A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

Zhang, Jianping

doi:https://doi.org/10.1155/2022/1372184

Journal of Function Spaces

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1372184 | https://doi.org/10.1155/2022/1372184

A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

Jianping Zhang¹

Academic Editor: Alberto Fiorenza

Received21 Aug 2021

Accepted06 Dec 2021

Published19 Jan 2022

Abstract

In recent years, dual wavelet frames derived from a pair of refinable functions have been widely studied by many researchers. However, the requirement of the Bessel property of wavelet systems is always required, which is too technical and artificial. In present paper, we will relax this restriction and only require the integer translation of the wavelet functions (or refinable functions) to form Bessel sequences. For this purpose, we introduce the notion of weak dual wavelet frames. And for generality, we work under the setting of reducing subspaces of Sobolev spaces, we characterize a pair of weak dual wavelet frames, and by using this characterization, we obtain a mixed oblique extension principle for such weak dual wavelet frames.

1. Introduction

Let be a separable Hilbert space. An at most countable sequence in is called a frame for if there exist two constants such that where and are called lower and upper frame bounds; it is called a Bessel sequence in if the right-hand side inequality in (1) holds, where is called a Bessel bound. Given a frame for , a sequence is called a dual of if it is a frame such that

It is well-known that is also a dual of if is a dual of . So, in this case, we also say is a pair of dual frames. The fundamentals of frames can be found in [1–3]. We denote by , , and the set of integers, the set of positive integers, and the set of nonnegative integers, respectively. Let and a expansive matrix (an integer matrix with all its eigenvalues being greater than in modulus). The dilation operator and the shift operator with are, respectively, defined by for , and naturally extended to tempered distribution. In what follows, we denote by the transpose of , and by a set of representatives of distinct cosets of with .

The Fourier transform of a function is defined by and naturally extended to tempered distribution, where denotes the Euclidean inner product in . Similarly, the inverse Fourier transform of a function is defined by and extended to tempered distribution as usual. For a real number , we denote by the Sobolev space consisting of all distributions such that where denotes its Euclidean norm for . It is easy to check that is a Hilbert space under the inner product

In particular, by the Plancherel theorem. For with nonzero measure, we write

Then, . Obviously, for each , defines a continuous linear function on . Then, forms a pair of dual spaces; so does by taking . Write

It is well known that is dense in and is dense in for every .

Definition 1. Given a real number and a expansive matrix , a nonzero closed linear subspace of is called a reducing subspace if and for each , and where .

In particular, when , (12) is trivial and Definition 1 reduces to one in [4], which is characterized in Fourier domain as follows.

Proposition 2 ([4], (Theorem 1)). For a expansive matrix , is a reducing subspace of if and only if for some with nonzero measure satisfying .

Proposition 3 ([4], (Theorem 2.1)). Let be a real number and a expansive matrix. Then, is a reducing subspace of if and only if for some with nonzero measure satisfying .

So, to be specific, we denote a reducing subspace of by instead of . Given a distribution , we write for and . Let and be reducing subspaces of and , respectively, and for , , and finite subsets of and , respectively, the wavelet systems and are defined as

We say is a pair of dual wavelet frames for if (1) is a wavelet frame for and is a wavelet frame for (2)The identity

holds for and .

We say is a pair of weak dual wavelet frames for if (1) are Bessel sequences in and , respectively(2)There exist dense subsets of and of such that

where the series converges in the following sense: both and converge unconditionally (equivalently, both and converge), and

It is obvious that the convergence of series in (16) is weaker than that in (15). And a pair of dual wavelet frames must be a pair of weak dual wavelet frames, whereas the converse is not true. Also, observe that in the above definition of weak dual wavelet frames, need not belong to , and need not belong to .

Due to the great design freedom and the potential applications in signal denoising, image restoration, numerical analysis, etc., the study of wavelet frames for and Sobolev spaces has been attracting many researchers and seen great achievements (see [5–16] for details). In particular, Bownik in [6] obtained the following important characterization for homogeneous dual wavelet frames:

Proposition 4. Let and be Bessel sequences in . Then, is a pair of dual wavelet frames for if and only if where is defined by Definition 6 in Section 2.

