Nonhomogeneous Wavelet Dual Frames and Extension Principles in Reducing Subspaces

Zhang, Jianping; Jia, Huifang

doi:https://doi.org/10.1155/2020/1716525

Journal of Mathematics

On this page

Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 1716525 | https://doi.org/10.1155/2020/1716525

Nonhomogeneous Wavelet Dual Frames and Extension Principles in Reducing Subspaces

Jianping Zhang¹and Huifang Jia²

Academic Editor: Efthymios G. Tsionas

Received27 Mar 2020

Accepted21 May 2020

Published08 Jun 2020

Abstract

It can be seen from the literature that nonhomogeneous wavelet frames are much simpler to characterize and construct than homogeneous ones. In this work, we address such problems in reducing subspaces of . A characterization of nonhomogeneous wavelet dual frames is obtained, and by using the characterization, an MOEP and an MEP are derived under general assumptions for such wavelet dual frames.

1. Introduction

In view of the great design freedom and the potential applications in signal processing and many other fields, wavelet frames have been extensively investigated by many researchers (see [1–9] for details). In particular, the homogeneous wavelet dual frames (or called affine dual frames) in were original characterized by Han [10] and then characterized by Bownik [11], some of their variations can be found in [12–14]. Mixed extension principles (MEP) give us an important method to construct homogeneous wavelet dual frames from refinable functions, which were proposed by Ron and Shen [15, 16]. When the systems generated by the dilation and integer translation of two functions with refinable structure cannot form frames for their closed linear span, respectively, except two Bessel sequences, construction of homogeneous wavelet dual frames cannot be carried out like the multiresolution analysis (MRA) of Mallat. In such case, the mixed extension principles give an ideal answer to the problem and provide an effective design strategy for stable wavelet filters. Subsequently, mixed extension principles were developed by Daubechies et al. [17] in the form of mixed oblique extension principles (MOEP), which presented a more general method for constructing wavelet dual frames. From then on, the study of MEP and MOEP has interested many researchers [12, 13, 18–25]. Nonhomogeneous wavelet frames have natural connections with refinable structures and filter banks; for this type of wavelet frames, Han in [26–28] and Romero et al. in [29] extensively studied them both in theory and application. Also, notice that all of the above works are focused on the whole space or . In this work, for generality, we address nonhomogeneous wavelet dual frames and extension principles in reducing subspaces of . A characterization of nonhomogeneous wavelet dual frames is obtained, and by using the characterization, an MOEP and an MEP are derived under general assumptions for such wavelet dual frames.

We begin with some notations and notions. We use to represent the set of integers, to represent the set of positive integers, and . Let . We use to represent a -dimensional unit torus, and, given a set on , to represent the Lebesgue measure of , and to represent the characteristic function on . A mapping is defined by

For a function , its Fourier transform is defined byand is naturally extended to , where is the usual inner product on . Similarly, for a function , its inverse Fourier transform is defined byand is naturally extended to . The spectrum of a function is defined to befor , and write as the Dirac sequences such that and for . We use to represent the conjugate transpose for a matrix and to represent the full set of containing the 0, i.e., a set of representatives distinct cosets of containing the 0. A matrix is said to be expansive if the modulus of all its eigenvalues are greater than 1. In many literature studies of wavelet frame theory, is required to be an expansive integer matrix include ( is the identity matrix of order ). The shift operator with and the dilation operator related to on are, respectively, defined by

For a finite set and , the homogeneous wavelet system and the nonhomogeneous wavelet system are, respectively, defined to be

Let be a closed subspace of . Let , , and two finite subsets and of . If there are such thatthen is said to be a homogeneous wavelet frame (HWF) for ; herein and are called frame bounds. In particular, if in (8), then is said to be a tight wavelet frames. If only the inequality on the right side of (8) is valid, then is said to be a Bessel sequence in , herein is called a Bessel bound. In addition, if the homogeneous wavelet systems and are both Bessel sequences in and the identityholds in -sense, then is said to be a pair of homogeneous wavelet dual frames (HWDF) for . Similarly, if there are such thatthen is said to be a nonhomogeneous wavelet frames (NWF) for , herein and are called frame bounds. In particular, if in (10), then is said to be a tight wavelet frames. If only the inequality on the right side of (10) is valid, then is said to be a Bessel sequence in , herein is called a Bessel bound. In addition, if the nonhomogeneous wavelet systems and are both Bessel sequences in and the identityholds in -sense, thenis said to be a pair of nonhomogeneous wavelet dual frames (NWDF) for .

