On Excesses and Duality in Woven Frames

TL;DR

This paper investigates the properties of woven frames in Hilbert spaces, focusing on excess, duality, and stability under perturbations, with implications for distributed signal processing.

Contribution

It introduces the concept of excess for woven frames, proves that woven frames share the same excess, and explores conditions under which duals and perturbations preserve the woven property.

Findings

Woven frames in separable Hilbert spaces have the same excess.

Frames with certain duals are woven if their redundant elements are small.

Perturbations can preserve the woven property under specific conditions.

Abstract

Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor networks. In this paper, we study the notion of excess for woven frames and prove that any two frames in a separable Hilbert space that are woven have the same excess. We also show that every frame with a large class of duals is woven provided that its redundant elements have small enough norm. Also, we try to transfer the woven property from frames to their duals and vise versa. Finally, we look at which perturbations of dual frames preserve the woven property, moreover it is shown that under some conditions the canonical Pareseval frame of two woven frames are also woven.