On the Fourier transform on function algebras on locally compact abelian groups
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 examples are discussed in Section 2 of this paper. The main result of this paper is a generalization of the theorem of Alesker et al. to the context of locally compact abelian groups where the Schwartz space is replaced by the Fourier twin pair. This is Theorem 3.1. The proof of this theorem is obtained as a consequence of a more general theorem proved in this paper which is Theorem 3.2. More precisely, let \(\mathcal{A}_G\) denote a function algebra w.r.t. convolution product and pointwise product, on a locally compact abelian group \(G\), containing \(C_c(G)\) and contained in \(C_0(G)\cap L^1(G).\) If \(T:\mathcal{A}_G\rightarrow\mathcal{A}_G\) is a bijection such that \(\forall\;f,g\in\mathcal{A}_G\), {\parindent=8mm\begin{itemize}\item[(i)] \(T(f+g^\ast)=T(f)+T(g)^\ast\)\item[(ii)] \(T(f.g)=T(f)\ast T(g)\)\item[(iii)] \(T(f\ast g)=T(f).T(g)\) \end{itemize}} then there is a measure preserving multiplicative homeomorphism \(\psi:G\rightarrow G\) such that either \(T(f)=f\circ\psi\) or \(T(f)=\overline{f\circ\psi}\;\forall\;f\in\mathcal{A}_G\). The proof of this theorem is proved in ten steps following the ideas of Alesker et al. In the case of \(\mathbb{R}^n\), one gets a matrix while in a general locally compact abelian group one gets only a measure preserving transformation. This is because of the fact that \(\mathbb{R}^n\) has a natural vector space structure while a general locally compact abelian group doesn't have such a structure. Another observation is that in the theorem due to Alesker et al., they needed a condition that the bijection \(T\) on the Schwartz space \(\mathcal{S}(\mathbb{R}^n)\) has a continuous bijective extension to the space of tempered distributions. This condition is not needed to prove that \(T\) is essentially the Fourier transform.
scientific article; zbMATH DE number 6813917
scientific article