Abstract
This article studies the stability of woven frames by introducing some special limits. We show that there exist certain relations among the different types of convergence of frames and obtain some new and more general stability results. As an application of these results, we provide a method for constructing woven frames.
1 Introduction
Woven frames with some applications in coding and decoding [6, 9], distributed signal processing [6], and wireless sensor networks [1, 4] were first introduced in 2015 by Bemrose, Casazza, Grochenig, Lammers, and Lynch [1, 4]. Right now, it has been generalized to g-frames [10], K-frames [5, 11], fusion frames [8], etc.
Definition 1.1 A family for separable Hilbert space H is said to be a frame if there exist 0 < A ≤ B < ∞such thatwhereA, Bare the lower frame bound and upper frame bound, respectively.If only the second inequality is required, it is called a Bessel sequence, and the B is called the Bessel sequence bound. For a Bessel sequence , the synthesis operator is defined byis bounded. Its adjoint operator T∗ is called the analysis operator. The composite operator S = TT∗ is bounded, positive, and self-adjoint, and it is called the frame operator while is a frame for H.
Definition 1.2 The frames family for separable Hilbert space H is woven, if there are universal constants 0 < A ≤ B < ∞such thatfor every partitionof. The familyis called a weaving for every partitionof.We note that and m ≥ 2, [m] = {1, 2, … , m}, is a partition of , is any partition of . H is a separable Hilbert space, and are the Bessel sequences for H, and are the families of Bessel sequences. The synthesis operators of , , , are listed as follows:Moreover, and for any f ∈ H and .This article focuses on the stability of woven frames, i.e., answers the following question: Suppose that the frames family is woven for H with universal lower and upper bounds A and B. We want to find some conditions about such that the family is woven for H. In m = 2 case, this question was first considered by Bemrose, Casazza, Grochenig, Lammers, and Lynch [1, 4], after that it was reconsidered by Ghobadzadeh, Najati, Anastassiou, and Park [7], right now their results have been generalized to K-frames, fusion frames, g-frames, and so on. Analyzing the existing results, it is not difficult to find that they are all based on sufficiently small perturbation. In order to explore the relations among them and generalize them from m = 2 to 2 ≤ m < ∞, we introduce four types of convergence of frames .
1.1 Strong Convergence
In this case, the limit is unique. Naturally, if the frames family is woven for H, then all in a sufficiently small neighborhood of is woven for H.
1.2 Convergence in Terms of Synthesis Operator or Analysis Operator
Note that the two types of convergence are equivalent. In both cases, the limit is not necessarily unique, but the synthesis operators of frames in are unique, this means that the corresponding analysis operators are also unique. Similarly to the first case, if the frames family is woven for H, then is woven for all sufficiently big k.
1.3 Convergence in Terms of Frame Operator Sσ
In this case, the limit , the corresponding synthesis operators, and the analysis operators of frames in limit are not necessarily unique, but the universal infimum and supremum of are unique. This implies that the judgment theorem about still holds.
It can be proved that type 1 implies type 2 and 3, type 2 and 3 imply type 4, but the reverse is not true. More generally, we conjecture that there probably exist other types of convergence and some different results about the stability of woven frames can be obtained from the new type of convergence.
This article is organized as follows: In Section 2, we introduce some special limits for woven frames and show the relations among different types of convergence of frames. In Section 3, we show some new results about the stability of woven frames.
2 Convergence for Woven Frames
In this section, we introduce four types of convergence for woven frames and discuss the relations among them.
Definition 2.1 We say a point sequence strongly converges to the point if
Definition 2.2 We say a point sequence converges to the point in terms of synthesis operator if
Definition 2.3 We say a point sequence converges to the point in terms of analysis operator if It is known that , thus converges to in terms of synthesis operator if and only if converges to in terms of analysis operator.
Definition 2.4 We say a point sequence converges to the point in terms of frame operator S σ if Next, we show the relations among the four types of convergence in Theorem 2.5 and Theorem 2.6.
While strongly converges to or converges to in terms of analysis operator or synthesis operator, we have for all .
Proof. Fromfor all and j ∈ [m], we can obtain this theorem.
If strongly converges to then converges to in terms of analysis operator; converges to in terms of analysis operator if and only if converges to in terms of synthesis operator; If converges to in terms of synthesis operator then converges to in terms of frame operator Sσ.
Proof. It is obvious that converges to in terms of synthesis operator if and only if converges to in terms of analysis operator. For all f ∈ H and ‖f‖ = 1, we compute
From we have This means that if strongly converges to then converges to in terms of analysis operator. We compute
Note thatsofor some positive Mj. Furthermore,
From we have It means that if converges to in terms of synthesis operator then converges to in terms of frame operator Sσ.
