A QUALITATIVE UNCERTAINTY PRINCIPLE FOR UNIMODULAR GROUPS OF TYPE-I

It has long been known that if f is-an-element-of L2(R(n)) and the sup ports of f and its Fourier transform f are bounded then f = 0 almost e verywhere. More recently it has been shown that the same conclusion ca n be reached under the weaker condition that the supports of f and f h ave finite measure. These results may be thought of as qualitative unc ertainty principles since they limit the ''concentration'' of the Four ier transform pair (f, f). Little is known, however, of analogous resu lts for functions on locally compact groups. A qualitative uncertainty principle is proved here for unimodular groups of type I.