Stability of Woven Frames

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 ${\{f_i\}}_{i=1}^{\infty }$ for separable Hilbert space H is said to be a frame if there exist 0 < A ≤ B < ∞ such that

$A\Vert f{\Vert }^2\le \sum _{i=1}^{\infty }|⟨f,f_i⟩{|}^2\le B\Vert f{\Vert }^2,f\in H,$

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 ${\{f_i\}}_{i=1}^{\infty }$, the synthesis operator is defined by

$T:l^2\left(\mathbb{N}\right)\to H,T\left(c\right)=\sum _{i=1}^{\infty }{c}_i{f}_i,c={\left\{c_i\right\}}_{i=1}^{\infty }\in l^2\left(\mathbb{N}\right)$

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 ${\{f_i\}}_{i=1}^{\infty }$ is a frame for H.

Definition 1.2. The frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ for separable Hilbert space H is woven, if there are universal constants 0 < A ≤ B < ∞ such that

$A\Vert f{\Vert }^2\le \sum _{j=1}^m\sum _{i\in {\sigma }_j}|⟨f,f_{ij}⟩{|}^2\le B\Vert f{\Vert }^2,f\in H,$

for every partition ${\{{\sigma }_j\}}_{j\in [m]}$ of $\mathbb{N}$. The family ${\{f_{ij}\}}_{i\in {\sigma }_j,j\in [m]}$ is called a weaving for every partition ${\{{\sigma }_j\}}_{j\in [m]}$ of $\mathbb{N}$.We note that $i,j,k,m\in \mathbb{N}$ and m ≥ 2, [m] = {1, 2, … , m}, $\sigma ={\{{\sigma }_j\}}_{j\in [m]}$ is a partition of $\mathbb{N}$, $\mathrm{\Omega }=\left\{\sigma |\sigma ={\{{\sigma }_j\}}_{j\in [m]}\right.$ is any partition of $\left.\mathbb{N}\right\}$. H is a separable Hilbert space, ${\{f_{ij}\}}_{i=1}^{\infty }$ and ${\{f_{ij}^{(k)}\}}_{i=1}^{\infty }$ are the Bessel sequences for H, $\mathcal{F}=\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ and ${\mathcal{F}}_k=\{{\{f_{ij}^{(k)}\}}_{i=1}^{\infty }|j\in [m]\}$ are the families of Bessel sequences. The synthesis operators of ${\{f_{ij}^{(k)}\}}_{i=1}^{\infty }$, ${\{f_{ij}\}}_{i=1}^{\infty }$, ${\{f_{ij}^{(k)}\}}_{i\in {\sigma }_j}$, ${\{f_{ij}\}}_{i\in {\sigma }_j}$ are listed as follows:

$\begin{array}{ccc}\hfill T_j^{\left(k\right)}\left(c\right)=\sum _{i=1}^{\infty }{c}_i{f}_{ij}^{\left(k\right)},T_j\left(c\right)=\sum _{i=1}^{\infty }{c}_i{f}_{ij},c={\left\{c_i\right\}}_{i=1}^{\infty }\in l^2\left(\mathbb{N}\right);& \hfill & \hfill \\ \hfill T_{{\sigma }_j}^{\left(k\right)}\left(c\right)=\sum _{i\in {\sigma }_j}{c}_i{f}_{ij}^{\left(k\right)},T_{{\sigma }_j}\left(c\right)=\sum _{i\in {\sigma }_j}{c}_i{f}_{ij},c={\left\{c_i\right\}}_{i\in {\sigma }_j}\in l^2\left({\sigma }_j\right).& \hfill & \hfill \end{array}$

Moreover, $S_{{\sigma }_j}^{(k)}=T_{{\sigma }_j}^{(k)}{T}_{{\sigma }_j}^{(k)\ast },S_{{\sigma }_j}=T_{{\sigma }_j}{T}_{{\sigma }_j}^{\ast },S_{\sigma }^{(k)}=\sum _{j=1}^m{S}_{{\sigma }_j}^{(k)}$ and $S_{\sigma }=\sum _{j=1}^m{S}_{{\sigma }_j}$ for any f ∈ H and $\sigma ={\{{\sigma }_j\}}_{j\in [m]}\in \mathrm{\Omega }$.This article focuses on the stability of woven frames, i.e., answers the following question: Suppose that the frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with universal lower and upper bounds A and B. We want to find some conditions about $\{{\{f_{ij}-g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ such that the family $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ 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 ${\mathcal{F}}_k\to \mathcal{F}$.

1.1 Strong Convergence

In this case, the limit $\mathcal{F}$ is unique. Naturally, if the frames family $\mathcal{F}$ is woven for H, then all ${\mathcal{F}}_k$ in a sufficiently small neighborhood of $\mathcal{F}$ 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 $\mathcal{F}$ is not necessarily unique, but the synthesis operators of frames in $\mathcal{F}$ are unique, this means that the corresponding analysis operators are also unique. Similarly to the first case, if the frames family $\mathcal{F}$ is woven for H, then ${\mathcal{F}}_k$ is woven for all sufficiently big k.

1.3 Convergence in Terms of Frame Operator S_σ

In this case, the limit $\mathcal{F}$, the corresponding synthesis operators, and the analysis operators of frames in limit $\mathcal{F}$ are not necessarily unique, but the universal infimum and supremum of $\mathcal{F}$ are unique. This implies that the judgment theorem about ${\mathcal{F}}_k$ 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 ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ strongly converges to the point $\mathcal{F}$ if $\underset{k\to \infty }{\lim }\underset{j\in [m]}{\max }\sum _{i=1}^{\infty }\Vert f_{ij}^{(k)}-f_{ij}{\Vert }^2=0.$

Definition 2.2. We say a point sequence ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to the point $\mathcal{F}$ in terms of synthesis operator if $\underset{k\to \infty }{\lim }\underset{j\in [m]}{\max }\Vert T_j^{(k)}-T_j\Vert =0.$

Definition 2.3. We say a point sequence ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to the point $\mathcal{F}$ in terms of analysis operator if $\underset{k\to \infty }{\lim }\underset{j\in [m]}{\max }\Vert T_j^{(k)\ast }-T_j^{\ast }\Vert =0.$It is known that $\Vert T_j^{(k)}-T_j\Vert =\Vert T_j^{(k)\ast }-T_j^{\ast }\Vert$, thus ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of synthesis operator if and only if ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator.

Definition 2.4. We say a point sequence ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to the point $\mathcal{F}$ in terms of frame operator S_σ if $\underset{k\to \infty }{\lim }\underset{\sigma \in \mathrm{\Omega }}{\sup }\Vert S_{\sigma }^{(k)}-S_{\sigma }\Vert =0.$Next, we show the relations among the four types of convergence in Theorem 2.5 and Theorem 2.6.

Theorem 2.5. While ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ strongly converges to $\mathcal{F}$ or ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator or synthesis operator, we have $\underset{k\to \infty }{\lim }\Vert f_{ij}^{(k)}-f_{ij}\Vert =0$ for all $i\in \mathbb{N},j\in [m]$.Proof. From

$\Vert f_{ij}^{\left(k\right)}-f_{ij}{\Vert }^2\le \sum _{i=1}^{\infty }\Vert f_{ij}^{\left(k\right)}-f_{ij}{\Vert }^2,\Vert f_{ij}^{\left(k\right)}-f_{ij}\Vert \le \Vert T_j^{\left(k\right)}-T_j\Vert =\Vert T_j^{\left(k\right)\ast }-T_j^{\ast }\Vert$

for all $i,k\in \mathbb{N}$ and j ∈ [m], we can obtain this theorem.

