Real Algebraic and Analytic Geometry |

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.

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