A weak qualitative uncertainty principle for compact groups

Abstract

For locally compact abelian groups it is known that if the product of the measures of the support of an $L^1$-function $f$ and its Fourier transform is less than $1$, then $f = 0$ almost everywhere. This is a weak version of the classical qualitative uncertainty principle. In this paper we focus on compact groups. We obtain conditions on the structure of a compact group under which there exists a lower bound for all products of the measures of the support of an integrable function and its Fourier transform, and conditions under which this bound equals $1$. For several types of compact groups, we determine the exact set of values which the product can attain.