𝑏2(πœ‰)∈𝐿2(𝑇𝑛), so 𝑓(π‘₯)βˆˆπ‘‰1. Thus, for any 𝑓(π‘₯), we get 𝑉0βŠ‚π‘‰1, so π‘‰π‘—βŠ‚π‘‰π‘—+1. Therefore, for any ⋃𝑓(π‘₯)βˆˆπ‘—βˆˆβ„€π‘‰π‘—, we can choose 𝑗𝑓>0 such that 𝑓(π‘₯)βˆˆπ‘‰π‘—π‘“, that is, βˆ‘π‘“(π‘₯)=π‘βˆ‘π‘˜π‘π‘,π‘˜πœ‘(π‘π‘Žπ‘—π‘“π‘₯βˆ’π‘˜). Hence, for any π‘šβˆˆβ„€π‘›, we have π‘‡π‘šξ“π‘“(π‘₯)=π‘ξ“π‘˜π‘π‘,π‘˜πœ‘ξ‚€π‘π‘Žπ‘—π‘“ξ‚=(π‘₯βˆ’π‘š)βˆ’π‘˜π‘ξ“π‘˜π‘π‘,π‘˜πœ‘ξ‚€π‘π‘Žπ‘—π‘“π‘₯βˆ’π‘π‘Žπ‘—π‘“ξ‚=ξ“π‘šβˆ’π‘˜π‘ξ“π‘˜π‘π‘,π‘˜πœ‘ξ‚€π‘π‘Žπ‘—π‘“ξ‚,π‘₯βˆ’π‘˜(3.9) which implies π‘‡π‘š(β‹ƒπ‘—βˆˆβ„€π‘‰π‘—β‹ƒ)βŠ†π‘—βˆˆβ„€π‘‰π‘—. Thus, π‘‡π‘š(β‹ƒπ‘—βˆˆβ„€π‘‰π‘—β‹ƒ)=π‘—βˆˆβ„€π‘‰π‘— holds. Note that π‘‡π‘˜ is also a unitary operator. Hence, π‘‡π‘š(β‹ƒπ‘—βˆˆβ„€π‘‰π‘—)=π‘‡π‘š(β‹ƒπ‘—βˆˆβ„€π‘‰π‘—)=β‹ƒπ‘—βˆˆβ„€π‘‰π‘—. By Proposition 1.3, β‹ƒπ‘—βˆˆβ„€π‘‰π‘— is a reducing subspace. We notate β‹ƒπ‘—βˆˆβ„€π‘‰π‘—=𝐿2(𝑆)∨. Next we have to prove ⋃𝑆=π‘—βˆˆβ„€β‹ƒπ‘ξ‚π‘Žπ‘—Μƒπ‘supp(ξπœ‘). By πœ‘(π‘π‘Žπ‘—β‹…)βˆˆπ‘‰π‘—, we have supp(ξπœ‘((π‘π‘Žπ‘—)βˆ’1πœ‰))βŠ‚π‘†. Obviously, we will only prove that β‹ƒπ‘†β§΅π‘—βˆˆβ„€β‹ƒπ‘ξ‚π‘Žπ‘—Μƒπ‘supp(ξπœ‘) is a zero measurable set. If β‹ƒπ‘†β§΅π‘—βˆˆβ„€β‹ƒπ‘ξ‚π‘Žπ‘—Μƒπ‘supp(ξπœ‘) is a set with nonzero measure, then a contradiction is led. In fact, choosing a set β‹ƒπ‘€βŠ‚π‘†β§΅π‘—βˆˆβ„€β‹ƒπ‘ξ‚π‘Žπ‘—Μƒπ‘supp(ξπœ‘) with 0<|𝑀|<+∞ and using Plancherel theorem, we have ‖‖𝑃𝑗𝑓‖‖2=𝑃𝑗𝑓,𝑃𝑗𝑓=𝑓,𝑃𝑗𝑓=𝑓𝑓,ξ…žπ‘—ξ‚­,(3.10) for any π‘“βˆˆπΏ2(ℝ𝑛), where π‘“ξ…žπ‘—βˆΆ=𝑃𝑗𝑓. In particular, setting 𝑓(πœ‰)=πœ’π‘€(πœ‰) and taking 𝑗→+∞ in (3.10), we get ‖𝑃𝑗𝑓‖2𝑓=βŸ¨π‘“,ξ…žπ‘—βŸ©=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 π‘ŽβˆˆπΊ and {𝑉𝑗}π‘—βˆˆβ„€ be defined by (2.8). If conditions (1), (2), and (5) of AB-RMRA are satisfied, then the following results are equivalent: (1)β‹ƒπ‘—βˆˆβ„€π‘‰π‘—=𝐿2(𝑆)∨; (2)lim𝑗→+βˆžβ€–π‘ƒπ‘—π‘“β€–2=‖𝑓‖2, where π‘ƒπ‘—βˆΆπΏ2(ℝ𝑛)→𝑉𝑗 denotes an orthogonal projection operator;(3)lim𝑗→+βˆžβˆ‘π‘βˆˆπ΅ξƒ΄|ξπœ‘((π‘π‘Žπ‘—)βˆ’1πœ‰)|2=1, a.e. πœ‰βˆˆπ‘†;(4)⋃𝑆=π‘—βˆˆβ„€β‹ƒπ‘Μƒπ‘Žπ‘—Μƒπ‘suppξπœ‘.