Since the nonhomogeneous wavelet frames have many desired properties, which can be studied in theory and application when they can be linked to homogeneous ones. Following this idea, Han [26] and Atreas et al. [18, 19] have discussed the relations between homogeneous wavelet dual frames and nonhomogeneous ones, and their connection with the refinable structures.

Definition 1. Given a nonzero closed subspace of . If and hold true for all , then is said to be a reducing subspace.
A characterization of reducing subspace in Fourier domain was gained by Dai et al. (Theorem 1 in [30]), and it reads as follows.

Proposition 1. Let be a expansive matrix, and a closed subspace of . is a reducing subspace of if and only if for some and and satisfy , hereinand .
According to Proposition 1, we often use to represent a reducing subspace of rather than . In particular, , and it is a reducing subspace of for a given expansive matrix, and Hardy space is a reducing subspace of for a given expansive factor greater than 1. We refer the interested readers to [10, 31–34] for some related works on HWDFs in for the details.

Definition 2. Let be a expansive matrix. A function is defined to beand set .
Li and Zhang (Lemma 2.3 in [31]) have the following characterization for HWDF in .

Proposition 2. Let be a reducing subspace of , and two finite subsets and of . Suppose that and are both Bessel sequences in . Then, is a pair of HWDF in if and only if, for each , we haveProposition 2 was also proved in [32] without any decay assumptions on the elements of and . Some variations of Proposition 2 can be found in [35–37].
The rest of this work is arranged as below. In Section 2, we provide some necessary lemmas that will be applied in the following section. In Section 3, we give a characterization of NWDF in by a pair of equations, and using this characterization we derive an MOEP and an MEP under general assumptions for such wavelet dual frames.

2. Some Auxiliary Lemmas

By Definition 2, it is easy to check the following lemmas.

Lemma 1. .

Lemma 2. For , we havefor .

Proof. For a.e. with , there exist and such thatSince , there exists such thator equivalently,where . It follows that , and thusNow, let us prove the converse inclusion. Suppose satisfies thatfor some and . Then, there exists such that . This implies that , and thus . The converse inclusion therefore follows. This proof has been completed.

Lemma 3. (see Lemma 2.4 in [10]). Let . If is a Bessel sequence in and bound is , then we have

Lemma 4. Let and . Then, for and , the th Fourier coefficient of is . In particular,if is a Bessel sequence.

Proof. Since , we have , and thusby the Plancherel theorem.
If is a Bessel sequence, then , and thus, we obtain (23) directly from (24). This proof has been completed.

Lemma 5. If is a Bessel sequence in and bound is . Then, we have

Proof. By the Bessel sequence assumption of , we haveBy Lemma 4, we haveand the above equality can be written asReplacing by in the second part of (28) and then separating the series into two parts: and , we obtainby the definition of in (14).
Suppose that (25) is not valid, then there exists with such that on . As a result, we have on some with and . Take such that in (29); then, we obtainby the Plancherel theorem, which contradict with (26). Therefore, we prove that (25) holds. This proof has been completed.

Lemma 6. (see Lemma 3.2 in [36]). Let be dense in a Hilbert space . If and are both Bessel sequences in , then is a pair of dual frames for if and only iffor .

Lemma 7. (see Lemma 2.5 in [37]). Let , , , , we havefor , , where

3. Main Conclusions

The following theorem generalizes Proposition 2. A characterization of NWDF of form (12) in is obtained.

