A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS. Issue 1 (28th November 2018)

• Record Type: Journal Article
• Authors: NASSERDDINE, WASSIM
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 99:Issue 1(2019)
• Page Start: 114
• Page End: 120
• Publication Date: 2018-11-28
• Subjects: primary 43A30, -- secondary 43A25; Paley–Wiener property, -- qualitative uncertainty principle, -- Fourier algebra, -- Fourier transformation, -- Fourier cotransformation; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972718001119
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ 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, $\hat{f}$, the regularised Fourier cotransform of $f$, is compactly supported and $\text{Im}\, \hat{f}$ is finite dimensional, then $f=0$ . In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.