Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 850850, 13 pages

http://dx.doi.org/10.1155/2011/850850

## Characterization of Generators for Multiresolution Analyses with Composite Dilations

^{1}School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China^{2}Basic Department, Henan Quality and Engineering Vocational College, Pingdingshan 467000, China^{3}School 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.

#### 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 (AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspace . 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 [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 is a generator for an MRA if and only if (1), a.e. ; (2), a.e. ;(3)there exists such that , a.e. .

If condition (2) is replaced by or another condition that the function 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 invertible matrix , , and , we define the dilation operator and the shift operator on by The affine system with composite dilations is defined by where . By choosing , , and appropriately, we can make an orthonormal basis or, more generally, a Parseval frame (PF) for [5β7]. In this case, is called an AB-multiwavelet or a PFββAB-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 by .

Before proceeding, we need some conventions. We denote by the -dimensional torus. For a Lebesgue measurable set in , we denote by its measure, denote by the characteristic function of , and define . An matrix is called an expansive matrix if it is an integer matrix with all its eigenvalues greater than 1 in the module. denotes the set of all expansive matrices. We denote by the set is an integral matrix and , by the set is an integral matrix and , and by the set of the subgroups of , respectively. For a Lebesgue measurable function , we define its support by The Fourier transform of is defined by on , where denotes the inner product in . Let be a Lebesgue nonzero measurable set in . We denote by the closed subspace of of the form

*Definition 1.1 (see [8, 9]). *The sequence in a separable Hilbert space is called a semiorthogonal PF for if is a PF for and satisfies for any , , and , where , are two countable index sets. In particular, if is a semiorthogonal PF for , it is called a semiorthogonal sequence.

*Definition 1.2 (see [4, 10]). *A closed subspace of is called a reducing subspace if and for any , .

The following proposition provides a characterization of reducing subspace.

Proposition 1.3 (see [4, 10]). *A closed subspace of is a reducing subspace if and only if
**
for some measurable set with . So, to be specific, one denotes a reducing subspace by instead of . In particular, is a reducing subspace of .*

*Definition 1.4 (see [5β7]). *Let be a subgroup of the integral affine group (the semidirect product of and ). The closed subspace of is called a invariant subspace if for any .

*Definition 1.5 (see [2β4]). *Let be a countable subset of and where . We say that a sequence of closed subspaces of is an AB-MRA if the following holds: (1) is a invariant space; (2)for each , , and ; (3);
(4)(5)there exists such that is a semiorthogonal PF for .

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

Similarly, we say that a sequence is an AB-RMRA if it is an AB-MRA on , that is, conditions (1), (2), (4), (5), and (3)β² ββ 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 be a countable index set and the orthogonal projection operator from a Hilbert space to its proper subspace . If is a Parseval frame on , then is a Parseval frame on .*

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

Theorem 1.8. *Suppose that is an AB-MRA, then is an AB-RMRA for , where , , , and is the orthogonal projection operator from to .*

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 is a Parseval frame sequence by Propositions 1.6 and 1.7. Naturally, is a semi-Parseval frame for . Let . Thus . For any , we have namely, . So is a invariant space. On the other hand, So . Notice that . Then . Thus, conditions (1), (2), and (5) in Definition 1.5 have been proved. However, condition (3)β² 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 [5β7], will be presented. The first one can be obtained obviously by the definition of AB-RMRA as follows.

Proposition 2.1. *Suppose that is an AB-RMRA. Then *(1)*for each , is a semiorthogonal PF on ;*(2)* is a invariant subspace, while is a invariant subspace.*

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

Proposition 2.2. *Let . Then is a semiorthogonal PF sequence if and only if *(1)*, a.e., where ;*(2)*, a.e. , for each and .*

*Proof. **Necessity*. For any , we have
By Theoremββ1.6 in [1] and Theoremββ7.2.3 in [8], conclusion (1) holds clearly. Using Parseval theorem, we can deduce
where . Note that for any , , if and only if for any and ,

Then, we have
*Sufficiency. *By Theoremββ7.2.3 in [8] and conclusion (1), is a PF sequence. So is for any . It follows from (2.4), (2.5), and conclusion (2) that for any , , and , we get . Thus, for any , there exists and such that
By Theoremββ1.6 in [1], the proof of Proposition 2.2 is completed.

