The strong truncated Hamburger moment problem with and without gaps

TL;DR

This paper solves the strong truncated Hamburger moment problem with gaps and its relation to specific two-dimensional moment problems, providing necessary and sufficient conditions for measure existence using Hankel matrices.

Contribution

It extends solutions to the strong truncated Hamburger moment problem, including cases with missing moments, and links these to specific two-dimensional moment problems with algebraic varieties.

Findings

Established necessary and sufficient conditions for the existence of measures.

Connected the STHMP to 2D TMP with variety xy=1.

Extended solutions to cases with missing moments.

Abstract

The strong truncated Hamburger moment problem (STHMP) of degree $(-2k_1,2k_2)$ asks to find necessary and sufficient conditions for the existence of a positive Borel measure, supported on $\mathbb{R}\setminus \{0\}$, such that $\beta_i=\int x^id\mu\; (-2k_1\leq i\leq 2k_2)$. Using the solution of the truncated Hamburger moment problem and the properties of Hankel matrices we solve the STHMP. Then, using the equivalence with the STHMP of degree $(-2k,2k)$, we obtain the solution of the 2-dimensional truncated moment problem (TMP) of degree $2k$ with variety $xy=1$, first solved by Curto and Fialkow. Our addition to their result is the fact previously known only for $k=2$, that the existence of a measure is equivalent to the existence of a flat extension of the moment matrix. Further on, we solve the STHMP of degree $(-2k_1,2k_2)$ with one missing moment in the sequence, i.e., $\beta_{-2k_1+1}$ or $\beta_{2k_2-1}$, which also gives the solution of the TMP with variety $x^2y=1$ as a special case, first studied by Fialkow.