The truncated univariate rational moment problem

TL;DR

This paper advances the understanding of the rational truncated moment problem by providing necessary and sufficient conditions for the existence of representing measures, including for unbounded sets and special cases like the Hamburger and unit circle problems.

Contribution

It introduces new conditions ensuring the existence of measures for the rational truncated moment problem, extending previous results to unbounded sets and addressing gaps in earlier solutions.

Findings

Established necessary and sufficient conditions for the rational K-truncated moment problem

Extended solutions to unbounded sets and special cases like Hamburger and unit circle problems

Bounded the number of atoms in minimal representing measures

Abstract

Given a closed subset $K$ in $\mathbb{R}$, the rational $K$-truncated moment problem ($K$-RTMP) asks to characterize the existence of a positive Borel measure $\mu$, supported on $K$, such that a linear functional $\mathcal{L}$, 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 $\mu$. The case of a compact set $K$ was solved by Chandler in 1994, but there is no argument that ensures that $\mu$ vanishes on all real zeros of $q$. An obvious necessary condition for the solvability of the $K$-RTMP is that $\mathcal{L}$ is nonnegative on every $f$ satisfying $f|_{K}\geq 0$. If $\mathcal{L}$ is strictly positive on every $0\neq f|_{K}\geq 0$, we add the missing argument from Chandler's solution and also bound the number of atoms in a minimal representing measure. We show by an example that nonnegativity of $\mathcal{L}$ 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.