Representation and Approximation of Pseudodifferential Operators by Sums of Gabor Multipliers

TL;DR

This paper introduces a novel method to represent and approximate pseudodifferential operators using sums of shifted Gabor multipliers, enabling better analysis and approximation especially for operators with off-diagonal decay.

Contribution

The paper develops a new representation of operators via sums of shifted Gabor multipliers based on their Gabor matrix structure, useful for operators with off-diagonal decay.

Findings

Characterizes symbol classes through Gabor multiplier symbols.

Provides approximation theorems for Sjostrand class pseudodifferential operators.

Links matrix decay properties to operator symbol behavior.

Abstract

We investigate a new representation of general operators by means of sums of shifted Gabor multipliers. These representations arise by studying the matrix of an operator with respect to a Gabor frame. Each shifted Gabor multiplier corresponds to a side-diagonal of this matrix. This representation is especially useful for operators whose associated matrix possesses some off-diagonal decay. In this case one can completely characterize the symbol class of the operator by the size of the symbols of the Gabor multipliers. As an application we derive approximation theorems for pseudodifferential operators in the Sjostrand class.