The following Example 2.7 and Example 2.8 show that the inverse proposition of Theorem 2.6 is untenable.
Example 2.7Let be an orthonormal basis for H. If for all , then converges to in terms of analysis operator, but does not strongly converge to . Proof. Fromwe have , i.e., converges to in terms of analysis operator. Fromwe have that does not strongly converge to .
Example 2.8Let be a Parseval frame for H. If for all , then converges to in terms of frame operator Sσ, but does not converge to in terms of synthesis operator. Proof. We computefor all andFrom Definition 2.4 and Theorem 2.5, we complete the proof.
3 Stability of Woven Frames
This section discusses the stability of woven frames by limits in Section 2. We generalize the existing results from m = 2 to 2 ≤ m < ∞. Moreover, many new woven frames can be obtained by using the limits.
In Theorem 3.1, Corollary 3.2, and Theorem 3.4, we discuss the stability of woven frames in terms of frame operator Sσ.
Suppose that the frames family is woven for H with universal bounds A and B. If converges to in terms of frame operator Sσ then for any non-negative number ɛ < A there exists a natural number N such that for every k > N, this implies that is woven for H with universal bounds A − ɛ and B + ɛ for every k > N.
Proof. If converges to in terms of frame operator Sσ, from the Definition 2.4, for any ɛ < A there exists a natural number N such that for all k > N. Hence,andi.e., for all f ∈ H and σ ∈ Ω. This implies that the bounded linear operator is an injection. It is known that the operator is self-adjoint, thus is also a surjection. From we have is a surjection, this implies that is a frame for H with the frame operator . Hence is woven with universal bounds A− ɛ and B+ ɛ for every k > N.
Corollary 3.2Suppose that the frames family is woven for H with universal bounds A, B and is a family of Bessel sequences for H. If there exist non-negative numbers αj, μj satisfied such thatfor any j ∈ [m] and σ ∈ Ω, then is woven for H with the universal lower and upper bounds A− ɛ and B+ ɛ. Proof. Sinceandcombing with the inequality in Corollary 3.2, we haveTake and can be regarded as for some k > N, from Theorem 3.1, we obtain Corollary 3.2.
Example 3.3Suppose that the frames family is woven for H with the universal bounds A, B and is an orthonormal basis for H. If the number λ, δ satisfiedthen the family is woven for H with the universal lower and upper bounds . Proof. Let gij = δei − λfij for all . Then is a Bessel sequence for H with bound and λfij + gij = δei for .LetThen,and furthermore,wherei.e.,and by Corrolary 3.2, we have is woven for H with the universal lower and upper boundsThe proof is completed.Similarly to the classical perturbations of frames, we have the following Theorem 3.4.
Suppose that the frames family is woven for H with the universal bounds A, B and is a family of Bessel sequences for H. If there exist non-negative numbers α, β, μ satisfied such thatfor any f ∈ H and σ ∈ Ω, then is woven for H with the universal lower and upper bounds ((1 − α)A− μ)(1 + β)−1 and ((1 + α)B+ μ)(1 − β)−1
Proof. Let
Then,and this implies that for any f ∈ H, σ ∈ Ω. On the other hand,and this implies that for any f ∈ H, σ ∈ Ω.
Since is self-adjoint and , we have is a surjection, i.e., the synthesis operator is a surjection. This means that is woven for H. Furthermore, we can obtain the universal lower and upper bounds ((1 − α)A− μ)(1 + β)−1 and ((1 + α)B+ μ)(1 − β)−1 by the frame operator .
Example 3.5Suppose that the frames family is woven for H with the universal bounds A, B and is an orthonormal basis for H. If the number λ, η, δ satisfiedthen the family is woven for H with the universal lower and upper bounds Proof. Let gij = δη−1ei − λη−1fij for all . Then, λfij + ηgij = δei for all and is a Bessel sequence for H with the bound fromfor all and ‖c‖ = 1. LetandThen,and further more,wherei.e.,and by Theorem 3.4, we have is woven for H with the universal lower and upper boundsThe proof is completed. Lemma 3.6 is a remarkable result on the perturbation of frames.
Lemma 3.6[2] Suppose that is a frame for H with the bounds A and B, . If there exist non-negative numbers α, β, μ satisfied such that for any , we havethen is a frame for H with bounds and .From this lemma, we can obtain the following theorem.
