On the truncated multidimensional moment problems in $\mathbb{C}^n$

TL;DR

This paper investigates the truncated multidimensional moment problem in complex space, providing conditions for the existence of a non-negative measure that matches given moments, with implications for polynomial linear functionals.

Contribution

It introduces simple solvability conditions for the truncated complex moment problem, extending classical results and connecting to integral representations of linear functionals.

Findings

Derived simple solvability criteria for the problem

Extended the classical moment problem to complex space

Established integral representation results for linear functionals

Abstract

We consider the problem of finding a (non-negative) measure $\mu$ on $\mathfrak{B}(\mathbb{C}^n)$ such that $\int_{\mathbb{C}^n} \mathbf{z}^{\mathbf{k}} d\mu(\mathbf{z}) = s_{\mathbf{k}}$, $\forall \mathbf{k}\in\mathcal{K}$. Here $\mathcal{K}$ is an arbitrary finite subset of $\mathbb{Z}^n_+$, which contains $(0,...,0)$, and $s_{\mathbf{k}}$ are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. At first, one may consider this problem as an extension of the truncated multidimensional moment problem on $\mathbb{R}^n$, where the support of the measure $\mu$ is allowed to lie in $\mathbb{C}^n$. Secondly, the moment problem is a particular case of the truncated moment problem in $\mathbb{C}^n$, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials.