A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS. Issue 1 (28th November 2018) - NASSERDDINE, WASSIM
Abstract : Let G be a separable locally compact group with type I left regular representation, G its dual, A(G) its Fourier algebra and fA(G) with compact support. If G=R and the Fourier transform of f is compactly supported, then, by a classical Paley–Wiener theorem, f=0 . There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if G has no (nonempty) open compact subsets, f, the regularised Fourier cotransform of f, is compactly supported and Im f is finite dimensional, then f=0 . In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.