Uncertainty Principles for Time-Frequency Representations

K. Gröchenig

Department of Mathematics, The University of Connecticut, Storrs, CT 06269-3009, USA
email: groch@math.uconn.edu

We present a machine to produce new uncertainty principles from old ones. Traditionally an uncertainty principle is an inequality involving both a function $f$ and its Fourier transform $\hat{f} $. To generate new uncertainty principles, we interpret the pair $(f, \hat{f} )$ as a time-frequency representation, replace it by a different time-frequency representation, and formulate a corresponding inequality. We discuss a few recent uncertainty principles for the short-time Fourier transform and the Wigner distribution that can be obtained in this way and suggest further problems.