Extented Paley-Wiener type theorems for the Mellin and Fourier transforms on the half-real line

TL;DR

This paper extends Paley-Wiener theorems to Mellin and Fourier transforms for functions with specific asymptotic behaviors, broadening the understanding of their analytic properties on the half-real line.

Contribution

It generalizes classical Paley-Wiener theorems to new settings involving Mellin and Fourier transforms with log-polyhomogeneous asymptotics.

Findings

Extended Paley-Wiener theorems for Mellin and Fourier transforms.

Applicable to functions with positive support and specific asymptotic expansions.

Based on and generalizing previous foundational results.

Abstract

In this paper we present generalisations of Paley-Wiener type theorems to Mellin and (Laplace-)Fourier transforms of rapidly decreasing smooth functions with positive support and log-polyhomogeneous asymptotic expansion at zero. This article is based on the thesis \cite{CdC} and uses results borrowed from \cite{RS}, \cite{RS81}, \cite{P79}, \cite{FGD}, \cite{P79}, \cite{SZ}.