Characterization of Fourier transform of $H$-valued functions on the real line

Abstract

A characterization is obtained for the Fourier transform of functions belonging to ${\mathcal{ℒ}}^2\left(ℝ,H\right)$, where $H$ denotes a Hilbert $C^{\ast }$-module. But in the case of functions belonging to $L^1\left(ℝ,H\right)$ a similar result is proved when $H$ is a separable Hilbert space.