Representation of operators in the time-frequency domain and generalized Gabor multipliers

TL;DR

This paper develops a comprehensive framework for representing and approximating linear operators in the time-frequency domain using Gabor multipliers, including generalized forms, with efficient computation methods and error estimates.

Contribution

It introduces a characterization of operators as Gabor multipliers, discusses optimal approximation conditions, and proposes generalized Gabor multipliers for better handling overspread operators.

Findings

Characterization of operators realizable as Gabor multipliers

Efficient method for optimal Gabor multiplier approximation

Introduction of generalized Gabor multipliers for improved approximation

Abstract

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator's best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator's best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.