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 that
where A, B are 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 by
is 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 that
for every partition of . The family is called a weaving for every partition of .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 Sσ if Next, we show the relations among the four types of convergence in Theorem 2.5 and Theorem 2.6.
Theorem 2.5. While strongly converges to or converges to in terms of analysis operator or synthesis operator, we have for all .Proof. From
for all and j ∈ [m], we can obtain this theorem.
Theorem 2.6. 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 that
so
for 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 2.7. Let be an orthonormal basis for H. If for all , then converges to in terms of analysis operator, but does not strongly converge to .Proof. From
we have , i.e., converges to in terms of analysis operator. From
we have that does not strongly converge to .
Example 2.8. Let 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 compute
for all and
From 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.
Theorem 3.1. 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,
and
i.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 3.2. Suppose 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 that
for any j ∈ [m] and σ ∈ Ω, then is woven for H with the universal lower and upper bounds A− ɛ and B+ ɛ.Proof. Since
and
combing with the inequality in Corollary 3.2, we have
Take and can be regarded as for some k > N, from Theorem 3.1, we obtain Corollary 3.2.
Example 3.3. Suppose 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 satisfied
then 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 .Let
Then,
and furthermore,
where
i.e.,
and by Corrolary 3.2, we have is woven for H with the universal lower and upper bounds
The proof is completed.Similarly to the classical perturbations of frames, we have the following Theorem 3.4.
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 that
for 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)−1Proof. 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 3.5. Suppose 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 satisfied
then 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 from
for all and ‖c‖ = 1. Let
and
Then,
and further more,
where
i.e.,
and by Theorem 3.4, we have is woven for H with the universal lower and upper bounds
The 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 have
then is a frame for H with bounds and .From this lemma, we can obtain the following theorem.
Theorem 3.7. 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 with
for f ∈ H and
we 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 3.8. Suppose 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 have
By Theorem 3.7, we have that is woven for H with the universal lower and upper bounds and for every k > N.
Example 3.9. Suppose that is woven for H with the universal lower and upper bounds A, B and is an orthonormal basis for H. If
then is woven for H for every .Proof. From
we complete the proof.
Corollary 3.10. Suppose 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 have
then 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 have
i.e., , where
Computing
from Theorem 3.7, the frames family is woven for H with the universal lower and upper bounds and .
Example 3.11. Suppose 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 satisfied
then 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 . Then
where
satisfied
By Corollary 3.10, the family is woven for H with the universal lower and upper bounds
We 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 3.12. Suppose 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 have
and then, is woven for H with the universal lower and upper bounds and .
Corollary 3.13. Suppose 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 have
and then, is woven for H with the universal lower and upper bounds and .Proof. We compute
i.e., , where
Hence . From Theorem 3.7, we have that is woven for H with the universal lower and upper bounds and .
Corollary 3.14. Suppose 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 have
and then, is woven for H with the universal lower and upper bounds and .Proof. We compute
i.e., , where
Hence, . From Theorem 3.7, we have that is woven for H with the universal lower and upper bounds and .
Example 3.15. Suppose 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. Computing
and
for 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 3.16. Let 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. If
then is woven for with the universal lower and upper bounds
Proof. From , we have . Since
i.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 and
It 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 compute
and
and this implies that has the universal lower and upper bounds
We complete the proof.
Example 3.17. Let 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
then the family is woven for with the universal lower and upper bounds
Proof. From , we have . Since
i.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 and
It 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 . Computing
and
implies that has the universal lower and upper bounds
We complete the proof.Considering the Wiener space
which is a Banach space with respect to the norm
we can obtain the following example.
Example 3.18. Let 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 and
then the family is woven for with the universal lower and upper bounds and .Proof. From , we have . Since
i.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 and
It 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 . Computing
implies that has the universal lower and upper bounds and .
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
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