The Early History of Moment Problems and Non-Negative Polynomials with Gaps: Sparse Moment Problems, Sparse Positivstellens\"atze, and Sparse Nichtnegativstellens\"atze from a T-System Point of View

TL;DR

This paper explores sparse univariate positivity certificates and moment problems using T-systems, providing complete descriptions and solutions for sparse positive and non-negative polynomials and extending classical moment problem results.

Contribution

It offers new complete characterizations of sparse positive and non-negative polynomials and extends classical Hausdorff and Stieltjes moment problems using T-system methods.

Findings

Complete descriptions of sparse positive polynomials on [a,b] and [0,∞)

Extended and simplified solutions to sparse Hausdorff and Stieltjes moment problems

Utilized T-systems to unify and advance sparse positivity and moment problem theory

Abstract

We deal with and investigate sparse univariate Positivstellens\"atze, Nichtnegativstellens\"atze, and solutions to sparse moment problems. The paper relies heavily on results on T-system by Karlin in 1963 and by Karlin and Studden in 1966. We gain complete descriptions of all sparse strictly positive and sparse non-negative algebraic polynomials on $[a,b]$ with $a\geq 0$ and $[0,\infty)$. We extend, simplify, and solve the sparse Hausdorff and Stieltjes moment problem with these results and the methods of adapted spaces and T-systems.