Advances in Operator Theory

On the truncated two-dimensional moment problem

Sergey Zagorodnyuk

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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


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References

  • D. Cichoń, J. Stochel, and F. H. Szafraniec, Extending positive definiteness, Trans. Amer. Math. Soc. 363 (2011), no. 1, 545–577.
  • R. Curto and L. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc. 119 (1996), no. 568, 1–52.
  • R. Curto and L. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Mem. Amer. Math. Soc. 136 (1998), no. 648, 1–56.
  • L. A. Fialkow, Solution of the truncated moment problem with variety $y=x^3$, Trans. Amer. Math. Soc. 363 (2011), no. 6, 3133–3165.
  • J.-B. Lasserre, Moments, positive polynomials and their applications, World Scientific, Singapore, 2010.
  • F.-H. Vasilescu, Dimensional stability in truncated moment problems, J. Math. Anal. Appl. 388 (2012), no. 1, 219–230.
  • F.-H. Vasilescu, An idempotent approach to truncated moment problems, Integral Equations Operator Theory 79 (2014), no. 3, 301–335.
  • S. Yoo, Sextic moment problems on 3 parallel lines, Bull. Korean Math. Soc. 54 (2017), no. 1, 299–318.
  • S. M. Zagorodnyuk, On the complex moment problem on radial rays, Zh. Mat. Fiz. Anal. Geom. 1 (2005), no. 1, 74–92.
  • S. M. Zagorodnyuk, A description of all solutions of the matrix Hamburger moment problem in a general case, Methods Funct. Anal. Topology 16 (2010), no. 3, 271–288.