Proposition 2.3. *Let be a sequence of closed subspace of , where
**
If conditions (1), (2), and (5) of AB-RMRA are satisfied, then one has the following. *(1)*There exists such that
*(2)*There exists such that
for any , where *

*Proof. *By conditions (2) and (5) of AB-RMRA and the fact that , we obtain
where and . Therefore (2.9) holds. Taking Fourier transform on both sides of (2.9), we obtain (2.10), where for any , . In what follows, we will only prove . Indeed, for is a subgroup of and the quotient group has order . Thus, we can choose a complete set of representatives of , that is, the set so that each can be uniquely expressed in the form with , . For simplicity, we denote and by and , respectively. Then we have
where (2.12) is obtained by the periodicity of function sequence 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 , we get . Then, for , , so . 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 .

Lemma 3.1. *Let be a sequence of closed subspaces of 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)*;
*(2)*, a.e. .*

*Proof. *Theoremsββ1.7 andββ5.2 in [1] imply that for any , is equivalent to . Thus, for any , is equivalent to . Hence, we have to prove that is equivalent to , a.e. . First, we prove (1)*β*(2) For any , , where is the orthogonal projection operator. Set . Then, when is large enough, we have
Before proving the equivalence, we need to prove two assertions as follows:(i);
(ii) makes sense.Since
it follows that (i) holds.

For (ii), we will only prove that is a monotonic bounded sequence when is fixed. Indeed, by the orthogonality, for each , we have . In addition, we deduce from (2.10) that, for any ,

Set . Then by the orthogonality and Proposition 2.3, we obtain
Hence, is a monotonic sequence when is fixed. On the other hand, by the property of , we deduce that . Note that is a PF for for any . Then is a PF on , which implies that there exists a set such that . By the orthogonality, , consequently . Hence, holds for . So exists. Now we have proved the two assertions. By the Lebesgue dominant convergence theorem, we get . Thus, , a.e. .

Next, we prove (2)*β*(1). Let be the class of all functions such that and is compactly supported in . If we can show that for all , then, by Lemmaββ1.10 in [1], the proof is finished. Indeed, denoting by , we have
where . Since has compact support, when is large enough, , consequently, . Thus, taking in (3.5), we obtain

Lemma 3.2. *Let , satisfy (2.10), and let be defined by (2.8). Then
**
where .*

*Proof. *By the definition of , we have for any . It follows that . Note that is a unitary operator. Hence . It is obvious to see that , where . Then, for any , we have , . Notice that for any , . Thus
Put . Then, we obtain . Recalling that and , we have , so . Thus, for any , we get , so . Therefore, for any , we can choose such that , that is, . Hence, for any , we have
which implies . Thus, holds. Note that is also a unitary operator. Hence, . By Proposition 1.3, is a reducing subspace. We notate . Next we have to prove . By , we have . Obviously, we will only prove that is a zero measurable set. If is a set with nonzero measure, then a contradiction is led. In fact, choosing a set with and using Plancherel theorem, we have
for any , where . In particular, setting and taking in (3.10), we get , but , and this is a contradiction. Therefore, we complete the proof of Lemma 3.2.

By Lemmas 3.1 and 3.2 and Theoremsββ1.7 andββ5.2 in [1], we characterize the density condition of AB-RMRA as follows.

Theorem 3.3. *Let and be defined by (2.8). If conditions (1), (2), and (5) of AB-RMRA are satisfied, then the following results are equivalent: *(1)*;
*(2)*, where denotes an orthogonal projection operator;*(3)*, a.e. ;*(4)*. *

It is well known that can be deduced by the other conditions of MRA. Similarly, the condition 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 , and let be a subgroup of with . Suppose that and is defined by (2.8). Then is a generator of AB-RMRA if and only if *(1)*there exists such that
*(2)*, a.e., where ;*(3)*for any and , , a.e. ;*(4)*.*

*Remarks 1. *(1) Condition (4) in Theorem 3.4 can be replaced by any one of conditions in Theorem 3.3.

(2) If in Theorem 3.4, then we obtain the characterization of generator of AB-MRA.

(3) If , , and in Theorem 3.4, then we obtain the characterization of as a generator of MRA on .

Corollary 3.5. *Let , and let be a subgroup of with . Suppose that is a bounded nonzero measurable set satisfying with . Define and . Then generates an AB-RMRA if and only if *(1)* with , ;*(2)*there exists such that and ;*(3)* on for .*

#### 4. Some Examples

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

*Example 4.1. *Let , where , , and is a triangle region with vertices . Let , , , . Define . Then by Corollary 3.5, we get the following:(1)when , is a generator for an AB-RMRA;(2)when , is not a generator.

*Example 4.2. *Given , let , where , . If , then generates an AB-RMRA, where , .

*Example 4.3. *Let , where . Assume that . Then generates an AB-RMRA, where , .

#### 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).

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