Some Density Theorems in the Set of Continuous Functions with Values in the Unit Interval

Paltineanu, Gavriil; Bucur, Ileana

doi:10.1007/s00009-017-0870-5

Cited by 9 publications

(5 citation statements)

References 9 publications

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“…The following definition is analogous with Definition 4.6. of [9] Definition 5.1. A subset S ⊂ X is called antisymmetric with respect to the pair (M, C) if any function ϕ ∈ M with the properties:…”

Section: Stone-weierstrass Theorem For Convex Cones In Weighted Spacesmentioning

confidence: 99%

De Branges Type Lemma and Approximation in Weighted Spaces

Bucur

Paltineanu

2021

Mediterr. J. Math.

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The purpose of this paper is to give some generalizations of de Branges Lemma for weighted spaces to obtain different approximation theorems in weighted spaces for algebras, vector subspaces or convex cones. We recall that the (original) de Branges Lemma (Proc Am Math Soc 10(5):822–824, 1959) was demonstrated for continuous scalar function on a compact space while, the weighted spaces are classes of continuous scalar functions on a locally compact space (e.g. the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).

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Section: Stone-weierstrass Theorem For Convex Cones In Weighted Spacesmentioning

confidence: 99%

De Branges Type Lemma and Approximation in Weighted Spaces

Bucur

Paltineanu

2021

Mediterr. J. Math.

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“…Usually, this leads not only to the continuity of T, but also to evaluating (or even determining exactly) its norm, in terms of the norm of the sublinear operator P. Extension results on linear operators, constrained by sandwich conditions, have been recently applied in [27] in order to characterize the monotone increasing convex operators defined on convex cones, in terms of their subgradients. In [28], some density results in a different framework with respect to that discussed in the present Section 3.2 are highlighted.…”

Section: Introductionmentioning

confidence: 96%

“…Various aspects of the moment problem are discussed in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. For other applications, results on Markov moment problem and connections with other fields of analysis see [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. The moment problem has applications in physics, probability theory, and statistics, as discussed in the Introduction of [2].…”

Section: Introductionmentioning

confidence: 99%

On the Moment Problem and Related Problems

Olteanu

2021

Mathematics

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Firstly, we recall the classical moment problem and some basic results related to it. By its formulation, this is an inverse problem: being given a sequence (yj)j∈ℕn of real numbers and a closed subset F⊆ℝn, n∈{1,2,…}, find a positive regular Borel measure μ on F such that ∫Ftjdμ=yj, j∈ℕn. This is the full moment problem. The existence, uniqueness, and construction of the unknown solution μ are the focus of attention. The numbers yj, j∈ℕn are called the moments of the measure μ. When a sandwich condition on the solution is required, we have a Markov moment problem. Secondly, we study the existence and uniqueness of the solutions to some full Markov moment problems. If the moments yj are self-adjoint operators, we have an operator-valued moment problem. Related results are the subject of attention. The truncated moment problem is also discussed, constituting the third aim of this work.

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“…In [30], the author constructs a solution for the full moment problem, as a limit of solutions for truncated moment problems. Articles [31,32] provide interesting approximation results, not necessarily referring to the moment problem. References [4,5,18,19,[33][34][35][36][37] are devoted to, or contain significant results on, the Markov moment problem.…”

Section: Introductionmentioning

confidence: 99%

Symmetry and Asymmetry in Moment, Functional Equations, and Optimization Problems

Olteanu

2023

Symmetry

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The purpose of this work is to provide applications of real, complex, and functional analysis to moment, interpolation, functional equations, and optimization problems. Firstly, the existence of the unique solution for a two-dimensional full Markov moment problem is characterized on the upper half-plane. The issue of the unknown form of nonnegative polynomials on R×R+ in terms of sums of squares is solved using polynomial approximation by special nonnegative polynomials, which are expressible in terms of sums of squares. The main new element is the proof of Theorem 1, based only on measure theory and on a previous approximation-type result. Secondly, the previous construction of a polynomial solution is completed for an interpolation problem with a finite number of moment conditions, pointing out a method of determining the coefficients of the solution in terms of the given moments. Here, one uses methods of symmetric matrix theory. Thirdly, a functional equation having nontrivial solution (defined implicitly) and a consequence are discussed. Inequalities, the implicit function theorem, and elements of holomorphic functions theory are applied. Fourthly, the constrained optimization of the modulus of some elementary functions of one complex variable is studied. The primary aim of this work is to point out the importance of symmetry in the areas mentioned above.

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Some Density Theorems in the Set of Continuous Functions with Values in the Unit Interval

Cited by 9 publications

References 9 publications

De Branges Type Lemma and Approximation in Weighted Spaces

De Branges Type Lemma and Approximation in Weighted Spaces

On the Moment Problem and Related Problems

Symmetry and Asymmetry in Moment, Functional Equations, and Optimization Problems

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