Li and Zhang in [17] generalized Proposition 4 to Sobolev space pairs for nonhomogeneous dual wavelet frames:

Proposition 5. Given , let and be Bessel sequences in and , respectively. Then, is a pair of dual frames in if and only if

An important method to construct (dual) wavelet frames from refinable functions is extension principles. Ron and Shen in [15, 16] prosed the unitary extension principle (UEP) and the mixed extension principle (MEP). Subsequently, Daubechies et al. in [10] developed them in the form of the oblique extension principle (OEP) and the mixed oblique extension principle (MOEP). From then on, the study of the extension principles has interested many researchers [4, 5, 7, 8, 11, 18–20].

Observe that all above works, the wavelet systems (or the refinable functions) are required to be Bessel sequences. In order to achieve the Bessel property, some conditions have to be imposed on the wavelet systems (or the refinable functions) that are too technical and artificial. It is natural to ask what are expected from general refinable functions without too many restrictions. For this purpose, Jia and Li in [21] introduced the nation of weak wavelet biframes (weak dual wavelet frames). Starting from a pair of general refinable functions without smoothness restrictions, they obtained a construction of weak dual wavelet frames for reducing subspace of .

Inspired by all these works, in present paper, we investigate a class of weak dual wavelet frames for reducing subspaces of Sobolev spaces. In Section 2, we first give some necessary lemmas, and then, we give a Fourier-domain characterization of weak dual wavelet frames in associated with . In Section 3, by using the above characterization, we derive a mixed oblique extension principle for such weak dual wavelet frames.

Before proceeding, we introduce some necessary notations. We denote by the -dimensional torus and, for a Lebesgue measurable set in , by its Lebesgue measure and its characteristic function, respectively, by the Dirac sequence such that and for , and denote by the mapping from to defined by

For functions and on , we define if it is well defined, and the spectrum by

It is obvious that if and only if and for some . So is independent of . For simplicity, we use to replace .

2. The Characterization of Weak Dual Wavelet Frames

This section is devoted to characterizing weak dual wavelet frames in . Fist, we give some necessary lemmas for later use.

Definition 6. Let be a expansive matrix. Define a function by and set .

By a standard argument, we have the following three lemmas:

Lemma 7.

Lemma 8. For and , we have

Lemma 9. Let and be two complex sequences, and . Then,

Lemma 10 ([4], (Lemma 3.1)). Given , a expansive matrix , and , we have the following: (i) is a Bessel sequence in if and only if . In this case, is a Bessel bound(ii)If is a Bessel sequence in , then is a Bessel sequence in with the Bessel bound for , andfor , where .

Lemma 11. Let , , , and . Then, for , the -th Fourier coefficient of is . In particular, if is a Bessel sequence in .

Proof. Since and , we have , and thus, by the Plancherel theorem. So the -th Fourier coefficient of is .
If is a Bessel sequence in , then is a Bessel sequence in by Lemma 10 (ii). It follows that , and thus, (27) holds.☐☐

Lemma 12 ([4], (Lemma 3.5)). Let be a bounded set in . Then, there exist finite sets and such that

Lemma 13. Given , let and be Bessel sequences in and , respectively. Then, for , where is defined by (11).

Proof. By Lemma 11, we have Write Then, by the Cauchy-Schwarz inequality, we have By a similar procedure, we also have If , then and with are Bessel sequences in . Also, observe that and are Bessel sequences in , and thus, by Lemma 10 (i). It follows that by the Fubini-Tonelli theorem. Therefore, we get By using the Cauchy-Schwarz inequality, we have which belongs to by Lemma 10 (i) since is also a Bessel sequence in if , and thus, Since , then there exists a bounded set in such that . By Lemma 12, there exist finite sets and such that Therefore, we have Write Then, for and , we have Also, observe that and . It follows that Therefore, we can change the order of summation and integration in (40). where Lemma 9 is used in the last equality. And thus, the conclusion follows by collecting (36), (38), and (44).☐☐

The following theorem characterizes a pair of weak dual wavelet frames for .

