正性保持算子的表示与逼近
Representation and Approximation of Positivity Preservers
摘要：
我们考虑R^n中的一个闭集S和多项式代数R[X_1，...，X_n]，它保持非负多项式，在以下意义下：如果f \geq 0在S上，那么Φ（f）\geq 0也在S上.我们表明，每个这样的运营商是由积分的措施，采取非负函数作为其值。这可以看作是哈维兰定理的推广，该定理涉及多项式代数上的线性泛函。对于紧集S，我们利用这个结果证明了任何保非负算子都是有限维范围内非常简单的非负算子的逐点极限。
英文摘要：
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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