Vector-Valued (Super) Weaving Frames

TL;DR

This paper explores the properties and construction of vector-valued weaving frames in Hilbert spaces, extending the concept of weaving frames with applications in signal processing and providing new theoretical insights and examples.

Contribution

It introduces fundamental properties of vector-valued weaving frames, establishes conditions for their construction, and connects them to woven frames for atomic spaces.

Findings

Woven vector-valued frames imply woven atomic space frames.

A technique for constructing vector-valued woven frames is provided.

Necessary and sufficient conditions for vector-valued weaving Riesz sequences are established.

Abstract

Two frames $\{\phi_{i}\}_{i \in I}$ and $\{\psi_{i}\}_{i \in I}$ for a separable Hilbert space $H$ are woven if there are positive constants $A \leq B$ such that for every subset $\sigma \subset I$, the family $\{\phi_{i}\}_{i \in \sigma} \cup \{\psi_{i}\}_{i \in \sigma^{c}}$ is a frame for $H$ with frame bounds $A, B$. Bemrose et al. introduced weaving frames in separable Hilbert spaces and observed that weaving frames has potential applications in signal processing. Motivated by this, and the recent work of Balan in the direction of application of vector-valued frames (or superframes) in signal processing, we study vector-valued weaving frames. In this paper, first we give some fundamental properties of vector-valued weaving frames. It is shown that if a family of vector-valued frames is woven, then the corresponding family of frames for atomic spaces is woven, but the converse is not true. We present a technique for the construction of vector-valued woven frames from given woven frames for atomic spaces . Necessary and sufficient conditions for vector-valued weaving Riesz sequences are given. Several numerical examples are given to illustrate the results.