On the truncated two-dimensional moment problem

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${ s_{m,n} }{0\leq m\leq M,\ 0\leq n\leq N}$isaprescribedsequenceofrealnumbers;$M,N\in$\mathbb{Z}$+$\mathrm{.}Forthecases$M=N=1$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.