Benedicks-type uncertainty principle for metaplectic time-frequency representations

TL;DR

This paper characterizes which metaplectic Wigner distributions satisfy an uncertainty principle akin to Benedicks, determining when finite measure support implies the functions are zero, extending known results from the short-time Fourier transform.

Contribution

It provides a complete characterization of metaplectic Wigner distributions that obey an uncertainty principle similar to Benedicks and Amrein-Berthier.

Findings

Uncertainty principle holds for certain metaplectic Wigner distributions.

Full characterization of distributions satisfying the principle.

Distinction between sesquilinear and quadratic versions.

Abstract

Metaplectic Wigner distributions are joint time-frequency representations that are parametrized by a symplectic matrix and generalize the short-time Fourier transform and the Wigner distribution. We investigate the question which metaplectic Wigner distributions satisfy an uncertainty principle in the style of Benedicks and Amrein-Berthier. That is, if the metaplectic Wigner distribution is supported on a set of finite measure, must the functions then be zero? While this statement holds for the short-time Fourier transform, it is false for some other natural time-frequency representations. We provide a full characterization of the class of metaplectic Wigner distributions which exhibit an uncertainty principle of this type, both for sesquilinear and quadratic versions.