The truncated moment problem on the union of parallel lines

In this article we study the bivariate truncated moment problem (TMP) of degree 2k on union parallel lines. First present an alternative proof Fialkow's solution [28] to TMP two lines (TMP–2pl) using Hamburger (THMP). We add a new equivalent solvability condition, which is then used together with THMP, solve three (TMP–3pl), our second main result article. Finally, establish sufficient condition for existence n in pure case, i.e. when matrix Mk highest possible rank, or equivalently only column relations come from The based feasibility certain linear inequality, corresponding extension by adding rows and columns indexed some monomials k+1. induction n, where n≥2 base n=2 use TMP–2pl.