Theorem 2.6. If ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ strongly converges to $\mathcal{F}$ then ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator; ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator if and only if ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of synthesis operator; If ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of synthesis operator then ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of frame operator S_σ.Proof. It is obvious that ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of synthesis operator if and only if ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator. For all f ∈ H and ‖f‖ = 1, we compute

$\Vert T_j^{\left(k\right)\ast }f-T_j^{\ast }f{\Vert }^2=\sum _{i=1}^{\infty }|⟨f,f_{ij}^{\left(k\right)}-f_{ij}⟩{|}^2\le \sum _{i=1}^{\infty }\Vert f{\Vert }^2\Vert f_{ij}^{\left(k\right)}-f_{ij}{\Vert }^2=\sum _{i=1}^{\infty }\Vert f_{ij}^{\left(k\right)}-f_{ij}{\Vert }^2.$

From $\underset{k\to \infty }{\lim }\underset{j\in [m]}{\max }\sum _{i=1}^{\infty }\Vert f_{ij}^{(k)}-f_{ij}{\Vert }^2=0,$ we have $\underset{k\to \infty }{\lim }\underset{j\in [m]}{\max }\Vert T_j^{(k)}-T_j\Vert =0.$ This means that if ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ strongly converges to $\mathcal{F}$ then ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator. We compute

$\begin{array}{ccc}\hfill \Vert S_{{\sigma }_j}^{\left(k\right)}-S_{{\sigma }_j}\Vert & =\hfill & \Vert T_{{\sigma }_j}^{\left(k\right)}{T}_{{\sigma }_j}^{\left(k\right)\ast }-T_{{\sigma }_j}{T}_{{\sigma }_j}^{\ast }\Vert =\Vert \left(T_{{\sigma }_j}^{\left(k\right)}{T}_{{\sigma }_j}^{\left(k\right)\ast }-T_{{\sigma }_j}^{\left(k\right)}{T}_{{\sigma }_j}^{\ast }\right)+\left(T_{{\sigma }_j}^{\left(k\right)}{T}_{{\sigma }_j}^{\ast }-T_{{\sigma }_j}{T}_{{\sigma }_j}^{\ast }\right)\Vert \hfill \\ \hfill & \le \hfill & \Vert T_{{\sigma }_j}^{\left(k\right)}\left(T_{{\sigma }_j}^{\left(k\right)\ast }-T_{{\sigma }_j}^{\ast }\right)\Vert +\Vert \left(T_{{\sigma }_j}^{\left(k\right)}-T_{{\sigma }_j}\right)T_{{\sigma }_j}^{\ast }\Vert \hfill \\ \hfill & \le \hfill & \Vert T_{{\sigma }_j}^{\left(k\right)}\Vert \Vert T_{{\sigma }_j}^{\left(k\right)\ast }-T_{{\sigma }_j}^{\ast }\Vert +\Vert T_{{\sigma }_j}^{\left(k\right)}-T_{{\sigma }_j}\Vert \Vert T_{{\sigma }_j}^{\ast }\Vert \hfill \\ \hfill & =\hfill & \left(\Vert T_{{\sigma }_j}^{\left(k\right)}\Vert +\Vert T_{{\sigma }_j}\Vert \right)\Vert T_{{\sigma }_j}^{\left(k\right)}-T_{{\sigma }_j}\Vert .\hfill \end{array}$

Note that

$\Vert T_{{\sigma }_j}^{\left(k\right)}\Vert \le \Vert T_j^{\left(k\right)}\Vert ,\Vert T_{{\sigma }_j}\Vert \le \Vert T_j\Vert ,\Vert T_{{\sigma }_j}^{\left(k\right)}-T_{{\sigma }_j}\Vert \le \Vert T_j^{\left(k\right)}-T_j\Vert ,$

so

$\Vert S_{{\sigma }_j}^{\left(k\right)}-S_{{\sigma }_j}\Vert \le \left(\Vert T_j^{\left(k\right)}\Vert +\Vert T_j\Vert \right)\Vert T_j^{\left(k\right)}-T_j\Vert \le M_j\Vert T_j^{\left(k\right)}-T_j\Vert$

for some positive M_j. Furthermore,

$\Vert S_{\sigma }^{\left(k\right)}-S_{\sigma }\Vert =‖\sum _{j\in \left[m\right]}{S}_{{\sigma }_j}^{\left(k\right)}-\sum _{j\in \left[m\right]}{S}_{{\sigma }_j}‖\le \sum _{j\in \left[m\right]}\Vert S_{{\sigma }_j}^{\left(k\right)}-S_{{\sigma }_j}\Vert \le \sum _{j\in \left[m\right]}{M}_j\Vert T_j^{\left(k\right)}-T_j\Vert .$

From $\underset{k\to \infty }{\lim }\underset{j\in [m]}{\max }\Vert T_j^{(k)\ast }-T_j^{\ast }\Vert =0,$ we have $\underset{k\to \infty }{\lim }\underset{\sigma \in \mathrm{\Omega }}{\sup }\Vert S_{{\sigma }_j}^{(k)}-S_{{\sigma }_j}\Vert =0.$ It means that if ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of synthesis operator then ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of frame operator S_σ.The following Example 2.7 and Example 2.8 show that the inverse proposition of Theorem 2.6 is untenable.

Example 2.7. Let ${\{e_i\}}_{i=1}^{\infty }$ be an orthonormal basis for H. If $f_{ij}^{(k)}=(1+\frac{1}{\sqrt{k+i}})e_i,f_{ij}=e_i$ for all $i,k\in \mathbb{N},j\in [m]$, then ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator, but ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ does not strongly converge to $\mathcal{F}$.Proof. From

$\Vert \left(T_j^{\left(k\right)\ast }-T_j^{\ast }\right)f{\Vert }^2=\sum _{i=1}^{\infty }{\left|⟨f,\frac{1}{\sqrt{k+i}}{e}_i⟩\right|}^2\le \frac{1}{k+1}\Vert f{\Vert }^2,f\in H,$

we have $\underset{k\to \infty }{\lim }\underset{j\in [m]}{\max }\Vert T_j^{(k)\ast }-T_j^{\ast }\Vert =0$, i.e., ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator. From

$\sum _{i=1}^{\infty }\Vert f_{ij}^{\left(k\right)}-f_{ij}{\Vert }^2=\sum _{i=1}^{\infty }{‖\frac{e_i}{\sqrt{k+i}}‖}^2=\sum _{i=1}^{\infty }\frac{\Vert e_i{\Vert }^2}{k+i}=\sum _{i=1}^{\infty }\frac{1}{k+i}=\infty ,$

we have that ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ does not strongly converge to $\mathcal{F}$.

Example 2.8. Let ${\{f_i\}}_{i=1}^{\infty }$ be a Parseval frame for H. If $f_{ij}^{(k)}=-f_{ij}=f_i$ for all $i,k\in \mathbb{N},j\in [m]$, then ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of frame operator S_σ, but ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$does not converge to $\mathcal{F}$ in terms of synthesis operator.Proof. We compute

$\underset{k\to \infty }{\lim }\Vert f_{ij}^{\left(k\right)}-f_{ij}\Vert =\underset{k\to \infty }{\lim }\Vert 2f_i\Vert >0$

for all $i\in \mathbb{N},j\in [m]$ and

$\underset{k\to \infty }{\lim }\underset{\sigma \in \mathrm{\Omega }}{\sup }\Vert S_{\sigma }^{\left(k\right)}-S_{\sigma }\Vert =\underset{k\to \infty }{\lim }\underset{\sigma \in \mathrm{\Omega }}{\sup }\Vert \mathbf{0}\Vert =0.$

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 S_σ.