Suppose that the frames family is woven for H with the universal lower and upper bounds A and B. If converges to in terms of analysis operator or synthesis operator then for any non-negative number there exists a natural number N such that for every k > N, this implies that is woven for H with the universal lower and upper bounds and for every k > N.
Proof. For any and , we compute
Combining withfor f ∈ H andwe have for . Let and , then , this implies that . From Lemma 3.6, the frames family is woven for H with the universal lower and upper bounds and for every k > N.
Corollary 3.8Suppose that is woven for H with the universal lower and upper bounds A and B. If strongly converges to then for any non-negative number there exists a natural number N such that for every k > N, this implies that is woven for H with the universal lower and upper bounds and for every k > N. Proof. From the proof of Theorem 2.6, we haveBy Theorem 3.7, we have that is woven for H with the universal lower and upper bounds and for every k > N.
Example 3.9Suppose that is woven for H with the universal lower and upper bounds A, B and is an orthonormal basis for H. Ifthen is woven for H for every . Proof. Fromwe complete the proof.
Corollary 3.10Suppose that the frames family is woven for H with the universal lower and upper bounds A, B and is a family of sequences for H. If there exist non-negative numbers αj, βj, μj satisfied such that for any and j ∈ [m], we havethen is woven for H with the universal lower and upper bounds and . Proof. By Lemma 3.6, we have that for any j ∈ [m], is a frame for H with bounds and . Hence,and let n → ∞; then, we havei.e., , whereComputingfrom Theorem 3.7, the frames family is woven for H with the universal lower and upper bounds and .
Example 3.11Suppose that the frames family is woven for H with the universal bounds A, B and is an orthonormal basis for H. If the number λ, η, δ satisfiedthen the family is woven for H with the universal lower and upper bounds Proof. Let gij = λη−1fij − δη−1ei, i.e. λfij − ηgij = δeij for . ThenwheresatisfiedBy Corollary 3.10, the family is woven for H with the universal lower and upper boundsWe complete the proof.From Theorem 3.7 or Corollary 3.10, we can obtain Theorem 3.2, Theorem 3.3, Proposition 3.4, and Corollary 3.5 in [7], and obtain Theorem 6.1 in [1]. The following corollary is obvious from Corollary 3.10.
Corollary 3.12Suppose that the frames family is woven for H with the universal lower and upper bounds A, B and is a family of sequences for H. If there exist non-negative numbers αj, μj satisfied such that for any and j ∈ [m], we haveand then, is woven for H with the universal lower and upper bounds and .
Corollary 3.13Suppose that the frames family is woven for H with universal lower and upper bounds A, B and is a family of Bessel sequences for H. If there exist non-negative numbers αj, μj satisfied such that for any j ∈ [m], we haveand then, is woven for H with the universal lower and upper bounds and . Proof. We computei.e., , whereHence . From Theorem 3.7, we have that is woven for H with the universal lower and upper bounds and .
Corollary 3.14Suppose that the frames family is woven for H with the universal lower and upper bounds A, B and is a family of Bessel sequences for H. If there exist non-negative numbers αj, μj satisfied such that for any j ∈ [m], we haveand then, is woven for H with the universal lower and upper bounds and . Proof. We computei.e., , whereHence, . From Theorem 3.7, we have that is woven for H with the universal lower and upper bounds and .
Example 3.15Suppose that is a Parseval frame for H and fij = fi for all , then is woven for H with the universal lower and upper bounds 1. Take for all , we have is woven for H. Proof. Computingandfor f ∈ H, j ∈ [m] and σ ∈ Ω, from Theorem 3.1, Corollary 3.2, or Theorem 3.4, we have that is woven for H. Furthermore, we can obtain that the universal lower and upper bounds and .Note that Example 3.15 can be proved by Theorem 3.1, Corollary 3.2, or Theorem 3.4, but it cannot be proved from Theorem 3.7.
Example 3.16Let and a > 1, b > 0 be given, and assume that the wavelet frames family is woven for with the universal lower and upper bounds A, B. Ifthen is woven for with the universal lower and upper bounds Proof. From , we have . Sincei.e.,for all j ∈ [m], by Theorem 15.2.3 and Theorem 22.5.1 in [3], is a Bessel sequence for with bound R and is a wavelet frame for for all j ∈ [m]. Let Tj and be the synthesis operators of and respectively. Then, is the synthesis operators of andIt is known that there is a one-to-one correspondence between and , by Theorem 3.7 or Corollary 3.14, we have that is woven for with the universal lower and upper bounds and . We computeandand this implies that has the universal lower and upper boundsWe complete the proof.
