Integral transforms characterized by convolution

TL;DR

This paper introduces a new method to characterize Fourier, cosine, and Laplace transforms using convolution properties on locally compact abelian groups, extending previous results and unifying transform characterizations.

Contribution

The paper provides a general convolution-based characterization of Fourier, cosine, and Laplace transforms on locally compact abelian groups, broadening the scope of existing Fourier analysis results.

Findings

Characterization of Fourier transform on any locally compact abelian group.

Extension of Jaming's Fourier characterization to broader groups.

Convolution-based characterization of cosine and Laplace transforms.

Abstract

Inspired by Jaming's characterization of the Fourier transform on specific groups via the convolution property, we provide a novel approach which characterizes the Fourier transform on any locally compact abelian group. In particular, our characterization encompasses Jaming's results. Furthermore, we demonstrate that the cosine transform as well as the Laplace transform can also be characterized via a suitable convolution property.