Benedicks-type uncertainty principle for metaplectic time-frequency representations

Abstract Metaplectic time-frequency representations 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 of which metaplectic time-frequency representations satisfy an uncertainty principle in the style of Benedicks and Amrein–Berthier. That is, if the metaplectic time-frequency representation 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 time-frequency representations which exhibit an uncertainty principle of this type, both for sesquilinear and quadratic versions.