A characterization of Fourier transforms
Volume 118 / 2010
Colloquium Mathematicum 118 (2010), 569-580
MSC: 42A38, 42A85, 42B10, 43A25.
DOI: 10.4064/cm118-2-12
Abstract
The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ${\mathbb Z}/ n{\mathbb Z}$, the integers ${\mathbb Z}$, the torus ${\mathbb T}$ and the real line. We also ask a related question for the twisted convolution.
