On the truncated two-dimensional moment problem

Abstract

We study the truncated two-dimensional moment problem (with rectangular data): to find a non-negative measure $\mu (\delta )$, $\delta \in \mathfrak{B}({\mathbb{R}}^2)$, such that ${\int }_{{\mathbb{R}}^2}{x}_1^m{x}_2^{n}d\mu =s_{m,n}$, $0\le m\le M,0\le n\le N$, where $\{s_{m,n}{\}}_{0\le m\le M,0\le n\le N}$ is a prescribed sequence of real numbers; $M,N\in {\mathbb{Z}}_{+}$. For the cases $M=N=1$ and $M=1,N=2$ explicit numerical necessary and sufficient conditions for the solvability of the moment problem are given. In the cases $M=N=2$; $M=2,N=3$; $M=3,N=2$; $M=3,N=3$ some explicit numerical sufficient conditions for the solvability are obtained. In all the cases some solutions (not necessarily atomic) of the moment problem can be constructed.

14 pages

References in corpus (1)

• The Nevanlinna-type parametrization for the operator Hamburger moment problem