Uncertainty in time--frequency representations on finite Abelian groups and applications

Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transforms. We use these results and their proofs to obtain similar results relating the support sizes of functions and their short--time Fourier transforms. Further, we discuss applications of our results. For example, we use our results to construct a class of equal norm tight Gabor frames that are maximally robust to erasures and we discuss consequences of our findings to the theory of recovering and storing signals which have sparse time--frequency representations.