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A characterization of Fourier transforms. (English) Zbl 1193.43004

The author deals with two types of results. The first one concerns the characterization of the Fourier transform (on locally compact Abelian groups) as being essentially the only continuous linear functional that turns a convolution product into a pointwise product. Here, he succeeds to remove certain assumptions used by the previous authors. Also, he adapts the proof of a recent theorem by Alesker, Artstein-Avidan, and Milman to obtain an analogous result on the cyclic group. Secondly, the author extends a theorem by Cooper on intertwining translations and modulations by the Fourier transform to other groups than the real axis.

MSC:

43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A85 Convolution, factorization for one variable harmonic analysis
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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