The strong truncated Hamburger moment problem with and without gaps

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.