From the Hahn–Banach extension theorem to the isotonicity of convex functions and the majorization theory

Niculescu, Constantin P.; Olteanu, Octav

doi:10.1007/s13398-020-00905-4

Cited by 18 publications

(24 citation statements)

References 27 publications

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“…For more general extension types and controlled regularity of linear operators see [20,34] and the references given there.…”

Section: Constrained Extension Results For Linear Operatorsmentioning

confidence: 99%

On Markov Moment Problem and Related Results

Olteanu

2021

Symmetry

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We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued functions on a compact subset or an space, where is a positive moment determinate measure on a closed unbounded set. The existence and uniqueness of the operator solution are proved. Our solutions satisfy the interpolation moment conditions and are between two given linear operators on the positive cone of the domain space. The norm controlling of the solution is emphasized. The most part of the results are stated and proved in terms of quadratic forms. This type of result represents the first aim of the paper. Secondly, we construct a polynomial solution for a truncated multidimensional moment problem.

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“…For more general extension types and controlled regularity of linear operators see [20,34] and the references given there.…”

Section: Constrained Extension Results For Linear Operatorsmentioning

confidence: 99%

On Markov Moment Problem and Related Results

Olteanu

2021

Symmetry

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“…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.…”

Section: Methodsmentioning

confidence: 99%

“…General background can be found in chapters or paragraphs from monographs [4][5][6][7][8]. The present paper is directly related to references [9][10][11][12][13][14][15] and partially related to articles [16][17][18][19][20]. The case of multidimensional moment problem on compact semi-algebraic subsets in R n was intensively studied.…”

Section: Introductionmentioning

confidence: 99%

“…A topological version of it is also proved. Section 3.2 is devoted to a direct sharp proof of a generalization of Hahn-Banach theorem, motivated by solving Markov moment problems and by characterization of isotonicity of convex operators over convex cones [14,15] (recall that an operator P acting between two ordered vector spaces E, F is called isotone on the cone C ⊂ E if…”

Section: Introductionmentioning

confidence: 99%

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From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications

Olteanu

2020

Mathematics

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The aim of this review paper is to recall known solutions for two Markov moment problems, which can be formulated as Hahn–Banach extension theorems, in order to emphasize their relationship with the following problems: (1) pointing out a previously published sandwich theorem of the type f ≤ h ≤ g, where f, −g are convex functionals and h is an affine functional, over a finite-simplicial set X, and proving a topological version for this result; (2) characterizing isotonicity of convex operators over arbitrary convex cones; giving a sharp direct proof for one of the generalizations of Hahn–Banach theorem applied to the isotonicity; (3) extending inequalities assumed to be valid on a small subset, to the entire positive cone of the domain space, via Krein–Milman or Carathéodory’s theorem. Thus, we point out some earlier, as well as new applications of the Hahn–Banach type theorems, emphasizing the topological versions of these applications.

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“…Indeed, in any ordered Banach space F with generating cone, any inequality of the form u ≤ v is equivalent to w * (u) ≤ w * (v) for all w * ∈ F * + . See [18], Lemma 1 (c). As a consequence, a function Φ : C → F is ω-convex if and only if w * • Φ is (w * • ω)-convex whenever w * ∈ F * + .…”

Section: Classes Of Convex Functionsmentioning

confidence: 99%