scholarly journals Korovkin type approximation theorems via lacunary equistatistical convergence

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3641-3647 ◽  
Author(s):  
Abdullah Alotaibi ◽  
M. Mursaleen
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Approximation Theorem ◽  
Test Functions ◽  
Central European ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Korovkin Type Approximation Theorems

Aktu?lu and H. Gezer [Central European J. Math. 7 (2009), 558-567] introduced the concepts of lacunary equistatistical convergence, lacunary statistical pointwise convergence and lacunary statistical uniform convergence for sequences of functions. In this paper, we apply the notion of lacunary equistatistical convergence to prove a Korovkin type approximation theorem by using test functions 1, x/1-x,(x/1-x)2.

The uniform convergence of a double sequence of functions at a point and Korovkin-type approximation theorems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Fadime Dirik ◽  
Kamil Demirci ◽  
Sevda Yıldız ◽  
Ana Maria Acu
Keyword(s):  
Uniform Convergence ◽  
Approximation Theorem ◽  
Double Sequence ◽  
Korovkin Theorem ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
The Korovkin Theorem ◽  
Korovkin Type Approximation Theorems ◽  
Sequence Of Functions

AbstractIn this paper, we introduce an interesting kind of convergence for a double sequence called the uniform convergence at a point. We give an example and demonstrate a Korovkin-type approximation theorem for a double sequence of functions using the uniform convergence at a point. Then we show that our result is stronger than the Korovkin theorem given by Volkov and present several graphs. Finally, in the last section, we compute the rate of convergence.

scholarly journals A certain class of statistical probability convergence and its applications to approximation theorems

2020 ◽  
pp. 39-39
Author(s):  
H.M. Srivastava ◽  
Bidu Jena ◽  
Susanta Paikray
Keyword(s):  
Banach Space ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Random Variables ◽  
Test Functions ◽  
Convolution Operators ◽  
Statistical Probability ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation

In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the concept of statistical convergence for sequences of real numbers, which are defined over a Banach space via deferred weighted summability mean. We first establish a theorem presenting a connection between them. Based upon our proposed methods, we then prove a new Korovkin-type approximation theorem with periodic test functions for a sequence of random variables on a Banach space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in statistical versions). We also estimate the rate of deferred weighted statistical probability convergence and accordingly establish a new result. Finally, an illustrative example is presented here by means of the generalized Fej?r convolution operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

scholarly journals Statistical deferred Cesàro summability and its applications to approximation theorems

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2307-2319 ◽  
Author(s):  
Bidu Jena ◽  
Susanta Paikray ◽  
Umakanta Misra
Keyword(s):  
Banach Space ◽  
Approximation Theorem ◽  
Summability Method ◽  
Trivial Extension ◽  
Cesàro Summability ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Cesaro Summability ◽  
Korovkin Type Approximation Theorems

Statistical (C,1) summability and a Korovkin type approximation theorem has been proved by Mohiuddine et al. [20] (see [S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical summability (C,1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012), Article ID 172, 1-8). In this paper, we apply statistical deferred Ces?ro summability method to prove a Korovkin type approximation theorem for the set of functions 1, e-x and e-2x defined on a Banach space C[0;1) and demonstrate that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also establish a result for the rate of statistical deferred Ces?ro summability method. Some interesting examples are also discussed here in support of our definitions and results.

scholarly journals Approximation Theorems for Functions of Two Variables viaσ-Convergence

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Mohammed A. Alghamdi
Keyword(s):  
Approximation Theorem ◽  
Bernstein Polynomials ◽  
Test Functions ◽  
Double Sequences ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Functions Of Two Variables

Çakan et al. (2006) introduced the concept ofσ-convergence for double sequences. In this work, we use this notion to prove the Korovkin-type approximation theorem for functions of two variables by using the test functions 1,x,y, andx2+y2and construct an example by considering the Bernstein polynomials of two variables in support of our main result.

