General Stieltjes moment problems for rapidly decreasing smooth functions

TL;DR

This paper establishes necessary and sufficient conditions for solving generalized Stieltjes moment problems with rapidly decreasing smooth functions, extending to measure, distribution, vector, and multidimensional cases.

Contribution

It provides a comprehensive characterization of when such moment problems have solutions within the Schwartz space, including generalizations to vector and multidimensional problems.

Findings

Characterization of solvability conditions for generalized Stieltjes moment problems

Extension to measure and distribution sequences

Analysis of vector and multidimensional moment problems

Abstract

We give (necessary and sufficient) conditions over a sequence $\left\{ f_{n}\right\} _{n=0}^{\infty}$ of functions under which every generalized Stieltjes moment problem \[ \int_{0}^{\infty} f_{n}(x)\phi(x)\mathrm{d} x=a_{n}, \ \ \ n\in\mathbb{N}, \] has solutions $\phi\in\mathcal{S}(\mathbb{R})$ with $\operatorname*{supp} \phi\subseteq[0,\infty)$. Furthermore, we consider more general problems of this kind for measure or distribution sequences $\left\{ f_{n}\right\} _{n=0}^{\infty}$. We also study vector moment problems with values in Frechet spaces and multidimensional moment problems.