On Perturbations of Woven Pairs of Frames in Hilbert Spaces

Table of Contents

• 1. Introduction and Preliminaries
• 2. Frames and Woven Frames
• 3. Perturbation Results for Woven Pairs
• 4. Operator-Based Characterization
• 5. Statistical Overview
• 6. Key Insights
• 7. Conclusion

1. Introduction and Preliminaries

Woven families of frames were introduced by Bemrose et al. in 2015, motivated by distributed signal processing applications in wireless sensor networks. The core idea involves preprocessing signals using a family of frames corresponding to sensor nodes, ensuring robust signal reconstruction regardless of which subset of measurements is obtained. Mathematically, a family of frames {f_ij}_{i∈I, j∈I_n} for a separable Hilbert space H is woven if, for every partition {σ_j}_{j∈I_n} of the index set I, the set {f_ij}_{i∈σ_j, j∈I_n} forms a frame for H with uniform bounds. This note focuses on woven pairs of frames (F, G), where F = {f_i}_{i∈I} and G = {g_i}_{i∈I}, examining small perturbations that preserve the woven property. We leverage synthesis operators to simplify proofs and explore characterizations involving oblique projections and nullspace angles.

2. Frames and Woven Frames

Let H be a separable Hilbert space, and B(H) denote the algebra of bounded linear operators on H. For an operator T ∈ B(H), R(T) and N(T) represent its range and nullspace, respectively. A frame F = {f_i}_{i∈I} for H satisfies A∥x∥² ≤ ∑_{i∈I} |⟨x, f_i⟩|² ≤ B∥x∥² for all x ∈ H, with optimal bounds A_F and B_F. The synthesis operator T_F : H → H is defined via an orthonormal basis B = {e_i}_{i∈I} as T_F e_i = f_i, with the analysis operator T_F* and frame operator S_F = T_F T_F*. Key properties include: F is a frame iff T_F is surjective, and S_F is positive and invertible. The canonical dual frame S_F^{-1}(F) enables reconstruction: x = ∑_{i∈I} ⟨x, f_i⟩ S_F^{-1} f_i.

Woven frames, as per Definition 2, require that for any partition {σ_j}_{j∈I_n} of I, the weaving {f_ij}_{i∈σ_j, j∈I_n} is a frame with uniform bounds A and B. Weak weaving drops the uniformity requirement. Theorem 1 (from [2]) establishes that weakly woven pairs are woven, simplifying analysis. This note concentrates on pairs (F, G), using operator theory to derive perturbation conditions.

3. Perturbation Results for Woven Pairs

Our results complement existing literature by examining small perturbations δF = {f_i + δ_i}_{i∈I} of a frame F. Under certain conditions on ∥δ_i∥, the pair (F, δF) remains woven. Specifically, if ∥δ_i∥ < ε for all i and ε is sufficiently small relative to frame bounds, then the perturbation preserves the woven property. Proofs utilize synthesis operators: Let T_F and T_δF be the synthesis operators of F and δF. If ∥T_F - T_δF∥ < A_F / 2, then T_δF remains surjective, ensuring δF is a frame. For weaving, consider any partition σ; the synthesis operator T_σ for the weaving {f_i}_{i∈σ} ∪ {δ_i}_{i∈σ^c} must satisfy similar bounds. We show that ∥T_σ - T_F∥ can be controlled, maintaining invertibility of S_σ.

Key lemma: If F and G are woven with bounds A, B, and ∥T_F - T_G∥ < A/2, then small perturbations of either frame retain the woven property. This extends to sequences where perturbations are summable, generalizing prior results.

4. Operator-Based Characterization

We characterize woven pairs via the angle between the nullspace of the mixed synthesis operator and ranges of oblique projections. Define the operator T_{F,G} : H → H × H by T_{F,G} x = (T_F x, T_G x). The pair (F, G) is woven iff for every partition σ, the restricted operator T_σ = (T_F|_σ, T_G|_{σ^c}) is surjective. This surjectivity is equivalent to the condition that the angle between N(T_{F,G}) and R(P_σ) is bounded below, where P_σ is an oblique projection onto the subspace corresponding to σ.

Specifically, let θ_σ be the minimal angle between N(T_{F,G}) and R(P_σ). Then, (F, G) is woven iff inf_σ θ_σ > 0. This resembles characterizations of Riesz frames, where uniformity across partitions is crucial. Applications include verifying wovenness for frames related through compact perturbations or finite-rank differences.

5. Statistical Overview

6. Key Insights

• Weakly woven pairs are woven, simplifying analysis to existence of frame bounds per partition.
• Synthesis operators provide a unified approach for perturbation proofs, avoiding coordinate-based arguments.
• The angle condition generalizes to Banach spaces and fusion frames, as indicated in prior works.
• Small perturbations in operator norm preserve the woven property, with explicit bounds derived from frame constants.
• Applications in sensor networks require robustness to node failures, modeled by partitions.

7. Conclusion

This note advances the theory of woven frames by establishing perturbation results through operator-theoretic methods. We show that small perturbations in the synthesis operator norm preserve the woven property, with bounds expressed in terms of frame constants. The characterization via angles between nullspaces and projection ranges offers a new perspective, linking wovenness to geometric properties in Hilbert spaces. Future work may extend these results to K-frames and continuous frames, further enhancing applications in distributed processing.