Metaplectic Wigner distributions

TL;DR

This paper surveys metaplectic Wigner distributions, a generalization of the classical Wigner distribution, exploring their applications in time-frequency analysis and introducing new results on uncertainty principles and operator continuity.

Contribution

It provides a comprehensive survey of metaplectic Wigner distributions and extends theoretical results on uncertainty principles and pseudodifferential operator continuity.

Findings

Generalized Lieb's uncertainty principle to matrix Wigner distributions.

Proved continuity of metaplectic pseudodifferential operators on modulation spaces.

Enhanced understanding of time-frequency analysis using metaplectic structures.

Abstract

Metaplectic Wigner distributions were recently investigated as natural generalizations of the classical Wigner distribution, and provide a wide class of time-frequency representations that exploits the structure of the symplectic group. This work serves as a survey on metaplectic Wigner distributions and their applications to the time-frequency analysis of modulation spaces and pseudodifferential operators, topics that are all still poorly understood. We also give some new results, generalizing Lieb's uncertainty principle to the so-called matrix Wigner distributions and proving the continuity on $M^p_{v_s}(\mathbb{R}^d)$ spaces of metaplectic pseudodifferential operators with symbols in $M^{p'}_{v_{-s}}(\mathbb{R}^d)$.