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18 pages, 354 KiB

Open AccessReview

Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation

by Octav Olteanu

Mathematics 2024, 12(18), 2878; https://doi.org/10.3390/math12182878 - 15 Sep 2024

Cited by 1 | Viewed by 1040

We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem [...] Read more.

We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval

[0, + \infty)

. Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved. Full article

(This article belongs to the Section E6: Functional Interpolation)

17 pages, 671 KiB

Open AccessReview

Markov Moment Problems on Special Closed Subsets of

R^{n}

by Octav Olteanu

Symmetry 2023, 15(1), 76; https://doi.org/10.3390/sym15010076 - 27 Dec 2022

Cited by 1 | Viewed by 1222

Abstract

First, this paper provides characterizing the existence and uniqueness of the linear operator solution

T

for large classes of full Markov moment problems on closed subsets

F

of

R^{n} .

One uses approximation by special nonnegative polynomials. The case when

F

is [...] Read more.

First, this paper provides characterizing the existence and uniqueness of the linear operator solution

T

for large classes of full Markov moment problems on closed subsets

F

of

R^{n} .

One uses approximation by special nonnegative polynomials. The case when

F

is compact is studied. Then the cases when

F = R^{n}

and

F = R_{+}^{n}

are under attention. Here, the main findings consist in proving and applying the density of special polynomials, which are sums of squares, in the positive cone of

L_{ν}^{1} (R^{n}),

and respectively of

L_{ν}^{1} (R_{+}^{n}),

for a large class of measures

ν

. One solves the important difficulty created by the fact that on

R^{n}, n \geq 2,

there exist nonnegative polynomials which are not expressible in terms of sums of squares. This is the second aim of the paper. On the other hand, two types of symmetry are outlined. Both these symmetry properties appear naturally from the thematic mentioned above. This is the third aim of the paper. They lead to new statements, illustrated in corollaries, and supported by a few examples. Full article

(This article belongs to the Special Issue Symmetry and Approximation Methods)

16 pages, 359 KiB

Open AccessReview

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

by Octav Olteanu

Mathematics 2022, 10(18), 3288; https://doi.org/10.3390/math10183288 - 10 Sep 2022

Cited by 2 | Viewed by 1448

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 [...] Read more.

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} \leq T \leq 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. Full article

(This article belongs to the Special Issue Variational Problems and Applications)

12 pages, 318 KiB

Open AccessReview

On Special Properties for Continuous Convex Operators and Related Linear Operators

by Octav Olteanu

Symmetry 2022, 14(7), 1390; https://doi.org/10.3390/sym14071390 - 6 Jul 2022

Cited by 2 | Viewed by 1831

Abstract

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of [...] Read more.

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis)

24 pages, 406 KiB

Open AccessReview

On Markov Moment Problem, Polynomial Approximation on Unbounded Subsets, and Mazur–Orlicz Theorem

by Octav Olteanu

Symmetry 2021, 13(10), 1967; https://doi.org/10.3390/sym13101967 - 18 Oct 2021

Cited by 1 | Viewed by 1783

Abstract

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 [...] Read more.

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

H

to

H

; 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. Full article

(This article belongs to the Special Issue Symmetry and Approximation Methods)

17 pages, 337 KiB

Open AccessArticle

On Four Classical Measure Theorems

by Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López

Mathematics 2021, 9(5), 526; https://doi.org/10.3390/math9050526 - 3 Mar 2021

Cited by 3 | Viewed by 1938

Abstract

A subset

B

of an algebra

A

of subsets of a set

Ω

has property

(N)

if each

B

-pointwise bounded sequence of the Banach space

b a (A)

is bounded in

b a (A)

, where [...] Read more.

A subset

B

of an algebra

A

of subsets of a set

Ω

has property

(N)

if each

B

-pointwise bounded sequence of the Banach space

b a (A)

is bounded in

b a (A)

, where

b a (A)

is the Banach space of real or complex bounded finitely additive measures defined on

A

endowed with the variation norm.

B

has property

(G)

[

(V H S)

] if for each bounded sequence [if for each sequence] in

b a (A)

the

B

-pointwise convergence implies its weak convergence.

B

has property

(s N)

[

(s G)

or

(s V H S)

] if every increasing covering

{B_{n} : n \in N}

of

B

contains a set

B_{p}

with property

(N)

[

(G)

or

(V H S)

], and

B

has property

(w N)

[

(w G)

or

(w V H S)

] if every increasing web

{B_{n_{1} n_{2} \dots n_{m}} : n_{i} \in N, 1 \leq i \leq m, m \in N}

of

B

contains a strand

{B_{p_{1} p_{2} \dots p_{m}} : m \in N}

formed by elements

B_{p_{1} p_{2} \dots p_{m}}

with property

(N)

[

(G)

or

(V H S)

] for every

m \in N

. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every

σ

-algebra has properties

(N)

,

(s N)

,

(G)

and

(V H S)

. Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every

σ

-algebra has property

(w N)

and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property

(w N)

of a

σ

-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset

B

of an algebra

A

has property

(w W H S)

if and only if

B

has property

(w N)

and

A

has property

(G)

. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

16 pages, 317 KiB

Open AccessReview

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

by Octav Olteanu

Mathematics 2020, 8(8), 1328; https://doi.org/10.3390/math8081328 - 10 Aug 2020

Cited by 6 | Viewed by 2424

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Hahn-Banach Theorem, Polynomial Approximation, Moment Problems, and Related Inverse Problems)

47 pages, 431 KiB

Open AccessArticle

Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces

by Sergey Ajiev

Axioms 2013, 2(2), 224-270; https://doi.org/10.3390/axioms2020224 - 23 May 2013

Cited by 5 | Viewed by 5267

Abstract

We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions [...] Read more.

We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hölder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration. Full article

(This article belongs to the Special Issue Wavelets and Applications)

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