Example 3.17Let and a, b > 0 be given, and assume that the Gabor frames family is woven for with the universal lower and upper bounds A, B. Ifthen the family is woven for with the universal lower and upper bounds Proof. From , we have . Sincei.e.,for all j ∈ [m], by Theorem 11.4.2 and Theorem 22.4.1 in [3], is a Bessel sequence for with bound R and is a Gabor frame for for all j ∈ [m]. Let Tj and be the synthesis operators of and respectively. Then, is the synthesis operators of andIt is known that there is a one-to-one correspondence between and , by Theorem 3.7 or Corollary 3.14, we have that is woven for with the universal lower and upper bounds and . Computingandimplies that has the universal lower and upper boundsWe complete the proof.Considering the Wiener spacewhich is a Banach space with respect to the normwe can obtain the following example.
Example 3.18Let and a, b > 0 be given, and assume that the Gabor frames family is woven for with the universal lower and upper bounds A, B. If ab ≤ 1 andthen the family is woven for with the universal lower and upper bounds and . Proof. From , we have . Sincei.e.,by Proposition 11.5.2 and Theorem 22.4.1 in [3], is a Bessel sequence for with bound R2 and is a Gabor frame for for all j ∈ [m]. Let Tj and be the synthesis operators of and respectively. Then, is the synthesis operators of andIt is known that there is a one-to-one correspondence between and , by Theorem 3.7 or Corollary 3.14, we have that is woven for with the universal lower and upper bounds and . Computingimplies that has the universal lower and upper bounds and .
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.873955/full#supplementary-material
References
1.
Bemrose T Casazza PG K. Lammers MC Lynch RG . Weaving Frames. Oper Matrices (2016) 10(4):1093–116. 10.7153/oam-10-61
2.
Christensen O . A Paley-Wiener Theorem for Frames. Proc Amer Math Soc (1995) 123(7):2199–201. 10.1090/s0002-9939-1995-1246520-x
3.
Christensen O . An Introduction to Frames and Riesz Bases. In: Applied and Numerical Harmonic Analysis. 2nd ed.Basel, Switzerland: Birkhäuser/Springer (2016). 10.1007/978-3-319-25613-9
4.
Casazza PG Lynch RG . Weaving Properties of Hilbert Space Frames. In: The 2015 International Conference on Sampling Theory and Applications (SampTA 2015); 25-29 May 2015; Washington, DC. New York: IEEE (2015). p. 110–4. 10.1109/sampta.2015.7148861
5.
Deepshikha B Vashisht LK . Weaving K-Frames in Hilbert Spaces. Results Math (2018) 73:20. 10.1007/s00025-018-0843-4
6.
Bibak Hafshejani A Dehghan MA . P-woven Frames. J Math Anal Appl (2019) 479:673–87. 10.1016/j.jmaa.2019.06.044
7.
Ghobadzadeh F Najati A Anastassiou GA Park C . Woven Frame in Hilbert C* Modules. J Comp Anal Appl (2018) 25:1220–32.
8.
Khosravi A Banyarani JS . Weaving G-Frames and Weaving Fusion Frames. Bull Malays Math Sci Soc (2019) 42(6):3111–29. 10.1007/s40840-018-0647-4
9.
Leng J Han D . Optimal Dual Frames for Erasures II. Linear Algebra its Appl (2011) 435:1464–72. 10.1016/j.laa.2011.03.043
10.
Vashisht LK Garg S Deepshikha B Das PK . On Generalized Weaving Frames in Hilbert Spaces. Rocky Mountain J Math (2018) 48(2):661–85. 10.1216/rmj-2018-48-2-661
11.
Xiao XC Yan K Zhao GP Zhu Y . Tight K-Frames and Weaving of K-Frames. J Pseudo-differ Oper Appl (2021) 12:14. 10.1007/s11868-020-00371-x
Summary
Keywords
frames, woven frames, convergence of frames, stability, weaving frames
Citation
Li X-B and Zhu Y-C (2022) Stability of Woven Frames. Front. Phys. 10:873955. doi: 10.3389/fphy.2022.873955
Received
11 February 2022
Accepted
25 April 2022
Published
24 June 2022
Volume
10 - 2022
Edited by
Kh S. Mekheimer, Al-Azhar University, Egypt
Reviewed by
Arran Fernandez, Eastern Mediterranean University, Turkey
Saikat Mukherjee, National Institute of Technology Meghalaya, India
Updates
Copyright
© 2022 Li and Zhu.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Xue-Bin Li, 1531388050@qq.com
This article was submitted to Statistical and Computational Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.