Representation and Approximation of Positivity Preservers

2009/02/02 by Tim Netzer, Netzer, Tim · 2 citations
Computer Science · Mathematics · #12E05 #15A04 #31B10 #41A36 #44A60 #47B38 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Polynomial and algebraic computation #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.0902.0279

openalex publication_date 2009/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a closed set S in Rⁿ and a linear operator Φon the polynomial algebra R[X₁,...,X_n] that preserves nonnegative polynomials, in the following sense: if f≥ 0 on S, then Φ(f)≥ 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.

Cited by

• Characterization of Matrix K-Positivity Preserver for K=ℝⁿ and for Compact Sets K⊆ℝⁿ
• Making Non-Negative Polynomials into Sums of Squares

Related

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