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arXiv · 0902.0279

Representation and Approximation of Positivity Preservers

Tim Netzer
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Abstract

We consider a closed set S in R^n and a linear operator Φon the polynomial algebra R[X_1,...,X_n] that preserves nonnegative polynomials, in the following sense: if f\geq 0 on S, then Φ(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.

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Tim Netzer. 2009-02-02. Representation and Approximation of Positivity Preservers. https://arxiv.org/abs/0902.0279

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