A characterization of product preserving maps with applications to a characterization of the Fourier transform

Abstract

It is shown that a product preserving bijective (not necessarily real linear or continuous) operator on an appropriate class of complex valued functions must have either the form $[\phi\mapsto \phi\circ u]$ or $[\phi\mapsto\overline{\phi\circ u}]$ where $u$ is a fixed diffeomorphism of the base.