On degenerate Hamburger moment problem and extensions of nonnegative Hankel block matrices

In this paper we consider two related objects: a singular finite Hankel block-matrix and an associated degenerate truncated matrix Hamburger moment problem. The description of all solutions of the degenerate matrix Hamburger moment problem is given in terms of a Linear fractional transformation. The case of interest is the Hamburger moment problem whose Hankel block-matrix admits a nonnegative Hankel extension (it turns out that a singular finite nonnegative Hankel block-matrix in general does not admit a nonnegative Hankel block-matrix extension). We give necessary and sufficient conditions for a Hankel block-matrix to be extensible and describe all one-step nonnegative Hankel extensions.