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Convexity, Markov Operators, Approximation, and Related Optimization

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  • Octav Olteanu

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

Abstract

The present review paper provides recent results on convexity and its applications to the constrained extension of linear operators, motivated by the existence of subgradients of continuous convex operators, the Markov moment problem and related Markov operators, approximation using the Krein–Milman theorem, related optimization, and polynomial approximation on unbounded subsets. In many cases, the Mazur–Orlicz theorem also leads to Markov operators as solutions. The common point of all these results is the Hahn–Banach theorem and its consequences, supplied by specific results in polynomial approximation. All these theorems or their proofs essentially involve the notion of convexity.

Suggested Citation

  • Octav Olteanu, 2022. "Convexity, Markov Operators, Approximation, and Related Optimization," Mathematics, MDPI, vol. 10(15), pages 1-17, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2775-:d:880646
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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. Laurent Gosse & Olof Runborg, 2008. "Existence, uniqueness and a constructive solution algorithm for a class of finite Markov moment problems," Post-Print hal-00323346, HAL.
    3. Laurent Gosse & Olof Runborg, 2008. "Existence, uniqueness and a constructive solution algorithm for a class of finite Markov moment problems," Papers 0809.3714, arXiv.org.
    4. 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.
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    Cited by:

    1. 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.

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    1. 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.
    2. Octav Olteanu, 2021. "On the Moment Problem and Related Problems," Mathematics, MDPI, vol. 9(18), pages 1-26, September.
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    5. Octav Olteanu, 2013. "New Results on Markov Moment Problem," International Journal of Analysis, Hindawi, vol. 2013, pages 1-17, February.

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