On the Fourier transform on function algebras on locally compact abelian groups
type: Item
instance of: scholarly article
title: On the Fourier transform on function algebras on locally compact abelian groups
zbMATH Open document ID: 1382.43003
author: R. Lakshmi Lavanya
published in: Monatshefte für Mathematik
publication date: 2017-11-30T00:00:00Z
review text: Fourier analysis on \(\mathbb{R}^n\) is one of the classical topics in analysis. It is well-known that the Fourier transform intertwines convolution and pointwise product, i.e., \(\forall\;f,g\in\mathcal{S}(\mathbb{R}^n)\) \(\mathcal{F}(f*g)=\mathcal{F}(f).\mathcal{F}(g)\) and \(\mathcal{F}(f.g)=\mathcal{F}(f)\ast\mathcal{F}(g).\) Further, the Fourier transform is a bijection from \(\mathcal{S}(\mathbb{R}^n)\) on to \(\mathcal{S}(\mathbb{R}^n)\). \textit{S. Alesker} et al. [in: Linear and complex analysis. Dedicated to V. P. Havin on the occasion of his 75th birthday. Providence, RI: American Mathematical Society (AMS). 11--26 (2009; Zbl 1184.42009)] showed that the only bijection from \(\mathcal{S}(\mathbb{R}^n)\) to \(\mathcal{S}(\mathbb{R}^n)\) which intertwines convolution and pointwise product is essentially the Fourier transform. More precisely, if \(T:\mathcal{S}(\mathbb{R}^n)\rightarrow\mathcal{S}(\mathbb{R}^n)\) is a bijection which admits a bijective extension \(T:\mathcal{S}^\prime(\mathbb{R}^n)\rightarrow\mathcal{S}^\prime(\mathbb{R}^n)\) such that \(\forall\;f\in \mathcal{S}(\mathbb{R}^n)\) and \(\varphi\in \mathcal{S}^\prime(\mathbb{R}^n),\) \[ T(f\ast\varphi) =T(f).T(\varphi) \] and \[ T(f.\varphi) = T(f)\ast T(\varphi) \] then \(\exists\) \(B\in SL(n,\mathbb{R})\) such that \(T(f)=\mathcal{F}(f\circ B)\) or \(T(f)=\mathcal{F}(\overline{f\circ B})\;\forall\;f\in\mathcal{S}(\mathbb{R}^n).\) The paper under review proves the above result in a more general context of locally compact abelian groups. The author defines a notion of Fourier twins on a locally compact abelian group with dual group \(\Gamma\). The classical Schwartz space \(\mathcal{S}(\mathbb{R}^n)\) is an example for a Fourier twin. More generally, the Schwartz-Bruhat space, defined and studied by Bruhat on general locally compact groups, is an example for a Fourier twin. Further examples are the Feichtinger's Segal algebra and the Wiener algebra on a locally compact abelian group. These exa…
reviewed by: N. Shravan Kumar
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 6813917
zbMATH Keywords: locally compact abelian groups
zbMATH Keywords: Schwartz-Bruhat space
zbMATH Keywords: Segal algebra
zbMATH Keywords: Feichtinger algebra
zbMATH Keywords: Fourier transform
MaRDI profile type: Publication
full work available at URL: s00605-017-1069-3
OpenAlex ID: W2617994794
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DOI: 10.1007/S00605-017-1069-3
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description: scientific article; zbMATH DE number 6813917