Equivalent norms for modulation spaces from positive Cohen's class distributions

TL;DR

This paper introduces a new class of equivalent norms for modulation spaces using positive Cohen's class distributions and Hilbert-Schmidt operators, expanding the tools for analyzing time-frequency representations.

Contribution

It establishes a novel framework for modulation space norms via positive Cohen's class distributions, linking them to time-frequency Wiener amalgam spaces and analyzing the nuclearity condition.

Findings

Equivalent norms for modulation spaces using Cohen's class distributions

Connection between modulation spaces and Wiener amalgam spaces

Characterization of Hilbert-Schmidt operators satisfying nuclearity

Abstract

We give a new class of equivalent norms for modulation spaces by replacing the window of the short-time Fourier transform by a Hilbert-Schmidt operator. The main result is applied to Cohen's class of time-frequency distributions, Weyl operators and localization operators. In particular, any positive Cohen's class distribution with Schwartz kernel can be used to give an equivalent norm for modulation spaces. We also obtain a description of modulation spaces as time-frequency Wiener amalgam spaces. The Hilbert-Schmidt operator must satisfy a nuclearity condition for these results to hold, and we investigate this condition in detail.