ON THE TRUNCATED TWO-DIMENSIONAL MOMENT PROBLEM

We study the truncated two-dimensional moment problem (with rectangular data) to find a non-negative R measure mu(delta), delta is an element of B(R-2), such that integral(R2) x(1)(m) x(2)(n) d mu = s(m,n), 0 <= m <= M, 0 <= n <= N, where {s(m,n)} 0 <= m <= M, 0 <= n <= N is a prescribed sequence of real numbers; M, N is an element of 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.