Theorem 3.1. Suppose that the frames family $\mathcal{F}$ is woven for H with universal bounds A and B. If ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of frame operator S_σ then for any non-negative number ɛ < A there exists a natural number N such that $\underset{\sigma \in \mathrm{\Omega }}{\sup }\Vert S_{\sigma }^{(k)}-S_{\sigma }\Vert \le \epsilon$ for every k > N, this implies that ${\mathcal{F}}_k$ is woven for H with universal bounds A − ɛ and B + ɛ for every k > N.Proof. If ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of frame operator S_σ, from the Definition 2.4, for any ɛ < A there exists a natural number N such that $\underset{\sigma \in \mathrm{\Omega }}{\sup }\Vert S_{\sigma }^{(k)}-S_{\sigma }\Vert \le \epsilon$ for all k > N. Hence,

$\Vert S_{\sigma }^{\left(k\right)}f\Vert \ge \Vert S_{\sigma }f\Vert -\Vert S_{\sigma }^{\left(k\right)}f-S_{\sigma }f\Vert \ge \Vert S_{\sigma }f\Vert -\Vert S_{\sigma }^{\left(k\right)}-S_{\sigma }\Vert \Vert f\Vert \ge \left(A-\epsilon \right)\Vert f\Vert$

and

$\Vert S_{\sigma }^{\left(k\right)}f\Vert \le \Vert S_{\sigma }f\Vert +\Vert S_{\sigma }^{\left(k\right)}f-S_{\sigma }f\Vert \le \Vert S_{\sigma }f\Vert +\Vert S_{\sigma }^{\left(k\right)}-S_{\sigma }\Vert \Vert f\Vert \le \left(B+\epsilon \right)\Vert f\Vert ,$

i.e., $(A-\epsilon )\Vert f\Vert \le \Vert S_{\sigma }^{(k)}f\Vert \le (B+\epsilon )\Vert f\Vert$ for all f ∈ H and σ ∈ Ω. This implies that the bounded linear operator $S_{\sigma }^{(k)}$ is an injection. It is known that the operator $S_{\sigma }^{(k)}$ is self-adjoint, thus $S_{\sigma }^{(k)}$ is also a surjection. From $S_{\sigma }^{(k)}=T_{\sigma }^{(k)}{T}_{\sigma }^{(k)\ast }$ we have $T_{\sigma }^{(k)}$ is a surjection, this implies that ${\{f_{ij}^{(k)}\}}_{i\in {\sigma }_j,j\in [m]}$ is a frame for H with the frame operator $S_{\sigma }^{(k)}$. Hence ${\mathcal{F}}_k$ is woven with universal bounds A− ɛ and B+ ɛ for every k > N.

Corollary 3.2. Suppose that the frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with universal bounds A, B and $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is a family of Bessel sequences for H. If there exist non-negative numbers α_j, μ_j satisfied $\epsilon =\sum _{j=1}^m({\alpha }_{j}B+{\mu }_j)<A$ such that

$\Vert \sum _{i\in {\sigma }_j}⟨f,f_{ij}⟩f_{ij}-\sum _{i\in {\sigma }_j}⟨f,g_{ij}⟩g_{ij}\Vert \le {\alpha }_j\Vert \sum _{i=1}^{\infty }⟨f,f_{ij}⟩f_{ij}\Vert +{\mu }_j\Vert f\Vert ,f\in H,$

for any j ∈ [m] and σ ∈ Ω, then $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds A− ɛ and B+ ɛ.Proof. Since

$\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}-\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij}\Vert \le \sum _{j=1}^m\Vert \sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}-\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij}\Vert$

and

$\sum _{j=1}^m\left({\alpha }_j\Vert \sum _{i=1}^{\infty }⟨f,f_{ij}⟩f_{ij}\Vert +{\mu }_j\Vert f\Vert \right)\le \sum _{j=1}^m\left({\alpha }_{j}B+{\mu }_j\right)\Vert f\Vert ,f\in H,$

combing with the inequality in Corollary 3.2, we have

$\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}-\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij}\Vert \le \sum _{j=1}^m\left({\alpha }_{j}B+{\mu }_j\right)\Vert f\Vert <A\Vert f\Vert ,f\in H.$

Take $\epsilon =\sum _{j=1}^m({\alpha }_{j}B+{\mu }_j)<A$ and $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ can be regarded as ${\mathcal{F}}_k$ for some k > N, from Theorem 3.1, we obtain Corollary 3.2.

Example 3.3. Suppose that the frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal bounds A, B and ${\{e_i\}}_{i=1}^{\infty }$ is an orthonormal basis for H. If the number λ, δ satisfied

$\sqrt{1-m^{-1}A{B}^{-1}}<\lambda \le 1,0\le \delta <\sqrt{m^{-1}A+\left(2{\lambda }^2-1\right)B}-\lambda \sqrt{B},$

then the family $\{{\{\delta e_i-\lambda f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds $A-m[(1-{\lambda }^2)B+2\lambda \delta \sqrt{B}+{\delta }^2],B+m[(1-{\lambda }^2)B+2\lambda \delta \sqrt{B}+{\delta }^2]$.Proof. Let g_ij = δe_i − λf_ij for all $i\in \mathbb{N},j\in [m]$. Then ${\{g_{ij}\}}_{i\in {\sigma }_j,j\in [m]}$ is a Bessel sequence for H with bound ${(\delta +\lambda \sqrt{B})}^2$ and λf_ij + g_ij = δe_i for $i\in \mathbb{N},j\in [m]$.Let

$M_1=\sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}\rangle \lambda f_{ij}-\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij},M=\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}-\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij},f\in H.$

Then,

$\begin{array}{ccc}\hfill \Vert M_1\Vert & =\hfill & \Vert \sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}\rangle \left(\lambda f_{ij}+g_{ij}\right)-\sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}+g_{ij}\rangle g_{ij}\Vert \hfill \\ \hfill & =\hfill & \Vert \sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}\rangle \delta e_i-\sum _{i\in {\sigma }_j}\langle f,\delta e_i\rangle g_{ij}\Vert \hfill \\ \hfill & \le \hfill & \Vert \sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}\rangle \delta e_i\Vert +\Vert \sum _{i\in {\sigma }_j}\langle f,\delta e_i\rangle g_{ij}\Vert \hfill \\ \hfill & \le \hfill & \lambda \delta \sqrt{B}\Vert f\Vert +\delta \left(\delta +\lambda \sqrt{B}\right)\Vert f\Vert \hfill \\ \hfill & =\hfill & \left(2\lambda \delta \sqrt{B}+{\delta }^2\right)\Vert f\Vert ,\hfill \end{array}$

and furthermore,

$\Vert M\Vert =\Vert \left(1-{\lambda }^2\right)\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}+M_1\Vert \le \left(1-{\lambda }^2\right)\Vert \sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}\Vert +\Vert M_1\Vert ,$

where

$\sqrt{1-m^{-1}A{B}^{-1}}<\lambda \le 1,0\le \delta <\sqrt{m^{-1}A+\left(2{\lambda }^2-1\right)B}-\lambda \sqrt{B},$

i.e.,

$\epsilon =m\left(1-{\lambda }^2\right)B+m\left(2\lambda \delta \sqrt{B}+{\delta }^2\right)<A,$

and by Corrolary 3.2, we have $\{{\{\delta e_i-\lambda f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds

$A-m\left[\left(1-{\lambda }^2\right)B+2\lambda \delta \sqrt{B}+{\delta }^2\right],B+m\left[\left(1-{\lambda }^2\right)B+2\lambda \delta \sqrt{B}+{\delta }^2\right].$

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 $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal bounds A, B and $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is a family of Bessel sequences for H. If there exist non-negative numbers α, β, μ satisfied $\max \left\{\alpha +\frac{\mu }{A},\beta \right\}<1$ such that

$\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}-\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij}\Vert \le \alpha \Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}\Vert +\beta \Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij}\Vert +\mu \Vert f\Vert$

for any f ∈ H and σ ∈ Ω, then $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ 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

$S_{\sigma }f=\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij},S_{\sigma }^{\prime }f=\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij},f\in H,\sigma \in \mathrm{\Omega }.$

Then,

$\begin{array}{ccc}\hfill \Vert S_{\sigma }^{\prime }f\Vert & \le \hfill & \Vert S_{\sigma }f\Vert +\Vert S_{\sigma }^{\prime }f-S_{\sigma }f\Vert \le \left(1+\alpha \right)\Vert S_{\sigma }f\Vert +\beta \Vert S_{\sigma }^{\prime }f\Vert +\mu \Vert f\Vert \hfill \\ \hfill & \le \hfill & \left(\left(1+\alpha \right)B+\mu \right)\Vert f\Vert +\beta \Vert S_{\sigma }^{\prime }f\Vert ,f\in H,\sigma \in \mathrm{\Omega },\hfill \end{array}$

and this implies that $\Vert {S^{\prime }}_{\sigma }f\Vert \le ((1+\alpha )B+\mu ){(1-\beta )}^{-1}\Vert f\Vert$ for any f ∈ H, σ ∈ Ω. On the other hand,

