Enhanced Laplace transform and holomorphic Paley-Wiener-type theorems

TL;DR

This paper develops explicit holomorphic Paley-Wiener-type theorems using enhanced Laplace transforms and contour integration, connecting classical results with modern microlocal analysis techniques.

Contribution

It introduces a method to derive explicit Paley-Wiener theorems from enhanced Laplace transforms, linking classical theorems to contemporary microlocal analysis.

Findings

Reformulation of Paley-Wiener theorems via enhanced Laplace transform

Explicit contour integral representations in the new framework

Connection of classical theorems to modern microlocal analysis

Abstract

Starting from a remark about the computation of Kashiwara-Schapira's enhanced Laplace transform by using the Dolbeault complex of enhanced distributions, we explain how to obtain explicit holomorphic Paley-Wiener-type theorems. As an example, we get back some classical theorems due to Polya and M\'{e}ril as limits of tempered Laplace-isomorphisms. In particular, we show how contour integrations naturally appear in this framework.