Representation and Approximation of Positivity Preservers

TL;DR

This paper characterizes linear operators on polynomial algebras that preserve nonnegativity on a set, showing they are represented by measures and can be approximated by simple operators, extending classical results like Haviland's Theorem.

Contribution

It provides a measure-theoretic characterization of positivity-preserving operators and demonstrates their approximation by finite-dimensional operators for compact sets.

Findings

Operators are represented by measures with nonnegative function values.

Any such operator on compact sets is a limit of simple finite-dimensional operators.

Generalizes Haviland's Theorem to polynomial algebra operators.

Abstract

We consider a closed set S in R^n and a linear operator \Phi on the polynomial algebra R[X_1,...,X_n] that preserves nonnegative polynomials, in the following sense: if f\geq 0 on S, then \Phi(f)\geq 0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland's Theorem, which concerns linear functionals on polynomial algebras. For compact sets S we use the result to show that any nonnegativity preserving operator is a pointwise limit of very simple nonnegativity preservers with finite dimensional range.