scholarly journals Approximation Theorem for New Modification of q -Bernstein Operators on (0,1)

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yun-Shun Wu ◽  
Wen-Tao Cheng ◽  
Feng-Lin Chen ◽  
Yong-Hui Zhou
Keyword(s):  
Approximation Theorem ◽  
Local Approximation ◽  
Modulus Of Smoothness ◽  
Central Moment ◽  
Bernstein Operators ◽  
Lipschitz Classes ◽  
Korovkin Type Approximation Theorem ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Quantitative Properties

In this work, we extend the works of F. Usta and construct new modified q -Bernstein operators using the second central moment of the q -Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’s K -functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.

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 Generalized Weighted Statistical Convergence for Double Sequences of Fuzzy Numbers and Associated Korovkin-Type Approximation Theorem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Abdullah Alotaibi
Keyword(s):  
Statistical Convergence ◽  
Fuzzy Numbers ◽  
Approximation Theorem ◽  
Bernstein Operators ◽  
Double Sequences ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Functions Of Two Variables ◽  
Sequences Of Fuzzy Numbers ◽  
Bivariate Functions

We define the notions of weighted λ,μ-statistical convergence of order γ1,γ2 and strongly weighted λ,μ-summability of γ1,γ2 for fuzzy double sequences, where 0<γ1,γ2≤1. We establish an inclusion result and a theorem presenting a connection between these concepts. Moreover, we apply our new concept of weighted λ,μ-statistical convergence of order γ1,γ2 to prove Korovkin-type approximation theorem for functions of two variables in a fuzzy sense. Finally, an illustrative example is provided with the help of q-analogue of fuzzy Bernstein operators for bivariate functions which shows the significance of our approximation theorem.

scholarly journals A Korovkin Type Approximation Theorem and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Malik Saad Al-Muhja
Keyword(s):  
Approximation Theorem ◽  
Modulus Of Smoothness ◽  
Linear Operators ◽  
Stieltjes Integral ◽  
Periodic Functions ◽  
Third Order ◽  
Korovkin Type Approximation Theorem ◽  
The Third ◽  
Type Approximation ◽  
Statistical Limit

We present a Korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic functions viaA-statistical approximation, for the rate of the third order Ditzian-Totik modulus of smoothness. Finally, we obtain an interleave between Riesz's representation theory and Lebesgue-Stieltjes integral-i, for Riesz's functional supremum formula via statistical limit.

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.

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 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 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 On Durrmeyer Type λ -Bernstein Operators via ( p , q )-Calculus

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Qing-Bo Cai ◽  
Guorong Zhou
Keyword(s):  
Convergence Theorem ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Numerical Examples ◽  
Korovkin Type Approximation Theorem ◽  
Lipschitz Continuous ◽  
Type Approximation ◽  
Second Moments

In the present paper, Durrmeyer type λ -Bernstein operators via ( p , q )-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in some λ cases than Durrmeyer type ( p , q )-Bernstein operators.

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.

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