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 ${\beta }_i=\int x^{i}d\mu (-2k_1\le i\le 2k_2)$. 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 $x^{2}y=1$ as a special case, first studied by Fialkow.

20 pages. arXiv admin note: text overlap with arXiv:2007.11846

References in corpus (5)

• Truncated K-moment problems in several variables
• The truncated Hamburger moment problems with gaps in the index set
• The Truncated Moment Problem for Unital Commutative R-Algebras
• On a minimal solution for the indefinite truncated multidimensional moment problem
• The moment problem on curves with bumps

Cited by in corpus (1)

• The truncated moment problem on the union of parallel lines