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

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  • Octav Olteanu

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

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.

Suggested Citation

  • Octav Olteanu, 2022. "Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms," Mathematics, MDPI, vol. 10(18), pages 1-16, September.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:18:p:3288-:d:911635
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    References listed on IDEAS

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    1. Stefan Cobzaş, 2005. "Geometric properties of Banach spaces and the existence of nearest and farthest points," Abstract and Applied Analysis, Hindawi, vol. 2005, pages 1-27, January.
    2. Yoon-Tae Kim & Hyun-Suk Park, 2022. "Fourth Cumulant Bound of Multivariate Normal Approximation on General Functionals of Gaussian Fields," Mathematics, MDPI, vol. 10(8), pages 1-17, April.
    3. Pier Luigi Novi Inverardi & Aldo Tagliani, 2021. "Stieltjes and Hamburger Reduced Moment Problem When MaxEnt Solution Does Not Exist," Mathematics, MDPI, vol. 9(4), pages 1-15, February.
    4. Octav Olteanu, 2022. "Convexity, Markov Operators, Approximation, and Related Optimization," Mathematics, MDPI, vol. 10(15), pages 1-17, August.
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    Cited by:

    1. Savin Treanţă, 2022. "Variational Problems and Applications," Mathematics, MDPI, vol. 11(1), pages 1-4, December.

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