Irrational series I Laplace transform in a neighborhood of $-\infty$

TL;DR

This paper investigates the behavior of Laplace transforms of exponential sums near negative infinity, focusing on convergence, decomposition, and continuity properties in complex neighborhoods.

Contribution

It develops a framework for decomposing holomorphic functions into exponential sums in neighborhoods of , extending classical Laplace transform theory.

Findings

Laplace transform and inverse are continuous in these neighborhoods

Partial sum operations are continuous under certain conditions

Resummation formulas are established for exponential sums

Abstract

Discrete sums of exponentials $g(w) = \sum a_{\beta} \mathrm{e}^{\beta w}$ with positive exponents may converge not normally in neighborhoods $H$ of $-\infty$ which do not contain half-planes. In order to obtain a decomposition of a holomorphic function $g$ in $H$ as a sum of exponentials we study the Laplace transform in general neighborhoods of $-\infty$. We adress questions such as continuity of Laplace and inverse Laplace transformations, continuity for the operation of taking partial sums, and resummation formulas.