Mixed-state localization operators: Cohen's class and trace class operators

TL;DR

This paper extends the theory of mixed-state localization operators within Cohen's class using Werner's operator convolutions, providing new insights into their properties, representations, and applications in phase space analysis.

Contribution

It generalizes known results from rank-one to trace class operators and characterizes mixed-state localization operators as positive correspondence rules.

Findings

Characterization of mixed-state localization operators as positive correspondence rules

Description of Cohen class via Werner's operator convolution

Results on positive Cohen class distributions and phase retrieval

Abstract

We study mixed-state localization operators from the perspective of Werner's operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore, we provide a description of the Cohen class in terms of Werner's convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen's class.