A Characterisation of the Euclidean Fourier transform on the Schwartz space

TL;DR

This paper characterizes the Fourier transform on Schwartz space as the unique bijection that interchanges convolution and pointwise multiplication, providing a fundamental understanding of its algebraic properties.

Contribution

It offers a new characterization of the Fourier transform on Schwartz space based on algebraic properties, specifically its role in interchanging convolution and multiplication.

Findings

Any additive bijection of Schwartz space that swaps convolution and pointwise product is essentially the Fourier transform.

The result provides a fundamental algebraic characterization of the Fourier transform.

The characterization applies to Schwartz class functions on .

Abstract

We obtain a characterisation of the Fourier transform on the space of Schwartz class functions on $\mathbb{R}^n.$ 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.