The truncated Hamburger moment problems with gaps in the index set

In this article we solve four special cases of the truncated Hamburger moment problem (THMP) of degree $2k$ with one or two missing moments in the sequence. As corollaries we obtain, by using appropriate substitutions, the solutions to bivariate truncated moment problems of degree $2k$ for special curves. Namely, for the curves $y=x^3$ (first solved by Fialkow), $y^2=x^3$, $y=x^4$ where a certain moment of degree $2k+1$ is known and $y^3=x^4$ with a certain moment given. The main technique is the completion of the partial positive semidefinite matrix (ppsd) such that the conditions of Curto and Fialkow's solution of the THMP are satisfied. The main tools are the use of the properties of positive semidefinite Hankel matrices and a result on all completions of a ppsd matrix with one unknown entry, proved by the use of the Schur complements for $2\times 2$ and $3\times 3$ block matrices.