Asymptotic Constant in Approximation of Twice Differentiable Functions by a Class of Positive Linear Operators

2018 ◽  
Vol 73 (2) ◽  
Author(s):  
Radu Păltănea
Keyword(s):  
Linear Operators ◽  
Positive Linear Operators ◽  
Differentiable Functions ◽  
Asymptotic Constant

Taylor's formula and preservation of generalized convexity for positive linear operators

2000 ◽  
Vol 37 (3) ◽  
pp. 765-777 ◽  
Author(s):  
José A. Adell ◽  
Alberto Lekuona
Keyword(s):  
Generalized Convexity ◽  
Positive Linear Operator ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Birth Process ◽  
Absolutely Continuous Functions ◽  
Taylor’S Formula ◽  
Differentiable Functions ◽  
Taylor's Formula

In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.

Taylor's formula and preservation of generalized convexity for positive linear operators

2000 ◽  
Vol 37 (03) ◽  
pp. 765-777 ◽  
Author(s):  
José A. Adell ◽  
Alberto Lekuona
Keyword(s):  
Generalized Convexity ◽  
Positive Linear Operator ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Birth Process ◽  
Absolutely Continuous Functions ◽  
Taylor’S Formula ◽  
Differentiable Functions ◽  
Taylor's Formula

In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.

scholarly journals On approximation of continuously differentiable functions by positive linear operators

1983 ◽  
Vol 27 (1) ◽  
pp. 73-81 ◽  
Author(s):  
Heinz H. Gonska
Keyword(s):  
Degree Of Approximation ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Differentiable Functions

The aim of this note is to prove a theorem on the pointwise degree of approximation of continuously differentiable functions by positive linear operators. As can be seen from the applications to Bernstein and Hermite-Fejér operators, our inequality yields better constants and sometimes even a higher degree of approximation than the known general results.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

On αβ-statistical convergence for sequences of fuzzy mappings and Korovkin type approximation theorem

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3749-3760 ◽  
Author(s):  
Ali Karaisa ◽  
Uğur Kadak
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Bernstein Operator ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Inclusion Relations ◽  
Prior Investigation

Upon prior investigation on statistical convergence of fuzzy sequences, we study the notion of pointwise ??-statistical convergence of fuzzy mappings of order ?. Also, we establish the concept of strongly ??-summable sequences of fuzzy mappings and investigate some inclusion relations. Further, we get an analogue of Korovkin-type approximation theorem for fuzzy positive linear operators with respect to ??-statistical convergence. Lastly, we apply fuzzy Bernstein operator to construct an example in support of our result.

Some properties of D-weak operator topology

2020 ◽  
Vol 70 (3) ◽  
pp. 753-758
Author(s):  
Marcel Polakovič
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Linear Subspace ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Weak Operator Topology ◽  
Generalized Effect Algebra ◽  
Weak Operator ◽  
Dense Linear Subspace

AbstractLet 𝓖D(𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D-weak operator topology (introduced by other authors) on 𝓖D(𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖D(𝓗) is the whole 𝓖D(𝓗). The closure of the set of all unbounded elements of 𝓖D(𝓗) is also the set 𝓖D(𝓗). If Q is arbitrary unbounded element of 𝓖D(𝓗), it determines an interval in 𝓖D(𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D-weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.

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