On the truncated two-dimensional moment problem

TL;DR

This paper investigates the truncated two-dimensional moment problem, providing explicit necessary and sufficient conditions for certain small cases and sufficient conditions for larger cases, along with methods to construct solutions.

Contribution

It offers explicit numerical criteria for solvability of the truncated 2D moment problem in specific cases and introduces solution construction methods.

Findings

Explicit necessary and sufficient conditions for M=N=1 and M=1, N=2.

Explicit sufficient conditions for M=N=2, M=2, N=3, M=3, N=2, and M=3, N=3.

Construction methods for solutions, not necessarily atomic.

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 $\int_{\mathbb{R}^2} x_1^m x_2^n d\mu = s_{m,n}$, $0\leq m\leq M,\quad 0\leq n\leq N$, where $\{ s_{m,n} \}_{0\leq m\leq M,\ 0\leq n\leq N}$ 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.