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A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  28 November 2018

WASSIM NASSERDDINE Open the ORCID record for WASSIM NASSERDDINE [Opens in a new window]
WASSIM NASSERDDINE*
Affiliation:
Faculty of Sciences – Section I, Lebanese University, 2905-3901 Hadath, Beirut, Lebanon email wassim.nasserddine@ul.edu.lb
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Abstract

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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.

Keywords

Paley–Wiener propertyqualitative uncertainty principleFourier algebraFourier transformationFourier cotransformation

MSC classification

Primary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
Secondary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 114 - 120
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2018 Australian Mathematical Publishing Association Inc.

Footnotes

This project has been funded with support from the Lebanese University.

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