A characterization of Fourier transforms

TL;DR

This paper demonstrates that Fourier transforms are uniquely characterized as the only continuous linear maps converting convolution into pointwise multiplication across various groups, including cyclic groups, integers, the torus, and the real line.

Contribution

It provides a comprehensive characterization of Fourier transforms as the sole continuous linear maps transforming convolution into pointwise product on multiple fundamental groups.

Findings

Fourier transforms uniquely characterize convolution to pointwise multiplication.

The results apply to cyclic groups, integers, the torus, and the real line.

Extension to twisted convolution is also discussed.

Abstract

The aim of this paper is to show that, in various situations, the only continuous linear map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups $\Z/nZ$, the integers $\Z$, the Torus $\T$ and the real line. We also ask a related question for the twisted convolution.