$\begin{array}{ccc}\hfill \Vert S_{\sigma }^{\prime }f\Vert & \ge \hfill & \Vert S_{\sigma }f\Vert -\Vert S_{\sigma }f-S_{\sigma }^{\prime }f\Vert \ge \left(1-\alpha \right)\Vert S_{\sigma }f\Vert -\beta \Vert S_{\sigma }^{\prime }f\Vert -\mu \Vert f\Vert \hfill \\ \hfill & \ge \hfill & \left(\left(1-\alpha \right)A-\mu \right)\Vert f\Vert -\beta \Vert S_{\sigma }^{\prime }f\Vert ,f\in H,\sigma \in \mathrm{\Omega },\hfill \end{array}$

and this implies that $\Vert S_{\sigma }^{\prime }f\Vert \ge ((1-\alpha )A-\mu ){(1+\beta )}^{-1}\Vert f\Vert$ for any f ∈ H, σ ∈ Ω.Since $S_{\sigma }^{\prime }=T_{\sigma }^{\prime }{T}_{\sigma }^{{\prime }^{\ast }}$ is self-adjoint and $\max \left\{\alpha +\frac{\mu }{A},\beta \right\}<1$, we have $S_{\sigma }^{\prime }$ is a surjection, i.e., the synthesis operator $T_{\sigma }^{\prime }$ is a surjection. This means that $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ 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 $S_{\sigma }^{\prime }$.

Example 3.5. Suppose that the frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal bounds A, B and ${\{e_i\}}_{i=1}^{\infty }$ is an orthonormal basis for H. If the number λ, η, δ satisfied

$0<{\lambda }^2\le 1,0<{\eta }^2\le 1,0\le {\delta }^2<{\lambda }^2{\left(\sqrt{A+B}-\sqrt{B}\right)}^2,$

then the family $\{{\{\delta {\eta }^{-1}{e}_i-\lambda {\eta }^{-1}{f}_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds

$\frac{{\lambda }^{2}A-\left(2|\lambda \Vert \delta |\sqrt{B}+{\delta }^2\right)}{2-{\eta }^2},\frac{\left(2-{\lambda }^2\right)B+\left(2|\lambda \Vert \delta |\sqrt{B}+{\delta }^2\right)}{{\eta }^2}.$

Proof. Let g_ij = δη⁻¹e_i − λη⁻¹f_ij for all $i\in \mathbb{N},j\in [m]$. Then, λf_ij + ηg_ij = δe_i for all $i\in \mathbb{N},j\in [m]$ and ${\{g_{ij}\}}_{i\in {\sigma }_j,j\in [m]}$ is a Bessel sequence for H with the bound ${(|\delta ||{\eta }^{-1}|+|\lambda ||{\eta }^{-1}|\sqrt{B})}^2$ from

$\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}{c}_i{g}_{ij}\Vert \le \Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}{c}_i\delta {\eta }^{-1}{e}_i\Vert +\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}{c}_i\lambda {\eta }^{-1}{f}_{ij}\Vert \le \left(|\delta ||{\eta }^{-1}|+|\lambda ||{\eta }^{-1}|\sqrt{B}\right)\Vert c\Vert$

for all $c={\{c_i\}}_{i=1}^{\infty }\in l^2(\mathbb{N})$ and ‖c‖ = 1. Let

$M_1=\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}\rangle \lambda f_{ij}-\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,\eta g_{ij}\rangle \eta g_{ij},f\in H$

and

$M=\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}-\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij},f\in H.$

Then,

$\begin{array}{ccc}\hfill \Vert M_1\Vert & =\hfill & \Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}\rangle \left(\lambda f_{ij}+\eta g_{ij}\right)-\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}+\eta g_{ij}\rangle \eta g_{ij}\Vert \hfill \\ \hfill & =\hfill & \Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}\rangle \delta e_i-\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,\delta e_i\rangle \eta g_{ij}\Vert \hfill \\ \hfill & \le \hfill & \Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,\lambda f_{ij}\rangle \delta e_i\Vert +\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,\delta e_i\rangle \eta g_{ij}\Vert \hfill \\ \hfill & \le \hfill & |\lambda \Vert \delta |\sqrt{B}\Vert f\Vert +|\delta \Vert \eta |\left(|\delta ||{\eta }^{-1}|+|\lambda ||{\eta }^{-1}|\sqrt{B}\right)\Vert f\Vert \hfill \\ \hfill & =\hfill & \left(2|\lambda \Vert \delta |\sqrt{B}+{\delta }^2\right)\Vert f\Vert ,\hfill \end{array}$

and further more,

$\begin{array}{ccc}\hfill \Vert M\Vert & =\hfill & \Vert \left(1-{\lambda }^2\right)\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}-\left(1-{\eta }^2\right)\sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij}+M_1\Vert \hfill \\ \hfill & \le \hfill & \left(1-{\lambda }^2\right)\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,f_{ij}\rangle f_{ij}\Vert +\left(1-{\eta }^2\right)\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}\langle f,g_{ij}\rangle g_{ij}\Vert +\Vert M_1\Vert ,\hfill \end{array}$

where

$0<{\lambda }^2,{\eta }^2\le 1,0\le {\delta }^2<{\lambda }^2{\left(\sqrt{A+B}-\sqrt{B}\right)}^2,$

i.e.,

$\max \left\{\left(1-{\lambda }^2\right)+\frac{2|\lambda \Vert \delta |\sqrt{B}+{\delta }^2}{A},1-{\eta }^2\right\}<1,$

and by Theorem 3.4, we have $\{{\{\delta {\eta }^{-1}{e}_i-\lambda {\eta }^{-1}{f}_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds

$\frac{{\lambda }^{2}A-\left(2|\lambda \Vert \delta |\sqrt{B}+{\delta }^2\right)}{2-{\eta }^2},\frac{\left(2-{\lambda }^2\right)B+\left(2|\lambda \Vert \delta |\sqrt{B}+{\delta }^2\right)}{{\eta }^2}.$

The proof is completed.Lemma 3.6 is a remarkable result on the perturbation of frames.

Lemma 3.6. [2] Suppose that ${\{f_i\}}_{i=1}^{\infty }$ is a frame for H with the bounds A and B, ${\{g_i\}}_{i=1}^{\infty }\subset H$. If there exist non-negative numbers α, β, μ satisfied $\max \left\{\alpha +\frac{\mu }{\sqrt{A}},\beta \right\}<1$ such that for any $c_1,c_2,\dots ,c_n(n\in \mathbb{N})$, we have

$\Vert \sum _{i=1}^n{c}_i{f}_i-\sum _{i=1}^n{c}_i{g}_i\Vert \le \alpha \Vert \sum _{i=1}^n{c}_i{f}_i\Vert +\beta \Vert \sum _{i=1}^n{c}_i{g}_i\Vert +\mu {\left(\sum _{i=1}^n|c_i{|}^2\right)}^{\frac{1}{2}}$

then ${\{g_i\}}_{i=1}^{\infty }$ is a frame for H with bounds $A{(1-\frac{\alpha +\beta +\mu /\sqrt{A}}{1+\beta })}^2$ and $B{(1+\frac{\alpha +\beta +\mu /\sqrt{B}}{1-\beta })}^2$.From this lemma, we can obtain the following theorem.

