scholarly journals Korovkin Second Theorem via -Statistical -Summability

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
M. Mursaleen ◽  
A. Kiliçman
Keyword(s):  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Real Interval ◽  
Test Functions ◽  
Approximation Process ◽  
The Real ◽  
Approximation Theorems ◽  
Whole Space ◽  
Korovkin Type Approximation Theorems

Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, , and in the space as well as for the functions 1, cos, and sin in the space of all continuous 2-periodic functions on the real line. In this paper, we use the notion of -statistical -summability to prove the Korovkin second approximation theorem. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from into .

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.

Deferred weighted 𝒜-statistical convergence based upon the (p,q)-Lagrange polynomials and its applications to approximation theorems

2018 ◽  
Vol 24 (1) ◽  
pp. 1-16 ◽  
Author(s):  
H. M. Srivastava ◽  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray ◽  
U. K. Misra
Keyword(s):  
Statistical Convergence ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Lagrange Polynomials ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Special Cases ◽  
Korovkin Type Approximation Theorems ◽  
Appl Math

AbstractRecently, the notion of positive linear operators by means of basic (orq-) Lagrange polynomials and{\mathcal{A}}-statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means ofq-Lagrange polynomials andA-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted{\mathcal{A}}-statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1,tand{t^{2}}defined on a Banach space{C[0,1]}for a sequence of (presumably new) positive linear operators based upon{(p,q)}-Lagrange polynomials. Furthermore, we investigate the deferred weighted{\mathcal{A}}-statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

scholarly journals Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 229
Author(s):  
Hari Mohan Srivastava ◽  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray
Keyword(s):  
Limit Theorems ◽  
Bernstein Polynomials ◽  
Linear Operators ◽  
Test Functions ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Korovkin Type Approximation Theorems ◽  
Algebraic Test ◽  
Lebesgue Summability ◽  
Lebesgue Integrability

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings.

scholarly journals ABSTRACT KOROVKIN THEOREMS VIA RELATIVE MODULAR CONVERGENCE FOR DOUBLE SEQUENCES OF LINEAR OPERATORS

2020 ◽  
pp. 561
Author(s):  
Sevda Yıldız ◽  
Kamil Demirci
Keyword(s):  
Positive Operators ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Double Sequences ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Modular Spaces ◽  
Abstract Version ◽  
Modular Convergence ◽  
Korovkin Type Approximation Theorems

We will obtain an abstract version of the Korovkin type approximation theorems with respect to the concept of statistical relative convergence in modular spaces for double sequences of positive linear operators. We will give an application showing that our results are stronger than classical ones. We will also study an extension to non-positive operators.

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

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Fadime Dirik ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Dimensional Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Primary Interest ◽  
The Real ◽  
Statistical Sense

AbstractOur primary interest in the present paper is to prove a Korovkintype approximation theorem for sequences of positive linear operators defined on the space of all real valued n-variate B-continuous functions on a compact subset of the real n-dimensional space via statistical convergence. Also, we display an example such that our method of convergence is stronger than the usual convergence.

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 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.

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.

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.

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