Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X3220180401On the truncated two-dimensional moment problem3883995118110.15352/aot.1708-1212ENSergey ZagorodnyukV. N. Karazin Kharkiv National University
School of Mathematics and Computer Sciences
Department of Higher Mathematics and Informatics
Svobody Square 4, 61022, Kharkiv, UkraineJournal Article20170804We study the truncated two-dimensional moment problem (with rectangular data) to find a non-negative measure $mu(delta)$, $deltainmathfrak{B}(mathbb{R}^2)$, such that $int_{mathbb{R}^2} x_1^m x_2^n dmu = s_{m,n}$, $0leq mleq M,quad 0leq nleq N$, where ${ s_{m,n} }_{0leq mleq M, 0leq nleq N}$ is a prescribed sequence of real numbers; $M,Ninmathbb{Z}_+$. 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.http://www.aot-math.org/article_51181_e83e76bde83920b1d8fc1a07b6244513.pdf