From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications
Abstract
:1. Introduction
2. Materials and Methods
- (1)
- Extension of linear operators such that the extension is dominated by a convex operator and minorated by a concave operator (see [9,10,11]). The convex and concave operators, respectively, are defined on arbitrary convex subsets of the domain space An important particular case is that when or (or both) is (are) equal to the positive cone The codomain space is always order complete, in order to allow applying Hahn–Banach results.
- (2)
- Characterizing existence of solutions for Markov moment problem in terms of the given moments
- (3)
- From one of the results mentioned at point (2), proving a sandwich result over a finite simplicial set. Adding the topological version of this result.
- (4)
- Giving a sharp direct proof for a generalization of Hahn–Banach theorem and applying it to a very recent result on isotonicity of a convex operator over a convex cone (see [14,15]). On the other hand, a variant of this generalized Hahn–Banach result was frequently applied to the Markov moment problem. It allows controlling the norm of the solution (see also Remark 5 below).
- (5)
- Using Krein–Milman and Caratheodory’s theorems in order to extend inequalities from a smaller set to the entire positive cone of the domain. Applying the notion of a quasiconvex operator [5].
3. Results
3.1. Simplexes, Finite-Simplicial Sets and Sandwich Theorems over Such Subsets
- (1)
- In any convex cone having a base that is a simplex (the corresponding order relation is laticial) is an unbounded finite simplicial set.
- (2)
- In for each the convex cone defined by
- (3)
- Let be an arbitrary infinite or finite dimensional vector space (of dimension , a non-null linear functional and Then the sets are finite-simplicial.
- (4)
- Let be as in Example (3), two real numbers such that The set
- (a)
- There is a linear operator such that
- (b)
- For any finite subsetand anythe following implication holds trueIfis a vector lattice, then assertions (a) and (b) are equivalent to (c), where
- (c)
- for alland for any finite subsetandwe have
3.2. A Direct Proof for a Generalization of Hahn–Banach Theorem and Its Motivation
- (a)
- There exists a positive linear extensionofsuch that
- (b)
- We havefor allsuch that
- (a)
- There exists a positive linear extensionofsuch thaton
- (b)
- We havefor allsuch that.
- (a)
- There exists a positive linear operatorsuch thaton
- (b)
- For any finite subsetand anythe following implication holds true
3.3. Extending Inequalities via Krein–Milman and Carathéodory’s Theorems
4. Discussion
Funding
Conflicts of Interest
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Olteanu, O. From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications. Mathematics 2020, 8, 1328. https://doi.org/10.3390/math8081328
Olteanu O. From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications. Mathematics. 2020; 8(8):1328. https://doi.org/10.3390/math8081328
Chicago/Turabian StyleOlteanu, Octav. 2020. "From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications" Mathematics 8, no. 8: 1328. https://doi.org/10.3390/math8081328