Sur le groupe affine d'un corps local

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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A class of Banach spaces

Nasserddine¹

2007

Proc. Japan Acad. Ser. A Math. Sci.

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Sur le groupe affine d'un corps local

Cited by 3 publications

References 5 publications

Une caractérisation de l'algèbre de Fourier pour certains groupes localement compacts

Une caractérisation de l'algèbre de Fourier pour certains groupes localement compacts

A Compact Qualitative Uncertainty Principle for Some Nonunimodular Groups

A class of Banach spaces

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