Moment problems in the Schwartz and Gelfand-Shilov spaces

TL;DR

This paper characterizes the geometric conditions on closed sets in Euclidean space that determine when every real sequence can be realized as moments of Schwartz or Gelfand-Shilov functions supported on those sets.

Contribution

It provides a new geometric characterization of supports for Schwartz and Gelfand-Shilov space moment problems, extending previous Radon measure results.

Findings

Characterization of sets for Schwartz moment sequences

Extension to Gelfand-Shilov spaces

Discussion of illustrative examples

Abstract

We provide a geometric characterization of the closed sets $K \subseteq \mathbb{R}^d$ such that every real $d$-sequence is the moment sequence of some Schwartz function on $\mathbb{R}^d$ with support in $K$. We obtain a similar result for Gelfand-Shilov spaces. Several illustrative examples are discussed. Our work is inspired by a recent result of Schm\"udgen [Expositiones Math. 43 (2025), 125657], who addressed the analogous problem for Radon measures.