## Advances in Operator Theory

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

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

Received: 4 August 2017

Accepted: 22 October 2017

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.15352/AOT.1708-1212

**Subjects**

Primary: 47A57: Operator methods in interpolation, moment and extension problems [See also 30E05, 42A70, 42A82, 44A60]

Secondary: 44A60: Moment problems

**Keywords**

Hankel matrix moment problem non-linear inequalities

#### 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