The uncertainty principle and a generalization of a theorem of Tao

Abstract

Let G be a finite abelian group. If f:G→C is a nonzero function with Fourier transform f̂, the classical uncertainty principle states that |supp(f)||supp(f̂)|≥|G|. Recently, Tao showed that, if G is cyclic of prime order p, then in fact a stronger inequality |supp(f)|+|supp(f̂) |≥p+1 holds. In this paper, we use representation theory of the unitary group and Weyl's character formula to derive a generalization of Tao's result for arbitrary finite cyclic groups. © 2012 Elsevier Inc. All rights reserved.