Functions and their Fourier transforms with supports of finite measure for certain locally compact groups

Abstract

It is well known that if the supports of a function f ε L¹(R^d) and its Fourier transform ∖ are contained in bounded rectangles, then f=0 almost everywhere. In 1974 Benedicks relaxed the requirements for this conclusion by showing that the supports of f and ∖ need only have finite measure. In this paper we extend the validity of this property to a wide variety of locally compact groups. These include R^d×K, where K is a compact connected Lie group, the motion group, the affine group, the Heisenberg group, SL(2, R), and all noncompact semisimple groups with some additional restrictions on the functions f.