Theorem 1. Let be a reducing subspace of . Let , , and two finite subsets and of . Suppose that and are both Bessel sequences in . Then, is a pair of NWDF for if and only if, for each , we have

Proof. Let be defined as in (33). Since is dense in , we can see from Lemma 6 that is a pair of NWDF for if and only iffor . By Lemma 7, (35) can be written aswhereClearly, (34) implies (36). Now, we need to show that (36) also implies (34) to complete the proof.
Suppose (36) holds. Since , and , , (34) always holds on , so we only need to prove that (34) holds a.e. on . By applying Lemma 5 and the Cauchy–Schwarz inequality, we know the series with converges absolutely a.e. on , as a result of this, belongs to . Therefore, almost every point in is a Lebesgue point. Let be such a point. For , take and such thatin ((36), where . Then, we haveand letting , we obtainFor , take and such thatin (36), where . Then, we haveand letting , we obtainBy the arbitrariness of and , then we obtain (34). This proof has been completed.
Next, by using Theorem 1, we derive an MOEP and an MEP for constructing NWDF in under the following setup: , and satisfy.

Assumption 1. a.e. on .

Assumption 2. and are both -refinable functions, that is, there are two -periodic measurable functions and such thatUnder this setup, define and bywith , and and being -periodic measurable functions.

Theorem 2. Let be a reducing subspace of . Let satisfy Assumptions 1 and 2, and two finite subsets and of defined as in (45). Suppose that and are both Bessel sequences in . Also, suppose there exists a -periodic function , and define by a.e. on . Then, is a pair of NWDF for if and only if

Proof. Since is a Bessel sequence in , especially, we have and are both Bessel sequences in , respectively. By the definition of and Lemma 3, we know is a Bessel sequence in . As a result, is also a Bessel sequence in . So, by Theorem 1, is a pair of NWDF in if and only if, for every , we haveNext, we will show that (48) is equivalent to (46) and (47).
First, we suppose that (46) and (47) hold. For , by Assumption 2 and (45) and the -periodicity of and , we havedue to for . By using (47), (49) can be written asand the last equality is deduced in the same procedure as (49). Continuing this way times, we conclude thatObserve that for some and by Lemma 1, and (51) therefore followsby (47) and the -periodicity of and .
Similarly, for , we have , and thenfor . By applying Lemma 5 and the Cauchy–Schwarz inequality, we know the seriesconverges absolutely for a.e. , and therefore, we haveLetting in (53), we obtainby Assumption 1 and (46).
Next, we prove the converse implication, i.e., (48) implies (46) and (47). Suppose that (48) holds. First, pick , and we have . For any , set , for the choice of , we have because . Since (48) is valid for all , by applying Assumption 2 and (45), we haveby the -periodicity of and . Take with and replace with in (57), where , and it follows thatdue to the -periodicity of and . Since and are arbitrary integers and with , by Lemma 2, there exist some and such thatThen, we obtainFinally, we need to prove (46) holds, and (47) holds for to finish the proof. Taking in (48), we have andReplacing with in (61), we havedue to . By applying Assumption 2 and (45), we haveand thus,Therefore, we obtainOn the contrary, collecting (53), (55), and (61), we conclude thatBy Assumption 1, then we obtain (46). This proof has been completed.

Remark 1. In the literature, the function is related to the notion of mixed fundamental function, which plays a significant role in MOEP.
Let a.e. on in the previous theorem, an immediate consequence is generalization of the MEP, which reads as follows.

Corollary 1. Let be a reducing subspace of . Let satisfy Assumptions 1 and 2, and two finite subsets and of defined as in (45). Suppose that and are both Bessel sequences in . Then, is a pair of NWDF for if and only if

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 11961072, 61861044, 61902339, and 11601290), Natural Science Basic Research Program of Shanxi (Grant no. 2020JM-547), and Doctoral Research Project of Yan’an University (Grant no. YDBK2017-21).