scholarly journals Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4549-4560 ◽  
Author(s):  
S.A. Mohiuddine ◽  
Bipan Hazarika ◽  
Mohammed Alghamdi
Keyword(s):  
Uniform Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Korovkin Theorem ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Convergence Of Sequences ◽  
Relative Uniform Convergence ◽  
Relatively Uniform Convergence

We introduce the notion of ideally relative uniform convergence of sequences of real valued functions. We then apply this notion to prove Korovkin-type approximation theorem, and then construct an illustrative example by taking (p,q)-Bernstein operators which proves that our Korovkin theorem is stronger than its classical version as well as statistical relative uniform convergence. The rate of ideal relatively uniform convergence of positive linear operators by means of modulus of continuity is calculated. Finally, the Voronovskaya-type approximation theorem is also investigated.

scholarly journals Korovkin type theorem for functions of two variables via lacunary equistatistical convergence

2017 ◽  
Vol 102 (116) ◽  
pp. 203-209
Author(s):  
M. Mursaleen
Keyword(s):  
Pointwise Convergence ◽  
Approximation Theorem ◽  
Type Theorem ◽  
Test Functions ◽  
Korovkin Type Theorem ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Functions Of Two Variables ◽  
Korovkin Approximation ◽  
Korovkin Approximation Theorem

Aktu?lu and Gezer [1] introduced the concepts of lacunary equistatistical convergence, lacunary statistical pointwise convergence and lacunary statistical uniform convergence for sequences of functions. Recently, Kaya and G?n?l [11] proved some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence by using test functions 1, x/1+x, y/1+y, (x/1+x)2 +(y/1+y)2. We apply the notion of lacunary equistatistical convergence to prove a Korovkin type approximation theorem for functions of two variables by using test functions 1, x/1?x, y/1?y, (x/1?x)2+(y/1?y)2.

scholarly journals Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 636 ◽  
Author(s):  
Hari Mohan Srivastava ◽  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray
Keyword(s):  
Banach Space ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Real Sequence ◽  
Test Functions ◽  
Real Numbers ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Algebraic Test

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.

scholarly journals Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 316 ◽  
Author(s):  
Hari Srivastava ◽  
Faruk Özger ◽  
S. Mohiuddine
Keyword(s):  
Uniform Convergence ◽  
Shape Parameter ◽  
Approximation Theorem ◽  
Modulus Of Smoothness ◽  
Bernstein Operators ◽  
Global Approximation ◽  
Approximation Result ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Direct Approximation

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter λ ∈ [ - 1 , 1 ] and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type λ -Bernstein operators and study their approximation behaviors.

scholarly journals TRIANGULAR A−STATISTICAL RELATIVE UNIFORM CONVERGENCE FOR DOUBLE SEQUENCES OF POSITIVE LINEAR OPERATORS

2021 ◽  
pp. 065
Author(s):  
Selin Çınar
Keyword(s):  
Uniform Convergence ◽  
Approximation Theorem ◽  
Dimensional Space ◽  
Linear Operators ◽  
Two Dimensional ◽  
Double Sequences ◽  
The Real ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Relative Uniform Convergence

In this paper, we introduce the concept of triangular A-statistical relative convergence for double sequences of functions defined on a compactsubset of the real two-dimensional space. Based upon this new convergencemethod, we prove Korovkin-type approximation theorem. Finally, we give some further developments.

scholarly journals I_2-Relative uniform convergence and Korovkin type approximation

2021 ◽  
Vol 25 (2) ◽  
pp. 189-200
Author(s):  
Sevda Yildiz
Keyword(s):  
Uniform Convergence ◽  
Approximation Theorem ◽  
Double Sequences ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Relative Uniform Convergence ◽  
New Type ◽  
First Time

In the present paper, an interesting type of convergence named ideal relative uniform convergence for double sequences of functions has been introduced for the first time. Then, the Korovkin type approximation theorem via this new type of convergence has been proved. An example to show that the new type of convergence is stronger than the convergence considered before has been given. Finally, the rate of  I2-relative uniform convergence has been computed.

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