Frame related operators for woven frames

TL;DR

This paper introduces and analyzes fundamental operators such as analysis, synthesis, Gram, and frame operators for woven frames, exploring their properties and effects on frame bounds, with applications and examples.

Contribution

It is the first to define and study these operators specifically for woven frames, expanding the theoretical framework of frame theory.

Findings

Operators for woven frames are defined and their properties reviewed.

Different operators influence the bounds of woven frames.

Conditions are provided under which sums of woven frames remain woven frames.

Abstract

A new notion in frame theory has been introduced recently under the name woven-weaving frames by Bemrose et. al. In the studying of frames, some operators like analysis, synthesis, Gram and frame operator play the central role. In this paper, for the first time, we introduce and define these operators for woven-weaving frames and review some properties of them. In continuation, we investigate the effect of different types of operators on the woven frames and their bounds. Also, we provide some conditions that shows sum of woven frames are also woven frames. Finally, we apply these properties with an example.