References

D.-Q. Chen, “Regularized generalized inverse accelerating linearized alternating minimization algorithm for frame-based poissonian image deblurring,” SIAM Journal on Imaging Sciences, vol. 7, no. 2, pp. 716–739, 2014.
View at: Publisher Site | Google Scholar
J.-F. Cai, B. Dong, and Z. Shen, “Image restoration: a wavelet frame based model for piecewise smooth functions and beyond,” Applied and Computational Harmonic Analysis, vol. 41, no. 1, pp. 94–138, 2016.
View at: Publisher Site | Google Scholar
R. H. Chan, S. D. Riemenschneider, L. Shen, and Z. Shen, “High-resolution image reconstruction with displacement errors: a framelet approach,” International Journal of Imaging Systems and Technology, vol. 14, no. 3, pp. 91–104, 2004.
View at: Publisher Site | Google Scholar
R. H. Chan, Z. Shen, and T. Xia, “A framelet algorithm for enhancing video stills,” Applied and Computational Harmonic Analysis, vol. 23, no. 2, pp. 153–170, 2007.
View at: Publisher Site | Google Scholar
B. Dong, “Sparse representation on graphs by tight wavelet frames and applications,” Applied and Computational Harmonic Analysis, vol. 42, no. 3, pp. 452–479, 2017.
View at: Publisher Site | Google Scholar
S. Didas, J. Weickert, and J. Weickert, “Integrodifferential equations for continuous multiscale wavelet shrinkage,” Inverse Problems & Imaging, vol. 1, no. 1, pp. 47–62, 2007.
View at: Publisher Site | Google Scholar
S. Mallat, A Wavelet Tour of Signal Processing, Elsevier/Academic Press, Amsterdam, Netherlands, 3rd edition, 2009.
R. Stevenson, “Adaptive solution of operator equations using wavelet frames,” SIAM Journal on Numerical Analysis, vol. 41, pp. 1074–1100, 2004.
View at: Google Scholar
H. Zhang, Y. Dong, and Q. Fan, “Wavelet frame based Poisson noise removal and image deblurring,” Signal Processing, vol. 137, pp. 363–372, 2017.
View at: Publisher Site | Google Scholar
B. Han, “On dual wavelet tight frames,” Applied and Computational Harmonic Analysis, vol. 4, no. 4, pp. 380–413, 1997.
View at: Publisher Site | Google Scholar
M. Bownik, “A characterization of affine dual frames in L²(ℝⁿ),” Applied and Computational Harmonic Analysis, vol. 8, no. 2, pp. 203–221, 2000.
View at: Publisher Site | Google Scholar
O. Christensen, H. O. Kim, and R. Y. Kim, “On extensions of wavelet systems to dual pairs of frames,” Advances in Computational Mathematics, vol. 42, no. 2, pp. 489–503, 2016.
View at: Publisher Site | Google Scholar
O. Christensen, H. Young Kim, and R. Y. Kim, “On Parseval wavelet frames with two or three generators via the unitary extension principle,” Canadian Mathematical Bulletin, vol. 57, no. 2, pp. 254–263, 2014.
View at: Publisher Site | Google Scholar
M. Ehler and B. Han, “Wavelet bi-frames with few generators from multivariate refinable functions,” Applied and Computational Harmonic Analysis, vol. 25, no. 3, pp. 407–414, 2008.
View at: Publisher Site | Google Scholar
A. Ron and Z. Shen, “Affine systems in L²(ℝ^d): the analysis of the analysis operator,” Journal of Functional Analysis, vol. 148, no. 2, pp. 408–447, 1997.
View at: Publisher Site | Google Scholar
A. Ron and Z. Shen, “Affine systems in L²(ℝ^d) II: dual systems,” The Journal of Fourier Analysis and Applications, vol. 3, no. 5, pp. 617–637, 1997.
View at: Publisher Site | Google Scholar
I. Daubechies, B. Han, A. Ron, and Z. Shen, “Framelets: MRA-based constructions of wavelet frames,” Applied and Computational Harmonic Analysis, vol. 14, no. 1, pp. 1–46, 2003.
View at: Publisher Site | Google Scholar
N. D. Atreas, M. Papadakis, and T. Stavropoulos, “Extension principles for dual multiwavelet frames of L²(ℝ^s) constructed from multirefinable generators,” Journal of Fourier Analysis and Applications, vol. 22, no. 4, pp. 854–877, 2016.
View at: Publisher Site | Google Scholar
