Abstract
It is well known that if the supports of a function f ϵ L 1 (R d ) and its Fourier transform \ ̂ tf 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 \ ̂ tf 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.
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