Some Conditions and Perturbation Theorem of Irregular Wavelet/Gabor Frames in Sobolev Space

Liu, Hui-Min; Tian, Yu

doi:https://doi.org/10.1155/2024/9932668

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Research Article | Open Access

Volume 2024 | Article ID 9932668 | https://doi.org/10.1155/2024/9932668

Some Conditions and Perturbation Theorem of Irregular Wavelet/Gabor Frames in Sobolev Space

Hui-Min Liu¹and Yu Tian²

Academic Editor: Ding-Xuan Zhou

Received12 Oct 2023

Revised19 Apr 2024

Accepted20 Apr 2024

Published10 May 2024

Abstract

Due to its potential applications in image restoration and deep convolutional neural networks, the study of irregular frames has interested some researchers. This paper addresses irregular wavelet systems (IWSs) and irregular Gabor systems (IGSs) in Sobolev space . We obtain the sufficient and necessary conditions for IWS and IGS to be frames. By applying these conditions, we also derive the characterizations of IWS and IGS to be frames. Finally, we discuss the perturbation theorem of irregular wavelet frames (IWFs) and irregular Gabor frames (IGFs). We also provided some examples to support our results.

1. Introduction

An at most countable sequence in a separable Hilbert space is called a Bessel sequence in if there exists such thatwhere is called a Bessel bound; it is called a frame for if there exists such thatwhere and are called frame bounds. The concept of frame was first proposed by Duffin and Schaeffer when studying the nonharmonic Fourier series in [1]. However, it did not attract people’s attention at that time. Until 1986, Daubechies et al. in [2] noticed that frames can represent the functions in in terms of series expansion. This expansion is very similar to the orthonormal basis expansion, but is more flexible than the orthonormal basis. Many scholars are beginning to realize the potential application of frame theory and frame theory is rapidly developing. So far, the frame theory is widely used in signal and image processing, biomedicine, applied mathematics, physical science, earth science, DCNNs, and many other fields. More details can be found in [2–18] and references therein.

Now the research on frame theory mainly focuses on regular wavelet frame (RWF) and regular Gabor frame (RGF) in and Sobolev space . We recall that for and , two sequences and are called RWS and RGS, if they form frames for , and we say that they are RWF and RGF for , respectively. For , we denote by the Sobolev space consisting of all tempered distributions such that

It is easy to check that is a Hilbert space under the inner product

In particular, by Plancherel theorem. Let denote the Schwartz space and by [19], satisfies the following property: a function .

We first have an overview of RWF and RGF.(i)RWF and RGF for and its subspaces. A core problem of wavelet/Gabor frame theory is what conditions we need to impose on the generator to make the wavelet/Gabor systems to be frames and dual frames. For relevant results about this, including the sufficient and necessary conditions for wavelet/Gabor systems to be frames and the characterizations of dual wavelet/Gabor frames, one can refer to [4, 20–23]. Li and Tian in [24] proposed the concept of partial Gabor systems (PGSs) and studied the conditions for PGS from Gabor frames. They also characterized the dual partial Gabor frames. For the latest research on wavelet/Gabor frame, see [25–28].(ii)RWF for Sobolev spaces. Ehler in [29] presented a method of constructing a pair of dual wavelet frames from any pair of multivariate refinable functions in a pair of Sobolev spaces. Han and Shen in [30] extended the mixed extension principle in to Sobolev Spaces. They in [31] also gave the characterization of the Sobolev spaces by using nonstationary tight wavelet frames for . Li and Zhang in [32] characterized the nonhomogeneous dual wavelet frames in Sobolev space and derived the mixed oblique extension principle. Li and Jia in [33] investigated the properties of weak nonhomogeneous wavelet bi-frames (WNWBF) in the reducing subspaces of a pair of dual Sobolev spaces and constructed the WNWBF. All the compactly supported th-order derivative-orthogonal Riesz wavelets in Sobolev space are completely depicted by Han and Michelle in [34]. For other studies on frames in Sobolev spaces, one can refer to [35–37].(iii)The perturbation of RWF and RGF. Zhang in [38] presented the conditions for a suitable perturbation of a wavelet/Gabor frame which is still a wavelet/Gabor frame. Christensen in [39] studied the stability frames and applied them to the perturbations of a Gabor frame. Sun and Zhou in [40] also obtained some results about the stability of Gabor frames. Bownik and Christensen in [41] characterized the Gabor frames with rational parameters, and as an application, they obtained results concerning the stability of Gabor frames under perturbation of the generators.