Theorem 3.7. Suppose that the frames family $\mathcal{F}$ is woven for H with the universal lower and upper bounds A and B. If ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ converges to $\mathcal{F}$ in terms of analysis operator or synthesis operator then for any non-negative number $\epsilon <\sqrt{A}$ there exists a natural number N such that $\sum _{j=1}^m\Vert T_j^{{(k)}^{\ast }}-T_j^{\ast }\Vert =\sum _{j=1}^m\Vert T_j^{(k)}-T_j\Vert \le \epsilon$ for every k > N, this implies that ${\mathcal{F}}_k$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$ for every k > N.Proof. For any $\sigma ={\{{\sigma }_j\}}_{j=1}^m\in \mathrm{\Omega }$ and $c={\{c_i\}}_{i=1}^{\infty }\in l^2(\mathbb{N})$, we compute

$\begin{array}{ccc}\hfill \Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}{c}_i{f}_{ij}-\sum _{j=1}^m\sum _{i\in {\sigma }_j}{c}_i{f}_{ij}^{\left(k\right)}\Vert & \le \hfill & \sum _{j=1}^m\Vert \sum _{i\in {\sigma }_j}{c}_i{f}_{ij}-\sum _{i\in {\sigma }_j}{c}_i{f}_{ij}^{\left(k\right)}\Vert \hfill \\ \hfill & \le \hfill & \sum _{j=1}^m\Vert T_{{\sigma }_j}-T_{{\sigma }_j}^{\left(k\right)}\Vert {\left(\sum _{i\in {\sigma }_j}|c_i{|}^2\right)}^{\frac{1}{2}}.\hfill \end{array}$

Combining with

${\left(T_{{\sigma }_j}-T_{{\sigma }_j}^{\left(k\right)}\right)}^{\ast }f={\left\{\langle f,f_{ij}-f_{ij}^{\left(k\right)}\rangle \right\}}_{i\in {\sigma }_j},{\left(T_j-T_j^{\left(k\right)}\right)}^{\ast }f={\left\{\langle f,f_{ij}-f_{ij}^{\left(k\right)}\rangle \right\}}_{i=1}^{\infty }$

for f ∈ H and

$\begin{array}{ccc}\hfill \Vert {\left(T_{{\sigma }_j}-T_{{\sigma }_j}^{\left(k\right)}\right)}^{\ast }\Vert & =\hfill & \underset{\Vert f\Vert =1}{\sup }{\left(\sum _{i\in {\sigma }_j}|\langle f,f_{ij}-f_{ij}^{\left(k\right)}\rangle {|}^2\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le \hfill & \underset{\Vert f\Vert =1}{\sup }{\left(\sum _{i=1}^{\infty }|\langle f,f_{ij}-f_{ij}^{\left(k\right)}\rangle {|}^2\right)}^{\frac{1}{2}}=\Vert {\left(T_j-T_j^{\left(k\right)}\right)}^{\ast }\Vert ,\hfill \end{array}$

we have

$\sum _{j=1}^m\Vert T_{{\sigma }_j}-T_{{\sigma }_j}^{\left(k\right)}\Vert {\left(\sum _{i\in {\sigma }_j}|c_i{|}^2\right)}^{\frac{1}{2}}\le \sum _{j=1}^m\Vert T_j-T_j^{\left(k\right)}\Vert {\left(\sum _{i=1}^{\infty }|c_i{|}^2\right)}^{\frac{1}{2}}\le \epsilon {\left(\sum _{i=1}^{\infty }|c_i{|}^2\right)}^{\frac{1}{2}}$

for $c={\{c_i\}}_{i=1}^{\infty }\in l^2(\mathbb{N})$. Let ${\{g_i\}}_{i=1}^{\infty }={\{f_{ij}^{(k)}\}}_{i\in {\sigma }_j,j\in [m]}$ and ${\{f_i\}}_{i=1}^{\infty }={\{f_{ij}\}}_{i\in {\sigma }_j,j\in [m]}$, then $\Vert \sum _{i=1}^{\infty }{c}_i{f}_i-\sum _{i=1}^{\infty }{c}_i{g}_i\Vert \le \epsilon {\left(\sum _{i=1}^{\infty }|c_i{|}^2\right)}^{\frac{1}{2}}$, this implies that $\Vert \sum _{i=1}^n{c}_i{f}_i-\sum _{i=1}^n{c}_i{g}_i\Vert \le \epsilon {\left(\sum _{i=1}^n|c_i{|}^2\right)}^{\frac{1}{2}}$. From Lemma 3.6, the frames family ${\mathcal{F}}_k$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$ for every k > N.

Corollary 3.8. Suppose that $\mathcal{F}$ is woven for H with the universal lower and upper bounds A and B. If ${\{{\mathcal{F}}_k\}}_{k=1}^{\infty }$ strongly converges to $\mathcal{F}$ then for any non-negative number $\epsilon <\sqrt{A}$ there exists a natural number N such that $\sum _{j=1}^m{\left(\sum _{i=1}^{\infty }\Vert f_{ij}^{(k)}-f_{ij}{\Vert }^2\right)}^{\frac{1}{2}}\le \epsilon$ for every k > N, this implies that ${\mathcal{F}}_k$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$ for every k > N.Proof. From the proof of Theorem 2.6, we have

$\sum _{j=1}^m\Vert T_j^{{\left(k\right)}^{*}}-T_j^{*}\Vert =\sum _{j=1}^m\Vert T_j^{\left(k\right)}-T_j\Vert \le \sum _{j=1}^m{\left(\sum _{i=1}^{\infty }\Vert f_{ij}^{\left(k\right)}-f_{ij}{\Vert }^2\right)}^{\frac{1}{2}}\le \epsilon <\sqrt{A}.$

By Theorem 3.7, we have that ${\mathcal{F}}_k$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$ for every k > N.

Example 3.9. Suppose that $\mathcal{F}$ is woven for H with the universal lower and upper bounds A, B and ${\{e_i\}}_{i=1}^{\infty }$ is an orthonormal basis for H. If

$f_{ij}^{\left(k\right)}=f_{ij}+2^{-\left(k+i\right)}{m}^{-1}\sqrt{A}{e}_{i}i\in \mathbb{N},j\in \left[m\right]$

then ${\mathcal{F}}_k$ is woven for H for every $k\in \mathbb{N}$.Proof. From

$\sum _{j=1}^m{\left(\sum _{i=1}^{\infty }\Vert f_{ij}^{\left(k\right)}-f_{ij}{\Vert }^2\right)}^{\frac{1}{2}}<\sum _{j=1}^m{m}^{-1}\sqrt{A}=\sqrt{A},$

we complete the proof.

Corollary 3.10. Suppose that the frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds A, B and $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is a family of sequences for H. If there exist non-negative numbers α_j, β_j, μ_j satisfied $\epsilon =\sum _{j=1}^m\left[({\alpha }_j+{\beta }_j)\sqrt{B}+{\mu }_j\right]{(1-{\beta }_j)}^{-1}<\sqrt{A}$ such that for any $c_1,c_2,\dots ,c_n(n\in \mathbb{N})$ and j ∈ [m], we have

$\Vert \sum _{i=1}^n{c}_i{f}_{ij}-\sum _{i=1}^n{c}_i{g}_{ij}\Vert \le {\alpha }_j\Vert \sum _{i=1}^n{c}_i{f}_{ij}\Vert +{\beta }_j\Vert \sum _{i=1}^n{c}_i{g}_{ij}\Vert +{\mu }_j{\left(\sum _{i=1}^n|c_i{|}^2\right)}^{\frac{1}{2}},$

then $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$.Proof. By Lemma 3.6, we have that for any j ∈ [m], ${\{g_{ij}\}}_{i=1}^{\infty }$ is a frame for H with bounds $A{(1-\frac{{\alpha }_j+{\beta }_j+{\mu }_j/\sqrt{A}}{1+{\beta }_j})}^2$ and $B{(1+\frac{{\alpha }_j+{\beta }_j+{\mu }_j/\sqrt{B}}{1-{\beta }_j})}^2$. Hence,

