On degenerate Hamburger moment problem and extensions of positive   semidefinite Hankel block matrices
  In this paper we consider two related objects: singular positive semidefinite Hankel block--matrices and associated degenerate truncated matrix Hamburger moment problems. The description of all solutions of a 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 positive semidefinite Hankel extension. 
Vladimir Bolotnikov