It is well known that β‹‚π‘—βˆˆβ„€π‘‰π‘—={0} can be deduced by the other conditions of MRA. Similarly, the condition β‹‚π‘—βˆˆβ„€π‘‰π‘—={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 π‘ŽβˆˆπΊ, and let 𝐡 be a subgroup of 𝑆𝐿𝑛(β„€) with π‘Žπ΅π‘Žβˆ’1βŠ†π΅. Suppose that πœ‘βˆˆπΏ2(𝑆)∨ and {𝑉𝑗}π‘—βˆˆβ„€ is defined by (2.8). Then πœ‘ is a generator of AB-RMRA if and only if (1)there exists {β„Žπ‘(πœ‰)}π‘βˆˆπ΅βŠ‚πΏβˆž(𝑇𝑛) such that ξ“ξπœ‘(πœ‰)=π‘π‘€π‘ξ‚€Μƒπ‘βˆ’1Μƒπ‘Žβˆ’1πœ‰ξ‚ξ‚€Μƒπ‘ξπœ‘βˆ’1Μƒπ‘Žβˆ’1πœ‰ξ‚;(3.11)(2)βˆ‘π‘˜|ξπœ‘(πœ‰+π‘˜)|2=πœ’πΉ(πœ‰), a.e., where 𝐹={πœ‰βˆˆπ‘‡π‘›βˆ©π‘†βˆΆξπœ‘(πœ‰+π‘˜)β‰ 0,π‘˜βˆˆβ„€π‘›};(3)for any π‘βˆˆπ΅ and 𝑏≠𝐼𝑛, βˆ‘π‘˜ξπœ‘(πœ‰+π‘˜)Μƒπ‘ξπœ‘(βˆ’1(πœ‰+π‘˜))=0, a.e. πœ‰βˆˆπ‘†;(4)⋃𝑆=π‘—βˆˆβ„€β‹ƒπ‘Μƒπ‘Žπ‘—Μƒπ‘supp(ξπœ‘).

Remarks 1. (1) Condition (4) in Theorem 3.4 can be replaced by any one of conditions in Theorem 3.3.
(2) If 𝑆=ℝ𝑛 in Theorem 3.4, then we obtain the characterization of generator of AB-MRA.
(3) If 𝑛=1, π‘Ž=(2), and 𝐡=(1) in Theorem 3.4, then we obtain the characterization of {𝑉𝑗}π‘—βˆˆβ„€ as a generator of MRA on 𝐿2(ℝ).