$\begin{array}{ccc}\hfill \Vert \sum _{i=1}^n{c}_i{f}_{ij}-\sum _{i=1}^n{c}_i{g}_{ij}\Vert & \le \hfill & {\alpha }_j\Vert \sum _{i=1}^n{c}_i{f}_{ij}\Vert +{\beta }_j\Vert \sum _{i=1}^n{c}_i{g}_{ij}\Vert +{\mu }_j{\left(\sum _{i=1}^n|c_i{|}^2\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le \hfill & \left[{\alpha }_j\sqrt{B}+{\beta }_j\sqrt{B}\left(1+\frac{{\alpha }_j+{\beta }_j+{\mu }_j/\sqrt{B}}{1-{\beta }_j}\right)+{\mu }_j\right]{\left(\sum _{i=1}^n|c_i{|}^2\right)}^{\frac{1}{2}}\hfill \\ \hfill & =\hfill & \left[\left({\alpha }_j+{\beta }_j\right)\sqrt{B}+{\mu }_j\right]{\left(1-{\beta }_j\right)}^{-1}{\left(\sum _{i=1}^n|c_i{|}^2\right)}^{\frac{1}{2}},\hfill \\ \hfill & \le \hfill & \left[\left({\alpha }_j+{\beta }_j\right)\sqrt{B}+{\mu }_j\right]{\left(1-{\beta }_j\right)}^{-1}{\left(\sum _{i=1}^{\infty }|c_i{|}^2\right)}^{\frac{1}{2}},\hfill \end{array}$

and let n → ∞; then, we have

$\Vert \sum _{i=1}^{\infty }{c}_i{f}_{ij}-\sum _{i=1}^{\infty }{c}_i{g}_{ij}\Vert \le \left[\left({\alpha }_j+{\beta }_j\right)\sqrt{B}+{\mu }_j\right]{\left(1-{\beta }_j\right)}^{-1}{\left(\sum _{i=1}^{\infty }|c_i{|}^2\right)}^{\frac{1}{2}},$

i.e., $\Vert T_j^{(k)}-T_j\Vert \le \left[({\alpha }_j+{\beta }_j)\sqrt{B}+{\mu }_j\right]{(1-{\beta }_j)}^{-1}$, where

$T_j\left({\left\{c_i\right\}}_{i=1}^{\infty }\right)=\sum _{i=1}^{\infty }{c}_i{f}_{ij},T_j^{\left(k\right)}\left({\left\{c_i\right\}}_{i=1}^{\infty }\right)=\sum _{i=1}^{\infty }{c}_i{g}_{ij},{\left\{c_i\right\}}_{i=1}^{\infty }\in l^2\left(\mathbb{N}\right).$

Computing

$\sum _{j=1}^m\Vert T_j^{\left(k\right)}-T_j\Vert \le \sum _{j=1}^m\left[\left({\alpha }_j+{\beta }_j\right)\sqrt{B}+{\mu }_j\right]{\left(1-{\beta }_j\right)}^{-1}=\epsilon <\sqrt{A},$

from Theorem 3.7, the frames family $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$.

Example 3.11. Suppose that the frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal bounds A, B and ${\{e_i\}}_{i=1}^{\infty }$ is an orthonormal basis for H. If the number λ, η, δ satisfied

$1-\frac{\sqrt{A}}{m\sqrt{B}}<\lambda \le 1,\frac{m\left(2-\lambda \right)\sqrt{B}}{m\sqrt{B}+\sqrt{A}}<\eta \le 1,0\le \delta <m^{-1}\eta \sqrt{A}-\left(2-\lambda -\eta \right)\sqrt{B},$

then the family $\{{\{\lambda {\eta }^{-1}{f}_{ij}-\delta {\eta }^{-1}{e}_i\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds

$A{\left[1-\frac{m\left(2-\lambda -\eta \right)\sqrt{B}+m\delta }{\eta \sqrt{A}}\right]}^2,B{\left[1+\frac{m\left(2-\lambda -\eta \right)\sqrt{B}+m\delta }{\eta \sqrt{B}}\right]}^2.$

Proof. Let g_ij = λη⁻¹f_ij − δη⁻¹e_i, i.e. λf_ij − ηg_ij = δe_ij for $i\in \mathbb{N},j\in [m]$. Then

$\begin{array}{ccc}\hfill \Vert \sum _{i=1}^n{c}_i{f}_{ij}-\sum _{i=1}^n{c}_i{g}_{ij}\Vert & =\hfill & \Vert \left(1-\lambda \right)\sum _{i=1}^n{c}_i{f}_{ij}-\left(1-\eta \right)\sum _{i=1}^n{c}_i{g}_{ij}+\delta \sum _{i=1}^n{c}_i{e}_i\Vert \hfill \\ \hfill & \le \hfill & \left(1-\lambda \right)\Vert \sum _{i=1}^n{c}_i{f}_{ij}\Vert +\left(1-\eta \right)\Vert \sum _{i=1}^n{c}_i{g}_{ij}\Vert +\delta {\left(\sum _{i=1}^n|c_i{|}^2\right)}^{\frac{1}{2}},\hfill \end{array}$

where

$1-\frac{\sqrt{A}}{m\sqrt{B}}\le \lambda \le 1,\frac{m\left(2-\lambda \right)\sqrt{B}}{m\sqrt{B}+\sqrt{A}}<\eta \le 1,0\le \delta <m^{-1}\eta \sqrt{A}-\left(2-\lambda -\eta \right)\sqrt{B}$

satisfied

$\epsilon =\sum _{j=1}^m\left[\left(1-\lambda +1-\eta \right)\sqrt{B}+\delta \right]{\eta }^{-1}=\left[m\left(2-\lambda -\eta \right)\sqrt{B}+m\delta \right]{\eta }^{-1}<\sqrt{A}.$

By Corollary 3.10, the family $\{{\{\lambda {\eta }^{-1}{f}_{ij}-\delta {\eta }^{-1}{e}_i\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds

$A{\left[1-\frac{m\left(2-\lambda -\eta \right)\sqrt{B}+m\delta }{\eta \sqrt{A}}\right]}^2,B{\left[1+\frac{m\left(2-\lambda -\eta \right)\sqrt{B}+m\delta }{\eta \sqrt{B}}\right]}^2.$

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 $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds A, B and $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is a family of sequences for H. If there exist non-negative numbers α_j, μ_j satisfied $\epsilon =\sum _{j=1}^m({\alpha }_j\sqrt{B}+{\mu }_j)<\sqrt{A}$ such that for any $c_1,c_2,\dots ,c_n(n\in \mathbb{N})$ and j ∈ [m], we have

$\Vert \sum _{i=1}^n{c}_i{f}_{ij}-\sum _{i=1}^n{c}_i{g}_{ij}\Vert \le {\alpha }_j\Vert \sum _{i=1}^n{c}_i{f}_{ij}\Vert +{\mu }_j{\left(\sum _{i=1}^n|c_i{|}^2\right)}^{\frac{1}{2}},$

and then, $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$.

Corollary 3.13. Suppose that the frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with universal lower and upper bounds A, B and $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is a family of Bessel sequences for H. If there exist non-negative numbers α_j, μ_j satisfied $\epsilon =\sum _{j=1}^m({\alpha }_j\sqrt{B}+{\mu }_j)<\sqrt{A}$ such that for any j ∈ [m], we have

${\left(\sum _{i=1}^{\infty }|\langle f,f_{ij}-g_{ij}\rangle {|}^2\right)}^{\frac{1}{2}}\le {\alpha }_j{\left(\sum _{i=1}^{\infty }|\langle f,f_{ij}\rangle {|}^2\right)}^{\frac{1}{2}}+{\mu }_j\Vert f\Vert ,f\in H,$

and then, $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$.Proof. We compute

${\left(\sum _{i=1}^{\infty }|\langle f,f_{ij}-g_{ij}\rangle {|}^2\right)}^{\frac{1}{2}}\le {\alpha }_j{\left(\sum _{i=1}^{\infty }|\langle f,f_{ij}\rangle {|}^2\right)}^{\frac{1}{2}}+{\mu }_j\Vert f\Vert \le \left({\alpha }_j\sqrt{B}+{\mu }_j\right)\Vert f\Vert ,f\in H,$

i.e., $\Vert T_j^{(k)\ast }-T_j^{\ast }\Vert \le ({\alpha }_j\sqrt{B}+{\mu }_j)$, where

$T_j^{\ast }\left(f\right)={\left\{\langle f,f_{ij}\rangle \right\}}_{i=1}^{\infty },T_j^{\left(k\right)\ast }\left(f\right)={\left\{\langle f,g_{ij}\rangle \right\}}_{i=1}^{\infty },f\in H.$

Hence $\sum _{j=1}^m\Vert T_j^{{(k)}^{\ast }}-T_j^{\ast }\Vert \le \sum _{j=1}^m({\alpha }_j\sqrt{B}+{\mu }_j)=\epsilon <\sqrt{A}$. From Theorem 3.7, we have that $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$.

