Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms

Octav Olteanu ()
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Octav Olteanu: Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania

Mathematics, 2022, vol. 10, issue 18, 1-16

Abstract: As is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by dominating special polynomials that are sums of squares. This also works in several-dimensional cases. To perform this, a Hahn–Banach-type theorem (Kantorovich theorem on an extension of positive linear operators), a Haviland theorem, and the notion of a moment-determinate measure are applied. Second, completions and other results on solving full Markov moment problems in terms of quadratic forms are proposed based on polynomial approximation. The existence and uniqueness of the solution are discussed. Third, the characterization of the constraints T 1 ≤ T ≤ T 2 for the linear operator T , only in terms of quadratic forms, is deduced. Here, T 1 , T , and T 2 are bounded linear operators. Concrete spaces, operators, and functionals are involved in our corollaries or examples.

Keywords: polynomial approximation; unbounded subsets; Markov moment problem; positive operators; solution; existence; uniqueness; sums of squares; Banach lattices (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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