The truncated Hamburger moment problems with gaps in the index set

TL;DR

This paper solves specific cases of the truncated Hamburger moment problem with missing moments, providing solutions for related bivariate moment problems on certain algebraic curves using matrix completion techniques.

Contribution

It introduces methods for solving truncated Hamburger moment problems with gaps and applies these to bivariate problems on special curves, expanding existing solution techniques.

Findings

Solved four special cases of the truncated Hamburger moment problem with missing moments.

Derived solutions for bivariate moment problems on specific algebraic curves.

Developed matrix completion techniques using positive semidefinite Hankel matrices.

Abstract

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.