The truncated univariate rational moment problem

Abstract

Given a closed subset K in $R$, the rational K–truncated moment problem (K–RTMP) asks to characterize the existence of a positive Borel measure μ, supported on K, such that a linear functional $L$, defined on all rational functions of the form $\frac{f}{q}$, where q is a fixed polynomial with all real zeros of even order and f is any real polynomial of degree at most 2k, is an integration with respect to μ. The case of a compact set K was solved in [4], but there is no argument that ensures that μ vanishes on all real zeros of q. An obvious necessary condition for the solvability of the K–RTMP is that $L$ is nonnegative on every f satisfying $f{|}_K\ge 0$. If $L$ is strictly positive on every $0\ne f{|}_K\ge 0$, we add the missing argument from [4] and also bound the number of atoms in a minimal representing measure. We show by an example that nonnegativity of $L$ is not sufficient and add the missing conditions to the solution. We also solve the K–RTMP for unbounded K and derive the solutions to the strong truncated Hamburger moment problem and the truncated moment problem on the unit circle as special cases.