Corollary 3.14. Suppose that the frames family $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds A, B and $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is a family of Bessel sequences for H. If there exist non-negative numbers α_j, μ_j satisfied $\epsilon =\sum _{j=1}^m\sqrt{{\alpha }_{j}B+{\mu }_j}<\sqrt{A}$ such that for any j ∈ [m], we have

$\sum _{i=1}^{\infty }|\langle f,f_{ij}-g_{ij}\rangle {|}^2\le {\alpha }_j\sum _{i=1}^{\infty }|\langle f,f_{ij}\rangle {|}^2+{\mu }_j\Vert f{\Vert }^2,f\in H,$

and then, $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$.Proof. We compute

$\sum _{i=1}^{\infty }|\langle f,f_{ij}-g_{ij}\rangle {|}^2\le {\alpha }_j\sum _{i=1}^{\infty }|\langle f,f_{ij}\rangle {|}^2+{\mu }_j\Vert f{\Vert }^2\le \left({\alpha }_{j}B+{\mu }_j\right)\Vert f{\Vert }^2,f\in H,$

i.e., $\Vert T_j^{(k)\ast }-T_j^{\ast }\Vert \le \sqrt{{\alpha }_{j}B+{\mu }_j}$, where

$T_j^{\ast }\left(f\right)={\left\{\langle f,f_{ij}\rangle \right\}}_{i=1}^{\infty },T_j^{\left(k\right)\ast }\left(f\right)={\left\{\langle f,g_{ij}\rangle \right\}}_{i=1}^{\infty },f\in H.$

Hence, $\sum _{j=1}^m\Vert T_j^{{(k)}^{\ast }}-T_j^{\ast }\Vert \le \sum _{j=1}^m\sqrt{{\alpha }_{j}B+{\mu }_j}=\epsilon <\sqrt{A}$. From Theorem 3.7, we have that $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$.

Example 3.15. Suppose that ${\{f_i\}}_{i=1}^{\infty }$ is a Parseval frame for H and f_ij = f_i for all $i\in \mathbb{N},j\in [m]$, then $\{{\{f_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H with the universal lower and upper bounds 1. Take $g_{ij}=-f_i+\frac{1}{m^2}{f}_i$ for all $i\in \mathbb{N},j\in [m]$, we have $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ is woven for H.Proof. Computing

$\Vert \sum _{i\in {\sigma }_j}⟨f,f_{ij}⟩f_{ij}-\sum _{i\in {\sigma }_j}⟨f,g_{ij}⟩g_{ij}\Vert =\frac{2m^2-1}{m^4}\Vert \sum _{i\in {\sigma }_j}⟨f,f_j⟩f_j\Vert \le \frac{2m^2-1}{m^4}\Vert f\Vert$

and

$\Vert \sum _{j=1}^m\sum _{i\in {\sigma }_j}⟨f,f_{ij}⟩f_{ij}-\sum _{j=1}^m\sum _{i\in {\sigma }_j}⟨f,g_{ij}⟩g_{ij}\Vert =\frac{2m^2-1}{m^4}\Vert \sum _{i=1}^{\infty }⟨f,f_j⟩f_j\Vert =\frac{2m^2-1}{m^4}\Vert f\Vert$

for f ∈ H, j ∈ [m] and σ ∈ Ω, from Theorem 3.1, Corollary 3.2, or Theorem 3.4, we have $\{{\{g_{ij}\}}_{i=1}^{\infty }|j\in [m]\}$ that is woven for H. Furthermore, we can obtain that the universal lower and upper bounds ${(1-\frac{1}{m^2})}^2$ and $1+\frac{2m^2-1}{m^4}$.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 ${\{{\psi }_j\}}_{j\in [m]},{\{{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$ and a > 1, b > 0 be given, and assume that the wavelet frames family $\left\{{\{a^{n/2}{\psi }_j(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds A, B. If

$R≔\frac{1}{b}{\max }_{j\in \left[m\right]}{\sup }_{|\gamma |\in \left[1,a\right]}\sum _{n,k\in \mathbb{Z}}\left|\left({\widehat{\psi }}_j-{\widehat{\phi }}_j\right)\left(a^j\gamma \right)\left({\widehat{\psi }}_j-{\widehat{\phi }}_j\right)\left(a^j\gamma +k/b\right)\right|<\frac{A}{m^2},$

then $\left\{{\{a^{n/2}{\phi }_j(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds

$A{\left(1-m\sqrt{\frac{R}{A}}\right)}^2,B{\left(1+m\sqrt{\frac{R}{B}}\right)}^2.$

Proof. From ${\{{\psi }_j\}}_{j\in [m]},{\{{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$, we have ${\{{\psi }_j-{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$. Since

$R=\frac{1}{b}{\max }_{j\in \left[m\right]}{\sup }_{|\gamma |\in \left[1,a\right]}\sum _{n,k\in \mathbb{Z}}\left|\left({\widehat{\psi }}_j-{\widehat{\phi }}_j\right)\left(a^j\gamma \right)\left({\widehat{\psi }}_j-{\widehat{\phi }}_j\right)\left(a^j\gamma +k/b\right)\right|<\frac{A}{m^2},$

i.e.,

$\frac{1}{b}{\sup }_{|\gamma |\in \left[1,a\right]}\sum _{n,k\in \mathbb{Z}}\left|\left({\widehat{\psi }}_j-{\widehat{\phi }}_j\right)\left(a^j\gamma \right)\left({\widehat{\psi }}_j-{\widehat{\phi }}_j\right)\left(a^j\gamma +k/b\right)\right|\le R<A$

for all j ∈ [m], by Theorem 15.2.3 and Theorem 22.5.1 in [3], ${\{a^{n/2}({\psi }_j-{\phi }_j)(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}$ is a Bessel sequence for $L^2(\mathbb{R})$ with bound R and ${\{a^{n/2}{\phi }_j(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}$ is a wavelet frame for $L^2(\mathbb{R})$ for all j ∈ [m]. Let T_j and $T_j^{\prime }$ be the synthesis operators of ${\{a^{n/2}{\psi }_j(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}$ and ${\{a^{n/2}{\phi }_j(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}$ respectively. Then, $T_j-T_j^{\prime }$ is the synthesis operators of ${\{a^{n/2}({\psi }_j-{\phi }_j)(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}$ and

$\epsilon =\sum _{j\in \left[m\right]}\Vert T_j-T_j^{\prime }\Vert \le m\sqrt{R}<m\sqrt{\frac{A}{m^2}}=\sqrt{A}.$

It is known that there is a one-to-one correspondence between $\mathbb{N}$ and ${\mathbb{Z}}^2$, by Theorem 3.7 or Corollary 3.14, we have that $\left\{{\{a^{n/2}{\phi }_j(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$. We compute

$A{\left(1-m\sqrt{\frac{R}{A}}\right)}^2={\left(\sqrt{A}-m\sqrt{R}\right)}^2\le {\left(\sqrt{A}-\epsilon \right)}^2$

and

$B{\left(1+m\sqrt{\frac{R}{B}}\right)}^2={\left(\sqrt{B}+m\sqrt{R}\right)}^2\ge {\left(\sqrt{B}+\epsilon \right)}^2,$

and this implies that $\left\{{\{a^{n/2}{\phi }_j(a^{n}x-kb)\}}_{n,k\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ has the universal lower and upper bounds

$A{\left(1-m\sqrt{\frac{R}{A}}\right)}^2,B{\left(1+m\sqrt{\frac{R}{B}}\right)}^2.$

We complete the proof.

