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 Generalized Kantorovich modifications of positive linear operators

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ana-Maria Acu ◽  
Ioan Cristian Buscu ◽  
Ioan Rasa
Keyword(s):  
Linear Operator ◽  
Generalized Convexity ◽  
Invariant Measures ◽  
Positive Linear Operator ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Voronovskaya Formula

<p style='text-indent:20px;'>Starting with a positive linear operator we apply the Kantorovich modification and a related modification. The resulting operators are investigated. We are interested in the eigenstructure, Voronovskaya formula, the induced generalized convexity, invariant measures and iterates. Some known results from the literature are extended.</p>

scholarly journals A Voronovskaya-type theorem for a positive linear operator

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Alexandra Ciupa
Keyword(s):  
Linear Operator ◽  
Exponential Growth ◽  
Positive Linear Operator ◽  
Type Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Voronovskaya Type Theorem

We consider a sequence of positive linear operators which approximates continuous functions having exponential growth at infinity. For these operators, we give a Voronovskaya-type theorem

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.

scholarly journals Opial typeLp-inequalities for fractional derivatives

2002 ◽  
Vol 31 (2) ◽  
pp. 85-95 ◽  
Author(s):  
G. A. Anastassiou ◽  
J. J. Koliha ◽  
J. Pecaric
Keyword(s):  
Fractional Derivatives ◽  
Continuous Functions ◽  
Taylor’S Formula ◽  
Integrable Functions ◽  
Generalized Fractional Derivatives ◽  
Taylor's Formula

This paper presents a class ofLp-type Opial inequalities for generalized fractional derivatives for integrable functions based on the results obtained earlier by the first author for continuous functions (1998). The novelty of our approach is the use of the index law for fractional derivatives in lieu of Taylor's formula, which enables us to relax restrictions on the orders of fractional derivatives.

scholarly journals A Generalization of Lacunary Equistatistical Convergence of Positive Linear Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Yusuf Kaya ◽  
Nazmiye Gönül
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Korovkin Approximation ◽  
Korovkin Approximation Theorem

In this paper we consider some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence. In particular we study lacunary equi-statistical convergence of approximating operators on spaces, the spaces of all real valued continuous functions de…ned on and satisfying some special conditions.

scholarly journals Approximation by Certain Linear Positive Operators of Two Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Afşin Kürşat Gazanfer ◽  
İbrahim Büyükyazıcı
Keyword(s):  
Convergence Rate ◽  
Modulus Of Continuity ◽  
Weighted Approximation ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Compact Set ◽  
Approximation Properties ◽  
Linear Positive Operators ◽  
Functions Of Two Variables

We introduce positive linear operators which are combined with the Chlodowsky and Szász type operators and study some approximation properties of these operators in the space of continuous functions of two variables on a compact set. The convergence rate of these operators are obtained by means of the modulus of continuity. And we also obtain weighted approximation properties for these positive linear operators in a weighted space of functions of two variables and find the convergence rate for these operators by using the weighted modulus of continuity.

Statistical σ approximation to Bögel-type continuous functions

2011 ◽  
Vol 48 (4) ◽  
pp. 475-488 ◽  
Author(s):  
Sevda Karakuş ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Approximation Result ◽  
The Real

In this paper, using the concept of statistical σ-convergence which is stronger than the statistical convergence, we obtain a statistical σ-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. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. Also we compute the rate of statistical σ-convergence of sequence of positive linear operators.

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