In practice, the sampling points may be irregular and it is desirable to have wavelet and Gabor systems in some Sobolev space. This inspires us to study irregular wavelet and Gabor systems in Sobolev space. Given and , we assume that and are two the subsets in . We define the IWS and IGS generated by and as

So, we have

Then, is a dense subset of .

For the research of the IWS and IGS in , Sun and Zhou in [42] constructed the IWF and IGF and gave the sufficient conditions for an IWS and IGS to be a frame. They in [43] also studied the density of IWF. Christensen in [44] gave the different sufficient conditions for IWS and IGS to be frames. For other relevant results, see [45, 46] and the references therein.

Motivated by the existing results mentioned above, we naturally raise a few questions: Are there similar necessary and sufficient conditions for IWS and IGS to be IWF and IGF in Sobolev spaces? How to provide the perturbation characterizations of IWF and IGF in Sobolev spaces? Is it possible to construct some examples to support the relevant results? In this paper, we will address these issues. It is nontrivial due to the more flexibility of the dilation and modulation factor and the complexity of Sobolev spaces.

1.1. Plan of Work

This paper addresses the IWF and IGF of the form (5) and (6) in and the rest of this paper is organized as follows. Section 2 is devoted to some lemmas for later use. In Section 3 and Section 4, we focus on the sufficient and necessary conditions of IWS and IGS to be frames. The characterization of IWS and IGS to be frames under certain restrictions is also obtained. In section 5, we present the perturbation theorem of IWF and IGF. Relevant examples are also presented. Finally, conclusions are drawn in Section 6.

2. Some Auxiliary Lemmas

We give some auxiliary lemmas in this section. The next lemma can be found in [4].

Lemma 1. We assume that and . Then, let be a frame with bounds . If there is a constant such thatthen is a frame for andare the frame bounds.

Lemma 2. Let , , , and . Then, we havefor .

Proof. For arbitrary , we haveBy the periodic process, we haveUsing variable substitution of , we getAfter a simple calculation, we havewhich belongs to , and is an orthonormal basis for , and we havewhere . So, we can obtainThis finishes the proof.
The proof of the following lemma is similar to Lemma 1 and we omit it.

Lemma 3. Let , , , and . Then we havefor .

3. The Sufficient Conditions of IWF and IGF

This section is devoted to the sufficient conditions of IWS and IGS to be frames for and some examples are given. We begin with the sufficient condition of IWS to be a frame.

Theorem 4. Let , , , and . Ifthen forms a Bessel sequence in with bound . If furthermore,then forms a IWF for with bounds and .

Proof. By Lemma 2, we haveWe first estimate asBy Cauchy–Schwartz inequality, we can getActually, , due toThen, we haveTogether with (3.1), (3.3), and (3.4), we haveIn addition, by (19) and (20), we getSo, forms a IWF for with bounds and by (25) and (26). The proof is thus finished.

Corollary 5. Suppose and supp with for each , then we get the following.

(i)Ifthen, forms a IWF in with bounds .(ii)Besides, if forms a IWF in with bounds , then we get

Proof. By Theorem 4, the statement (i) is correct. For (ii), we assume that forms a IWF. By (20), we can getfor all . It follows thatIf there exists with such thaton , then by taking in (30) and the reduction to absurdity, we geta.e. on . By a standard argument, we getwhich holds a.e. on . The other inequality in (28) is similarly provable. We thus finish the proof.

Example 1. Suppose . For each , we assume that almost everywhere with and supp with . We take . Then, if and for every , we can get that forms an IWF for .

Proof. We suppose thatandAlso, supp , so , i.e., , and by , we can obtain . Then, we have only , where is not equal to 0, and this implies that . It follows thatBy a standard argument, if , the series is convergent, and its sum is . Then, we can getThis implies that forms an IWF for by Corollary 5.

