A characterization of product preserving maps with applications to a characterization of the Fourier transform
This paper presents novel results on product-preserving maps and maps that turn products into convolutions. It also strengthens the characterizations from a previous article by \textit{S. Alesker, S. Artstein-Avidan} and \textit{V. Milman} [in: Linear and complex analysis, Providence, RI: AMS. Translations. Series 2. American Mathematical Society 226; Advances in the Mathematical Sciences 63, 11--26 (2009; Zbl 1184.42009)]. To formulate the first result, take a bijective map on the Schwartz space \(\mathcal{T}:\mathcal{S}\to\mathcal{S}\) that admits an extension to the space of tempered distributions \(\mathcal{T}':\mathcal{S}'\to\mathcal{S}'\) satisfying \(\mathcal{T}'(f\cdot g)=(\mathcal{T}f)\cdot(\mathcal{T}'g)\) for \(f\in\mathcal{S}\) and \(g\in\mathcal{S}'\). Here ``\(\cdot\)'' denotes the usual pointwise product. One can conclude that \(\mathcal{T}\) must be either of the form \(\mathcal{T}f=f\circ u\), or of the form \(\mathcal{T}f=\overline{f\circ u}\) for some \(C^\infty\)-diffeomorphism \(u\) of \(\mathbb{R}^n\). A real-valued variant of this theorem already appeared in the previously mentioned article, but the version with complex-valued functions has wider potential applicability and is required in the following characterization of the Fourier transform. For the second result, the authors assume that a bijection \(\mathcal{F}:\mathcal{S}\to\mathcal{S}\) admits a bijective extension \(\mathcal{F}':\mathcal{S}'\to\mathcal{S}'\) such that \(\mathcal{F}'(f\cdot g)=(\mathcal{F}f)\ast(\mathcal{F}'g)\) for \(f\in\mathcal{S}\) and \(g\in\mathcal{S}'\). Here ``\(\ast\)'' denotes the operation of convolution. Then they conclude that \(\mathcal{F}\) has to be either of the form \(\mathcal{F}f=\mathbb{F}(f\circ w)\), or of the form \(\mathcal{F}f=\overline{\mathbb{F}(f\circ w)}\), where \(w\) is a \(C^\infty\)-diffeomorphism of \(\mathbb{R}^n\) and \(\mathbb{F}\) denotes the Fourier transform. It is important to observe that no linearity of \(\mathcal{F}\) is assumed, but real-linearity follows as a consequence of the characterization. A special case of this theorem, where \(\mathcal{F}'\) is also required to turn convolutions into products, already appeared in the previously mentioned paper. The authors also discuss the natural generalization of their first result to product-preserving maps on the dual of the space of test-functions on general smooth manifolds.
scientific article; zbMATH DE number 6122087