$\tau$-quantization and $\tau$-Cohen classes distributions of Feichtinger operators

TL;DR

This paper explores $ au$-quantizations and Cohen's class distributions for Feichtinger operators, establishing their properties and showing their usefulness as a substitute for Schwartz operators in time-frequency analysis.

Contribution

It introduces the concept of Feichtinger operators and demonstrates their role as a practical operator-analog to Schwartz functions, extending time-frequency analysis tools.

Findings

Feichtinger operators serve as a convenient substitute for Schwartz operators.

Cohen's classes are characterized as convolutions of Wigner functions with distributions.

Schwartz operators are characterized as intersections of weighted Feichtinger operator classes.

Abstract

We investigate the $\tau$-quantizations and Cohen's class distributions of a suitable class of trace-class operators, called Feichtinger's operators, and show that it is a convenient substitute for the class of Schwartz operators. Many well-known concepts and results for functions in time-frequency analysis have an operator-analog in our setting, e.g. that Cohen's classes are convolutions of Wigner functions with distributions or characterization of the class of Schwartz operators as an intersection of weighted variants of the class of Feichtinger operators.