Corollary 3.5. Let π‘ŽβˆˆπΊ, and let 𝐡 be a subgroup of 𝑆𝐿𝑛(β„€) with π‘Žπ΅π‘Žβˆ’1βŠ†π΅. Suppose that 𝐸 is a bounded nonzero measurable set satisfying Μƒπ‘Žβˆ’1πΈβŠ‚πΈ with ⋃𝑆=π‘—βˆˆβ„€β‹ƒπ‘Μƒπ‘Žπ‘—Μƒπ‘πΈ. Define ξπœ‘(πœ‰)=πœ’πΈ(πœ‰) and 𝐹={πœ‰βˆˆπ‘‡π‘›βˆ©Ξ©βˆΆξπœ‘(πœ‰+π‘˜)β‰ 0,π‘˜βˆˆβ„€π‘›}. Then πœ‘ generates an AB-RMRA if and only if (1)⋃𝐹=π‘˜βˆˆβ„€π‘›(𝐸+π‘˜) with |𝐸∩(𝐸+π‘˜)|=0, |π‘π‘‡πΈβˆ©π‘ξ…žπ‘‡πΈ|=0(π‘β‰ π‘ξ…ž);(2)there exists π‘βˆˆπ΅ such that (π‘π‘Ž)βˆ’1πΈβŠ‚πΈ and [(π‘π‘Ž)βˆ’1𝐸]βˆΌξƒ€βˆ©πΈ=(π‘π‘Ž)βˆ’1𝐸;(3)π‘‡π‘˜πœ’ξƒ€(π‘π‘Ž)βˆ’1𝐸(πœ‰)=πœ’(π‘π‘Ž)βˆ’1𝐸(πœ‰) on (π‘π‘Ž)βˆ’1ξƒ€πΈβˆ©[(π‘π‘Ž)βˆ’1𝐸+π‘˜]β‰ βˆ… for π‘˜βˆˆβ„€π‘›β§΅{0}.

4. Some Examples

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

Example 4.1. Let ξπœ‘(πœ‰)=πœ’πΌ(πœ‰), where 𝐼=𝐼+βˆͺπΌβˆ’, πΌβˆ’={πœ‰βˆˆβ„2βˆ£βˆ’πœ‰βˆˆπΌ+}, and 𝐼+ is a triangle region with vertices (0,0),(𝛼,0),(𝛼,𝛼). Let ξ‚€π‘Ž=2002, 𝐡={1𝑖01ξ‚βˆΆπ‘–βˆˆβ„€}, 𝑆0={(πœ‰1,πœ‰2)βˆˆβ„2∢0β‰€πœ‰1≀𝛼,πœ‰2βˆˆβ„}, π‘†π‘—ξ‚π‘Ž=(𝑗)𝑆0. Define 𝑉𝑗=𝐿2(𝑆𝑗)∨. Then by Corollary 3.5, we get the following:(1)when 𝛼≀1, πœ‘(π‘₯) is a generator for an AB-RMRA;(2)when 𝛼>1, πœ‘(π‘₯) is not a generator.

Example 4.2. Given π‘˜βˆˆβ„€, let 𝐸=[0,𝛽0]2⋃βˆͺ(βˆ’1𝑗=βˆ’π‘˜[2𝑗𝛼,2𝑗𝛽]2), where 0<𝛽0≀2βˆ’π‘˜π›Ό, 𝛼<𝛽≀min{2𝛼,2π‘˜+1𝛽0,2}. If ξπœ‘=πœ’πΈ(β‹…), then πœ‘ generates an AB-RMRA, where 𝑆={(π‘₯,𝑦)∢π‘₯β‰₯0,𝑦β‰₯0}, ξ‚€π‘Ž=0220.

Example 4.3. Let 𝐸=[0,1/4]2βˆͺ[3/8,3/8+πœ€]2, where 0<πœ€β‰€1/16. Assume that ξπœ‘=πœ’πΈ(β‹…). Then πœ‘ generates an AB-RMRA, where 𝑆={(π‘₯,𝑦)∢π‘₯β‰₯0,𝑦β‰₯0}, ξ‚€π‘Ž=2002.


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).


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