On the Real Multidimensional Rational K-Moment Problem

TL;DR

This paper solves the real multidimensional rational K-moment problem by characterizing when linear functionals are represented by positive measures, extending classical moment problem results to rational functions over semialgebraic sets.

Contribution

It provides a new representation theorem for linear functionals on localized polynomial algebras, addressing the rational K-moment problem with finitely many polynomial inequalities.

Findings

Linear functionals nonnegative on a preordering are represented by positive measures.

The result applies when the localization contains an element with sufficient growth on K.

Extends classical moment problem solutions to rational functions over semialgebraic sets.

Abstract

We present a solution to the real multidimensional rational K-moment problem, where K is defined by finitely many polynomial inequalities. More precisely, let S be a finite set of real polynomials in X=(X_1,...,X_n) such that the corresponding basic closed semialgebraic set K_S is nonempty. Let E=D^{-1}R[X] be a localization of the real polynomial algebra, and T_S^E the preordering on E generated by S. We show that every linear functional L on E that is nonnegative on T_S^E is represented by a positive measure on a certain subset of K_S, provided D contains an element that grows fast enough on K_S.