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

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.

Corollary 1 was corrected

References in corpus (1)

• Complex moment problem and recursive relations