Example 3.17. Let ${\{{\psi }_j\}}_{j\in [m]},{\{{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$ and a, b > 0 be given, and assume that the Gabor frames family $\left.{\left\{E_{mb}{T}_{na}{\psi }_j\right\}}_{m,n\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds A, B. If

$R≔\frac{1}{b}{\max }_{j\in \left[m\right]}{\sup }_{x\in \left[0,a\right]}\sum _{k\in \mathbb{Z}}\left|\sum _{n\in \mathbb{Z}}\left({\psi }_j-{\phi }_j\right)\left(x-na\right)\overline{\left({\psi }_j-{\phi }_j\right)\left(x-na-k/b\right)}\right|<\frac{A}{m^2},$

then the family $\left.{\left\{E_{mb}{T}_{na}{\phi }_j\right\}}_{m,n\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds

$A{\left(1-m\sqrt{\frac{R}{A}}\right)}^2,B{\left(1+m\sqrt{\frac{R}{B}}\right)}^2.$

Proof. From ${\{{\psi }_j\}}_{j\in [m]},{\{{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$, we have ${\{{\psi }_j-{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$. Since

$R=\frac{1}{b}{\max }_{j\in \left[m\right]}{\sup }_{x\in \left[0,a\right]}\sum _{k\in \mathbb{Z}}\left|\sum _{n\in \mathbb{Z}}\left({\psi }_j-{\phi }_j\right)\left(x-na\right)\overline{\left({\psi }_j-{\phi }_j\right)\left(x-na-k/b\right)}\right|<\frac{A}{m^2},$

i.e.,

$\frac{1}{b}{\sup }_{x\in \left[0,a\right]}\sum _{k\in \mathbb{Z}}\left|\sum _{n\in \mathbb{Z}}\left({\psi }_j-{\phi }_j\right)\left(x-na\right)\overline{\left({\psi }_j-{\phi }_j\right)\left(x-na-k/b\right)}\right|\le R<A$

for all j ∈ [m], by Theorem 11.4.2 and Theorem 22.4.1 in [3], ${\{E_{mb}{T}_{na}({\psi }_j-{\phi }_j)\}}_{m,n\in \mathbb{Z}}$ is a Bessel sequence for $L^2(\mathbb{R})$ with bound R and ${\{E_{mb}{T}_{na}{\phi }_j\}}_{m,n\in \mathbb{Z}}$ is a Gabor frame for $L^2(\mathbb{R})$ for all j ∈ [m]. Let T_j and $T_j^{\prime }$ be the synthesis operators of ${\{E_{mb}{T}_{na}{\psi }_j\}}_{m,n\in \mathbb{Z}}$ and ${\{E_{mb}{T}_{na}{\phi }_j\}}_{m,n\in \mathbb{Z}}$ respectively. Then, $T_j-T_j^{\prime }$ is the synthesis operators of ${\{E_{mb}{T}_{na}({\psi }_j-{\phi }_j)\}}_{m,n\in \mathbb{Z}}$ and

$\epsilon =\sum _{j\in \left[m\right]}\Vert T_j-T_j^{\prime }\Vert \le m\sqrt{R}<m\sqrt{\frac{A}{m^2}}=\sqrt{A}.$

It is known that there is a one-to-one correspondence between $\mathbb{N}$ and ${\mathbb{Z}}^2$, by Theorem 3.7 or Corollary 3.14, we have that $\left.{\left\{E_{mb}{T}_{na}{\phi }_j\right\}}_{m,n\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$. Computing

$A{\left(1-m\sqrt{\frac{R}{A}}\right)}^2={\left(\sqrt{A}-m\sqrt{R}\right)}^2\le {\left(\sqrt{A}-\epsilon \right)}^2$

and

$B{\left(1+m\sqrt{\frac{R}{B}}\right)}^2={\left(\sqrt{B}+m\sqrt{R}\right)}^2\ge {\left(\sqrt{B}+\epsilon \right)}^2$

implies that $\left.{\left\{E_{mb}{T}_{na}{\phi }_j\right\}}_{m,n\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ has the universal lower and upper bounds

$A{\left(1-m\sqrt{\frac{R}{A}}\right)}^2,B{\left(1+m\sqrt{\frac{R}{B}}\right)}^2.$

We complete the proof.Considering the Wiener space

$W≔\left\{g:\mathbb{R}\to \mathbb{C}\left|g\text{\hspace{0.17em}measurable\hspace{0.17em}and\hspace{0.17em}}\right.\sum _{k\in \mathbb{Z}}\Vert g{\chi }_{\left[ka,\right.\left(k+1\right)a\left[\right.}{\Vert }_{\infty }<\infty \right\}$

which is a Banach space with respect to the norm

$\Vert g{\Vert }_{W,a}=\sum _{k\in \mathbb{Z}}\Vert g{\chi }_{\left[ka\right.,\left(k+1\right)a]}{\Vert }_{\infty },$

we can obtain the following example.

Example 3.18. Let ${\{{\psi }_j\}}_{j\in [m]},{\{{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$ and a, b > 0 be given, and assume that the Gabor frames family $\left.{\left\{E_{mb}{T}_{na}{\psi }_j\right\}}_{m,n\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds A, B. If ab ≤ 1 and

$R:=\sqrt{\frac{2}{b}}{\max }_{j\in \left[m\right]}\Vert {\psi }_j-{\phi }_j{\Vert }_{W,a}<\sqrt{\frac{A}{m^2}},$

then the family $\left.{\left\{E_{mb}{T}_{na}{\phi }_j\right\}}_{m,n\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds ${(\sqrt{A}-mR)}^2$ and ${(\sqrt{B}+mR)}^2$.Proof. From ${\{{\psi }_j\}}_{j\in [m]},{\{{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$, we have ${\{{\psi }_j-{\phi }_j\}}_{j\in [m]}\subset L^2(\mathbb{R})$. Since

$R=\sqrt{\frac{2}{b}}{\max }_{j\in \left[m\right]}\Vert {\psi }_j-{\phi }_j{\Vert }_{W,a}<\sqrt{\frac{A}{m^2}},$

i.e.,

$\sqrt{\frac{2}{b}}\Vert {\psi }_j-{\phi }_j{\Vert }_{W,a}\le R<\sqrt{A},j\in \left[m\right]$

by Proposition 11.5.2 and Theorem 22.4.1 in [3], ${\{E_{mb}{T}_{na}({\psi }_j-{\phi }_j)\}}_{m,n\in \mathbb{Z}}$ is a Bessel sequence for $L^2(\mathbb{R})$ with bound R² and ${\{E_{mb}{T}_{na}{\phi }_j\}}_{m,n\in \mathbb{Z}}$ is a Gabor frame for $L^2(\mathbb{R})$ for all j ∈ [m]. Let T_j and $T_j^{\prime }$ be the synthesis operators of ${\{E_{mb}{T}_{na}{\psi }_j\}}_{m,n\in \mathbb{Z}}$ and ${\{E_{mb}{T}_{na}{\phi }_j\}}_{m,n\in \mathbb{Z}}$ respectively. Then, $T_j-T_j^{\prime }$ is the synthesis operators of ${\{E_{mb}{T}_{na}({\psi }_j-{\phi }_j)\}}_{m,n\in \mathbb{Z}}$ and

$\epsilon =\sum _{j\in \left[m\right]}\Vert T_j-T_j^{\prime }\Vert \le mR<m\sqrt{\frac{A}{m^2}}=\sqrt{A}.$

It is known that there is a one-to-one correspondence between $\mathbb{N}$ and ${\mathbb{Z}}^2$, by Theorem 3.7 or Corollary 3.14, we have that $\left.{\left\{E_{mb}{T}_{na}{\phi }_j\right\}}_{m,n\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ is woven for $L^2(\mathbb{R})$ with the universal lower and upper bounds ${(\sqrt{A}-\epsilon )}^2$ and ${(\sqrt{B}+\epsilon )}^2$. Computing

${\left(\sqrt{A}-mR\right)}^2\le {\left(\sqrt{A}-\epsilon \right)}^2,{\left(\sqrt{B}+mR\right)}^2\ge {\left(\sqrt{B}+\epsilon \right)}^2$

implies that $\left.{\left\{E_{mb}{T}_{na}{\phi }_j\right\}}_{m,n\in \mathbb{Z}}\left|j\in [m]\right.\right\}$ has the universal lower and upper bounds ${(\sqrt{A}-mR)}^2$ and ${(\sqrt{B}+mR)}^2$.

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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