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