Complex moment problem and recursive relations

TL;DR

This paper presents a new approach to solving the truncated complex moment problem by analyzing recursive sequences and characteristic polynomials, providing explicit solutions for certain polynomial forms and recapturing recent results.

Contribution

It introduces a novel strategy involving recursive doubly indexed sequences and characteristic polynomials for the complex moment problem, including explicit solutions for specific polynomial cases.

Findings

Provided a computable solution for cubic harmonic characteristic polynomials.

Recaptured a recent result by Curto-Yoo for cubic column relations.

Solved the moment problem with column dependence relations of a general form.

Abstract

We introduce a new strategy in solving the truncated complex moment problem. To this aim we investigate recursive doubly indexed sequences and their characteristic polynomials. A characterization of recursive doubly indexed \emph{moment} sequences is provided. As a simple application, we obtain a computable solution to the complex moment problem for cubic harmonic characteristic polynomials of the form $z^3+az+b\overline{z}$, where $a$ and $b$ are arbitrary real numbers. We also recapture a recent result due to Curto-Yoo given for cubic column relations in $M(3)$ of the form $Z^3=itZ+u\overline{Z}$ with $t,u$ real numbers satisfying some suitable inequalities. Furthermore, we solve the truncated complex moment problem with column dependence relations of the form $Z^{k+1}= \sum\limits_{0\leq n+ m \leq k} a_{nm} \overline{Z}^n Z^m$ ($a_{nm} \in \mathbb{C}$).