A Characterisation of the Fourier transform on the Schwartz-Bruhat space of locally compact Abelian groups

TL;DR

This paper characterizes the Fourier transform on Schwartz-Bruhat functions over locally compact Abelian groups, showing that any bijection preserving convolution and pointwise multiplication is essentially the Fourier transform.

Contribution

It extends previous Euclidean Fourier transform characterizations to the broader setting of locally compact Abelian groups.

Findings

Any additive bijection exchanging convolution and pointwise product is the Fourier transform.

The proof parallels recent Euclidean Fourier transform characterizations.

The result generalizes Fourier transform properties to more abstract group settings.

Abstract

We obtain a characterisation of the Fourier transform on the space of Schwartz-Bruhat functions on locally compact Abelian groups. The result states that any appropriately additive bijection of the Schwartz space onto itself, which interchanges convolution and pointwise products is essentially the Fourier transform. The proof of this result is very similar to that obtained by the author recently for the Euclidean Fourier transform.