Moment Summation Methods and Non-Homogeneous Carleman Classes

TL;DR

This paper characterizes the images of smooth function spaces under a generalized Laplace transform using novel non-homogeneous Carleman classes, extending classical theorems and applying to multi-summability and differential equations.

Contribution

It introduces and analyzes non-homogeneous Carleman classes, extending classical theorems and providing new criteria for moment summation methods and quasianalytic continuation.

Findings

Characterization of function space images under generalized Laplace transform

Development of non-homogeneous Carleman classes

Extension of quasianalytic continuation concepts

Abstract

We extend the classical theorems of F. Nevanlinna and Beurling by characterizing the image of various spaces of smooth functions under the generalized Laplace transform. To achieve this, we introduce and analyze novel non-homogeneous Carleman classes, which generalize the traditional homogeneous definitions. This characterization allows us to derive necessary and sufficient conditions for the applicability of moment summation methods within a given class of functions. Furthermore, we establish an extension of \'{E}calle's concept of quasianalytic continuation and apply these results to the theory of multi-summability and Euler-type differential equations.