Moment problems in the Schwartz and Gelfand-Shilov spaces

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.