Moment problems in the Schwartz and Gelfand-Shilov spaces

TL;DR

The paper characterizes when the unrestricted $K$-moment problem has a solution in the Schwartz space $\mathcal{S}(\mathbb{R}^d)$ and in Gelfand–Shilov spaces $\mathcal{S}^{\{M\}}(\mathbb{R}^d)$ for regular closed sets $K$. It reduces solvability to thickness-at-infinity conditions expressed as finite-dimensionality of polynomial spaces with $d_K$-weighted decay, using Eidelheit's theorem and Grothendieck's factorization to connect moment interpolation to linear algebra in Fréchet spaces. The results reveal a genuine dependence on the GS index (e.g., Gevrey $\sigma>1$) and provide a suite of examples, including unbounded $K$ and segmented sets $K_{a,b}$, illustrating how geometry governs solvability. This framework extends Schmüdgen's measure-theoretic results to function spaces, offering precise criteria for unrestricted moment problems in ultradifferentiable contexts and informing applications in analysis and spectral theory.

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 Schmüdgen [Expositiones Math. 43 (2025), 125657], who addressed the analogous problem for Radon measures.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Eidelheit's theorem
• Lemma 2.2
• proof
• proof : Alternative proof of Theorem \ref{['main-schmudgen']}
• Proposition 3.1
• proof