Theorem 14. Let and be reducing subspaces of and , respectively, and , and , finite subsets of and . Suppose and are both Bessel sequences in and , respectively. Then, is a pair of weak dual wavelet frames for if and only if

Proof. Since is dense in for every , then is a pair of weak dual wavelet frames for if and only if for , or equivalently for due to . By the Bessel assumptions, we know that the series and are absolutely convergent for since is dense both in and . By Lemma 13, (48) can be written as Obviously, (45) and (46) imply (49). Next, we prove the converse implication to complete the proof. Suppose (49) holds. For , the function and thus, almost every point in is a Lebesgue point of all function with . Let be such a point. For and , take and such that in (49), where . Then, we have or equivalently, since if and belong to when . Letting in (53), we obtain By the arbitrariness of and , then we have And thus, (46) holds since the above function vanishes out of . Now, we prove (45) holds.
Arbitrarily fix , compact , then there exists a finite set such that . Define for . Then, we have , and thus, (45) holds for if it holds for each with . Next, we prove (45) holds for each with . Take and in (49) such that Then, we obtain (45) therefore holds for and holds for . By the arbitrariness of , we obtain (45). The proof is completed.☐☐

3. Constructing Weak Dual Wavelet Frames from a Pair of General Refinable Functions

This section is devoted to constructing weak dual wavelet frames for from a pair of general refinable functions under the following assumptions:

Assumption 1. and are -refinable functions with symbols in , i.e., there exists such that

Assumption 2. .

Assumption 3. for some , where denotes the -neighborhood of the origin .

Given , let with define and by

Remark 15. By Lemma 10 (i), Assumption 2 is equivalent to the fact that and are Bessel sequences in and , respectively. It is easy to check that and are also Bessel sequences in and , respectively.

Theorem 16. Let and be reducing subspaces of and , respectively, and two functions satisfy Assumptions 1–3, and let and be two finite subsets of and , respectively, defined by (59). Assume that there exists a function such that

Then, is a pair of weak dual wavelet frames for , where is defined by .

Proof. First, we claim that the following equation holds: a.e. on for and . Indeed, by Assumption 1 and (59) and the -periodicity of with , we have So (62) holds if a.e. on with . Fix with . Suppose for some and . Then, by Lemma 8, and thus, by (61). This is equivalent to (64) by the -periodicity of , , and with . Therefore, (62) holds.
By Theorem 14, to complete the proof, it is enough to prove that for , compact , and We first prove (67). For , by using (62), we have By Lemma 7, we have for some and . It follows that by (61), and (67) therefore holds.
Next, we prove (66) holds. By using (62), we have where (61) was used in the last equality. And thus, for with compact . Since is compact and , then there exists such that for and . And thus, there exists a constant such that by Assumption 3. This implies that by Assumptions 1 and 3. Therefore, (66) holds by collecting (71) and (60).☐☐

Remark 17. (i)In the literature, the function is related to the notion of mixed fundamental function, which plays a significant role in MOEP.(ii)In the above theorem, (60) is trivial and is a pair of weak dual wavelet frames for if a.e. on

4. Conclusion

In this paper, we introduce the notion of weak dual wavelet frames. And for generality, we work under the setting of reducing subspaces of Sobolev spaces, we characterize a pair of weak dual wavelet frames, and by using this characterization, we obtain a mixed oblique extension principle for such weak dual wavelet frames.

Data Availability

There is no data in our paper.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

The author read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11961072 and 62041212), the Natural Science Basic Research Program of Shanxi (Grant Nos. 2020JM-547 and 2020JM-548), and the Doctoral Research Project of Yan’an University (Grant No. YDBK2017-21).

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Copyright © 2022 Jianping Zhang. 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.

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