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 f ˆ , the classical uncertainty principle states that | supp ( f ) | | supp ( f ˆ ) | ⩾ | G | . Recently, Tao showed that, if G is cyclic of prime order p, then in fact a stronger inequality | supp ( f ) | + | supp ( f ˆ ) | ⩾ 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.