Papers › The truncated moment problem on the union of parallel lines
The truncated moment problem on the union of parallel lines
Aljaž Zalar
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In this article we study the bivariate truncated moment problem (TMP) of degree 2k on the union of parallel lines. First we present an alternative proof of Fialkow's solution \cite{Fia15} to the TMP on the union of two parallel lines (TMP--2pl) using the solution of the truncated Hamburger moment problem (THMP). We add a new equivalent solvability condition, which is then used together with the THMP, to solve the TMP on the union of three parallel lines (TMP--3pl), our second main result of the article. Finally, we establish a sufficient condition for the existence of a solution to the TMP on the union of n parallel lines in the pure case, i.e.\ when the moment matrix Mₖ is of the highest possible rank, or equivalently the only column relations come from the union of n lines. The condition is based on the feasibility of a certain linear matrix inequality, corresponding to the extension of Mₖ by adding rows and columns indexed by some monomials of degree k+1. The proof is by induction on n, where n≥2 and for the base of induction n=2 we use the solution of the TMP--2pl.
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