Weaving Frames

TL;DR

This paper investigates the properties of woven frames in Hilbert spaces, establishing fundamental characteristics, differences from Riesz bases, and potential applications in signal processing and wireless sensor networks.

Contribution

It introduces the concept of woven frames, explores their properties, and demonstrates key differences from Riesz bases, including the non-wovenness of Riesz bases with redundant frames.

Findings

Woven frames are defined with uniform frame bounds for all subsets.

A Riesz basis cannot be woven with a redundant frame.

Weaving is shown to be equivalent to a weaker but related concept.

Abstract

We study an intriguing question in frame theory we call "Weaving Frames" that is partially motivated by preprocessing of Gabor frames. Two frames $\{\varphi_i\}_{i\in I}$ and $\{\psi_i \}_{i\in I}$ for a Hilbert space ${\mathbb H}$ are woven if there are constants $0<A \le B $ so that for every subset $\sigma \subset I$, the family $\{\varphi_i\}_{i\in \sigma} \cup \{\psi_i\}_{i\in \sigma^c}$ is a frame for $\mathbb H$ with frame bounds $A,B$. Fundamental properties of woven frames are developed and key differences between weaving Riesz bases and weaving frames are considered. In particular, it is shown that a Riesz basis cannot be woven with a redundant frame. We also introduce an apparently weaker form of weaving but show that it is equivalent to weaving. Weaving frames has potential applications in wireless sensor networks that require distributed processing under different frames, as well as preprocessing of signals using Gabor frames.