Real Algebraic and Analytic Geometry

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271. Tim Netzer:
Representation and Approximation of Positivity Preservers.

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Submission: 2009, February 19.

Abstract:
We consider a closed set $S\subseteq\R^n$ and a linear operator $$\Ph\colon\R[X_1,\ldots,X_n]\rightarrow \R[X_1,\ldots,X_n]$$ that preserves nonnegative polynomials, in the following sense: if $f\geq 0$ on $S$, then $\Ph(f)\geq 0$ 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 \textit{functionals} on $\R[X_1,\ldots,X_n]$. 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.

Mathematics Subject Classification (2000): 12E05, 15A04, 47B38, 44A60, 31B10, 41A36.

Keywords and Phrases: Positive and non-negative polynomials, linear preservers, moment problems, integral representations, approximation of operators.

Full text, 17p.: dvi 108k, ps.gz 184k, pdf 235k.


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