A survey on the unconditional convergence and the invertibility of multipliers with implementation

TL;DR

This survey reviews the properties of frame multipliers, focusing on their unconditional convergence and invertibility, including recent theoretical results and practical algorithms for Gabor and wavelet multipliers.

Contribution

It provides a comprehensive overview of conditions for convergence and invertibility of multipliers, extending recent results and including implementation details for specific frame types.

Findings

Conditions for unconditional convergence of multipliers

Criteria for invertibility of multipliers

Implementation of inverse representations in MATLAB

Abstract

The paper presents a survey over frame multipliers and related concepts. In particular, it includes a short motivation of why multipliers are of interest to consider, a review as well as extension of recent results, devoted to the unconditional convergence of multipliers, sufficient and/or necessary conditions for the invertibility of multipliers, and representation of the inverse via Neumann-like series and via multipliers with particular parameters. Multipliers for frames with specific structure, namely, Gabor and wavelet multipliers, are also considered. Some of the results for the representation of the inverse multiplier are implemented in Matlab codes and the algorithms are described.