### On the truncated two-dimensional moment problem

Sergey Zagorodnyuk

#### 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.

#### Article information

Source
Adv. Oper. Theory Volume 3, Number 2 (2018), 388-399.

Dates
Accepted: 22 October 2017
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.aot/1513328638

Digital Object Identifier
doi:10.15352/AOT.1708-1212

#### Citation

Zagorodnyuk, Sergey. On the truncated two-dimensional moment problem. Adv. Oper. Theory 3 (2018), no. 2, 388--399. doi:10.15352/AOT.1708-1212. https://projecteuclid.org/euclid.aot/1513328638

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