Markov Operators, Positive Semigroups and Approximation Processes

Altomare, Francesco; Cappelletti, Mirella; Leonessa, Vita; Raşa, Ioan

doi:10.1515/9783110366976

Cited by 59 publications

(64 citation statements)

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“…If f ∈ C [0, 1] is convex the inequality Inequalities of type (1) have important applications. They are useful when studying whether the Bernstein-Schnabl operators preserve convexity (see [2,3,4]).…”

Section: Introductionmentioning

confidence: 99%

A sharpening of a problem on Bernstein polynomials and convex functions

Abel¹,

Raşa²

2018

Mathematical Inequalities &Amp; Applications

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Section: Introductionmentioning

confidence: 99%

A sharpening of a problem on Bernstein polynomials and convex functions

Abel¹,

Raşa²

2018

Mathematical Inequalities &Amp; Applications

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“…During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Raşa [12] recalled his problem.Inequalities of type (1.1) have important applications. They are useful when studying whether the Bernstein-Schnabl operators preserve convexity (see [3,4]). Recently, J. Mrowiec, T. Rajba and S. Wąsowicz [10] affirmed the conjecture (1.1) in positive.…”

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confidence: 99%

A sharpening of a problem on Bernstein polynomials and convex functions and related results

Komisarski¹,

Rajba²

2018

Mathematical Inequalities &Amp; Applications

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We present a short proof of a conjecture proposed by I. Raşa (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Raşa (2017). The methods of our proof allow us to obtain some extended versions of this inequality as well as other inequalities given by I. Raşa. As a tool we use stochastic convex ordering relations. We propose also some generalizations of the binomial convex concentration inequality. We use it to insert some additional expressions between left and right sides of the Raşa inequalities.is valid for all x, y ∈ [0, 1].This inequality involving Bernstein basic polynomials and convex functions was stated as an open problems 25 years ago by I. Raşa. During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Raşa [12] recalled his problem.Inequalities of type (1.1) have important applications. They are useful when studying whether the Bernstein-Schnabl operators preserve convexity (see [3,4]).Recently, J. Mrowiec, T. Rajba and S. Wąsowicz [10] affirmed the conjecture (1.1) in positive. Their proof makes heavy use of probability theory. As a tool they applied a concept of stochastic convex orderings, as well as the so-called binomial convex concentration inequality. Later, U. Abel [1] gave an elementary proof of (1.1), which was much shorter than that given in [10]. Very recently, A. Komisarski and T. Rajba [6] gave a new, very short proof of (1.1), which is significantly simpler and shorter than that given by U. Abel [1]. As a tool the authors use both stochastic convex orders as well as the usual stochastic order.Let us recall some basic notations and results on stochastic ordering (see [14]). If µ and ν are two probability distributions such that ϕ(x)µ(dx) ≤ ϕ(x)ν(dx) for all convex functions ϕ : R → R, 2010 Mathematics Subject Classification. 60E15, 39B62.

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“…Moreover (see [5, In general, each convex function f ∈ C (K) is T -convex. Moreover Taxially convex functions are T -convex as well (see [5,Definition 3.5.1 and remarks on p. 148]).…”

Section: Preservation Propertiesmentioning

confidence: 99%

“…, pr d , δ) = ω(f, δ),(3.10)so that, setting r(K) := max{ x 2 | x ∈ K},Ω(f, δ)≤ ω(f, δr(K)),(3.11) the last inequality being an equality if d = 1. By using Proposition 1.6.5 in[5] (see also [2, Proposition 5.1.4]) we get the following result.…”

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confidence: 91%

A generalization of Kantorovich operators for convex compact subsets

Altomare¹,

Montano²,

Leonessa³

et al. 2017

Banach J. Math. Anal.

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In this article, we introduce and study a new sequence of positive linear operators acting on function spaces dened on a convex compact subset.\ud Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases\ud of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators dened in the 1-dimensional and multidimensional\ud setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators\ud by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity

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Markov Operators, Positive Semigroups and Approximation Processes

Cited by 59 publications

References 0 publications

A sharpening of a problem on Bernstein polynomials and convex functions

A sharpening of a problem on Bernstein polynomials and convex functions

A sharpening of a problem on Bernstein polynomials and convex functions and related results

A generalization of Kantorovich operators for convex compact subsets

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