Characterizing compact families via the Laplace transform

TL;DR

This paper explores the use of the Laplace transform as an alternative to the Fourier transform for characterizing compact families in function spaces, extending classical results to broader settings.

Contribution

It introduces the Laplace transform as a new tool for characterizing compact families, expanding the scope beyond Fourier-based methods.

Findings

Laplace transform effectively characterizes compact families.

Extension of classical Fourier results to broader groups.

Potential for new applications in harmonic analysis.

Abstract

In 1985, Robert L. Pego characterized compact families in $L^2(\reals)$ in terms of the Fourier transform. It took nearly 30 years to realize that Pego's result can be proved in a wider setting of locally compact abelian groups (works of G\'orka and Kostrzewa). In the current paper, we argue that the Fourier transform is not the only integral transform that is efficient in characterizing compact families and suggest the Laplace transform as a possible alternative.