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A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS

  • WASSIM NASSERDDINE (a1)
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.

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

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References
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[14] Nasserddine, W., ‘Some Fourier analysis results on certain non-abelian groups’, submitted.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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