N. Atreas, A. Melas, and T. Stavropoulos, “Affine dual frames and extension principles,” Applied and Computational Harmonic Analysis, vol. 36, no. 1, pp. 51–62, 2014.
View at: Publisher Site | Google Scholar
M. Bownik and Z. Rzeszotnik, “Construction and reconstruction of tight framelets and wavelets via matrix mask functions,” Journal of Functional Analysis, vol. 256, no. 4, pp. 1065–1105, 2009.
View at: Publisher Site | Google Scholar
C. K. Chui and W. He, “Compactly supported tight frames associated with refinable functions,” Applied and Computational Harmonic Analysis, vol. 8, no. 3, pp. 293–319, 2000.
View at: Publisher Site | Google Scholar
I. Daubechie and B. Han, “Pairs of dual wavelet frames from any two refinable functions,” Constructive Approximation, vol. 20, no. 3, pp. 325–352, 2004.
View at: Publisher Site | Google Scholar
S. S. Goh and K. M. Teo, “Extension principles for tight wavelet frames of periodic functions,” Applied and Computational Harmonic Analysis, vol. 25, no. 2, pp. 168–186, 2008.
View at: Publisher Site | Google Scholar
T. Stavropoulos, “The geometry of extension principles,” Houston Journal of Mathematics, vol. 38, pp. 833–853, 2012.
View at: Google Scholar
Z. Zhang and N. Saito, “Constructions of periodic wavelet frames using extension principles,” Applied and Computational Harmonic Analysis, vol. 27, no. 1, pp. 12–23, 2009.
View at: Publisher Site | Google Scholar
B. Han, “Homogeneous wavelets and framelets with the refinable structure,” Science China Mathematics, vol. 60, no. 11, pp. 2173–2198, 2017.
View at: Publisher Site | Google Scholar
B. Han, “Nonhomogeneous wavelet systems in high dimensions,” Applied and Computational Harmonic Analysis, vol. 32, no. 2, pp. 169–196, 2012.
View at: Publisher Site | Google Scholar
B. Han, “Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space,” Applied and Computational Harmonic Analysis, vol. 29, no. 3, pp. 330–353, 2010.
View at: Publisher Site | Google Scholar
J. R. Romero, S. K. Alexander, S. Baid, S. Jain, and M. Papadakis, “The geometry and the analytic properties of isotropic multiresolution analysis,” Advances in Computational Mathematics, vol. 31, no. 1–3, pp. 283–328, 2009.
View at: Publisher Site | Google Scholar
X. Dai, Y. Diao, Q. Gu, and D. Han, “Frame wavelets in subspaces of L²(ℝ^d),” Proceedings of the American Mathematical Society, vol. 130, no. 11, pp. 3259–3268, 2002.
View at: Publisher Site | Google Scholar
Y.-Z. Li and J.-P. Zhang, “Extension principles for affine dual frames in reducing subspaces,” Applied and Computational Harmonic Analysis, vol. 46, no. 1, pp. 177–191, 2019.
View at: Publisher Site | Google Scholar
Y. Z. Li and F. Y. Zhou, “Affine and quasi-affine dual wavelet frames in reducing subspaces of L²(ℝ^d),” Acta Mathematica Sinica,Chinese Series, vol. 53, pp. 551–562, 2010.
View at: Google Scholar
Y.-Z. Li and F.-Y. Zhou, “GMRA-based construction of framelets in reducing subspaces of L²(ℝ^d),” International Journal of Wavelets, Multiresolution and Information Processing, vol. 09, no. 02, pp. 237–268, 2011.
View at: Publisher Site | Google Scholar
H. Volkmer, “Frames of wavelets in Hardy space,” Analysis, vol. 15, pp. 405–421, 1995.
View at: Publisher Site | Google Scholar
H. Jia and Y. Li, “Weak (quasi-)affine bi-frames for reducing subspaces of L²(ℝ^d),” Science China Mathematics, vol. 58, no. 5, pp. 1005–1022, 2015.
View at: Publisher Site | Google Scholar
Y. Tian and Y. Li, “Subspace dual super wavelet and Gabor frames,” Science China Mathematics, vol. 60, no. 12, pp. 2429–2446, 2017.
View at: Publisher Site | Google Scholar
J. P. Zhang and Y. Z. Li, “On a class of weak nonhomogeneous affine bi-frames for reducing subspaces of L²(ℝ^d),” Acta Mathematica Sinica, English Series, vol. 33, no. 10, pp. 1339–1351, 2017.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Jianping Zhang and Huifang Jia. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

199

Downloads

652

Citations