Fourier transform of function on locally compact Abelian groups taking value in Banach spaces

TL;DR

This paper investigates the boundedness of the Fourier transform for vector-valued functions on locally compact Abelian groups, establishing conditions on the Banach space for the transform to be bounded.

Contribution

It characterizes when the Fourier transform is bounded on $L_2$ spaces for vector-valued functions, linking boundedness to the Banach space being isomorphic to a Hilbert space.

Findings

Fourier transform is bounded for finite groups.

Boundedness on infinite groups occurs only when the Banach space is Hilbert.

Provides a characterization of Banach spaces for bounded Fourier transforms.

Abstract

We consider Fourier transform of vector-valued functions on a locally compact group $G$, which take value in a Banach space $X$, and are square-integrable in Bochner sense. If $G$ is a finite group then Fourier transform is a bounded operator. If $G$ is an infinite group then Fourier transform $F: L_2(G,X)\to L_2(\widehat G,X)$ is a bounded operator if and only if Banach space $X$ is isomorphic to a Hilbert one.