Remark 6. Let and satisfy the conditions in Example 1. Taking as the general term of a convergent series and being bounded above, we can obtain that forms an IWF for by Corollary 5. From this, we can construct many useful examples.

Then, we give the sufficient condition of IGF.

Theorem 7. Let , , , and . Ifthen, is a Bessel sequence with bound . Besides, ifthen, forms IGF with bounds and .

Proof. By Lemma 3, , we haveWe first compute asThen, we prove that . Actually,So, we haveBy (40), we haveThus, together with (45), we can obtainHence, we finish the proof by (3.10), (3.11), and (3.17).

Corollary 8. Suppose and supp with for each , then we can get the following.

(i)If then is a IGF for with bounds .(ii)If forms a IGF for with bounds , then we can obtain

Proof. So, (i) is obviously right by Theorem 7. For (ii), by (40), we getfor all . Then, we haveSo,holds a.e. on by a standard argument. We can similarly prove another inequality in (49). Thus, the proof is completed.

Example 2. Suppose and supp with for each , and for arbitrary . Then we can get that forms a IGF for after a simple calculation.

4. The Necessary Condition

We present the necessary conditions of IWF and IGF in . First, we give a necessary condition for to be a IWF for . By Proposition 19.1.3 in [4], if the IWS forms a IWF for , then has to be a finite union of logarithmically separated sets. Then, we get the following theorem.

Theorem 9. Let , , , and . We assume that forms an IWF in and is the upper frame bound. Then, we have

Proof. We just need to prove that for arbitrary and ,So, for arbitrary fix , we getHence,For , is an o.n.b for , where is a closed interval with length . Suppose is a closed interval with and supp . Then, . By Plancherel theorem, we haveWe assume that , and by taking such that and . We choose in (57) such thatThen, we haveIf , we can getby (55). Then, we haveby and the arbitrariness of . The case can be proved similarly. The proof is thus completed.
The next theorem is devoted to the necessary condition of that forms IGF in .

Theorem 10. Let , , , and . We assume that is an IGF for with bounds and . Then, we can get

Proof. Since is an IGF for with bounds and . Then, we can obtainSuppose , then we getWe suppose that with length and supp . Since , so for , such that . Then, we haveSo, andThen by (63), we getSo, we getThis finishes the proof.

5. The Perturbation of IWF and IGF

In this section, we give the perturbation theorems of IWF and IGF.

Theorem 11. Let , , and be given. We assume that is an IWF with bounds . If and satisfy thatthen forms an IWF for with frame bounds

Proof. Fix , Similar to Lemma 2 proof, we getThen, we haveActually, , due toSo, by (69), we haveThus, we finish the proof by Lemma 1.

Example 3. Let and suppose satisfies with and supp with. Then, we assume that satisfies with and supp with and . Taking for each , we have which forms a frame for after calculations similar to Example 1.

Theorem 12. Let , , and be given. We assume that forms an IGF for with bounds . Ifthen forms a IGF for with bounds

Proof. If fix , then we haveBy the periodic process, we getA simple calculation implies thatand is an orthonormal basis for . It follows thatwhereThen, we can obtainSo, we getHence,Similar to the proof mentioned above and using condition (5.2), the proof is thus completed by Lemma 1.

Example 4. Suppose and supp with . We assume for arbitrary . Let and . Then, by taking the function such that and supp , we have which forms a IGF for .

6. Conclusion

In this paper, we introduced the concept of IWS and IGS in Sobolev space . Then, we provided the necessary and sufficient conditions for IWS and IGS to be IWF and IGF in . Using these conditions, we also constructed specific IWF and IGF. At last, we discussed the perturbation theorem of IWF and IGF. The obtained results can provide theoretical reference for the practical application of frame in image restoration and DCNNs [47–50]

Data Availability

No data were used to support the study.

Disclosure

A preprint has previously been published ((Yu Tian, 2022); see [51]).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Science and Technology Research Project of Henan Province (Grant no. 222102210335).

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Copyright © 2024 Hui-Min Liu and Yu Tian. 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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