On Markov Moment Problem, Polynomial Approximation on Unbounded Subsets, and Mazur–Orlicz Theorem
We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our solutions have a codomain spa...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/10/1967 |
| Summary: | We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our solutions have a codomain space a commutative algebra of (linear) symmetric operators acting from the entire real or complex Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>H</mi></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow></semantics></math></inline-formula>; this algebra of operators is also an order complete Banach lattice. In particular, Hahn–Banach type theorems for the extension of linear operators having a codomain such a space can be applied. The truncated moment problem is briefly discussed by means of reference citations. This is the second purpose of the paper. In the end, a general extension theorem for linear operators with two constraints is recalled and applied to concrete spaces. Here polynomial approximation plays no role. This is the third aim of this work. |
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| ISSN: | 2073-8994 |