scholarly journals A numerical comparative study of generalized Bernstein-Kantorovich operators

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Uğur Kadak ◽  
Faruk Özger
Keyword(s):  
Comparative Study ◽  
Rate Of Convergence ◽  
Statistical Convergence ◽  
Bernstein Operators ◽  
Shape Parameters ◽  
Convergence Behavior ◽  
Type Approximation ◽  
Summability Matrix ◽  
Kantorovich Operators ◽  
Regular Summability

<p style='text-indent:20px;'>In this paper, a new generalization of the Bernstein-Kantorovich type operators involving multiple shape parameters is introduced. Certain Voronovskaja and Grüss-Voronovskaya type approximation results, statistical convergence and statistical rate of convergence of proposed operators are obtained by means of a regular summability matrix. Some illustrative graphics that demonstrate the convergence behavior, accuracy and consistency of the operators are given via Maple algorithms. The proposed operators are comprehensively compared with classical Bernstein, Bernstein-Kantorovich and other new modifications of Bernstein operators such as <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein, <inline-formula><tex-math id="M2">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich, <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Bernstein and <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich operators.</p>

scholarly journals Approximation by Parametric Extension of Szász-Mirakjan-Kantorovich Operators Involving the Appell Polynomials

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Md. Nasiruzzaman ◽  
A. F. Aljohani
Keyword(s):  
Rate Of Convergence ◽  
Weighted Space ◽  
Modulus Of Smoothness ◽  
Linear Operators ◽  
Appell Polynomials ◽  
Global Approximation ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Dunkl Analogue ◽  
Kantorovich Operators

The purpose of this article is to introduce a Kantorovich variant of Szász-Mirakjan operators by including the Dunkl analogue involving the Appell polynomials, namely, the Szász-Mirakjan-Jakimovski-Leviatan-type positive linear operators. We study the global approximation in terms of uniform modulus of smoothness and calculate the local direct theorems of the rate of convergence with the help of Lipschitz-type maximal functions in weighted space. Furthermore, the Voronovskaja-type approximation theorems of this new operator are also presented.

scholarly journals Approximation properties of the new type generalized Bernstein-Kantorovich operators

2021 ◽  
Vol 7 (3) ◽  
pp. 3826-3844
Author(s):  
Mustafa Kara ◽  
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Lipschitz Function ◽  
Type Theorem ◽  
Numerical Results ◽  
Bernstein Operators ◽  
Approximation Properties ◽  
Voronovskaya Type Theorem ◽  
New Type ◽  
Kantorovich Operators

<abstract><p>In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.</p></abstract>

scholarly journals Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 335-343 ◽  
Author(s):  
Arun Kajla ◽  
Serkan Araci
Keyword(s):  
Asymptotic Formula ◽  
Rate Of Convergence ◽  
Modulus Of Smoothness ◽  
Stancu Operators ◽  
Rate Of Approximation ◽  
Type Approximation ◽  
Kantorovich Operators

AbstractIn the paper the authors introduce the Kantorovich variant of Stancu operators based on Pólya-Eggenberger distribution. By making use of this new operator, we obtain some indispensable auxiliary results. We also deal with a Voronovskaja type asymptotic formula and some estimates of the rate of approximation involving modulus of smoothness, such as Ditzian-Totik modulus of smoothness. The rate of convergence for differential functions whose derivatives are bounded is also obtained.

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.

A Voronovskaya-type formula for SMK operators via statistical convergence

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Ali Aral ◽  
Oktay Duman
Keyword(s):  
Statistical Convergence ◽  
Type Theorem ◽  
Type Formula ◽  
Voronovskaya Type Theorem ◽  
Summability Matrix ◽  
Regular Summability

AbstractIn this paper, we obtain a statistical Voronovskaya-type theorem for the Szász-Mirakjan-Kantorovich (SMK) operators by using the notion of A-statistical convergence, where A is a non-negative regular summability matrix.

scholarly journals Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3473-3486 ◽  
Author(s):  
Faruk Özger
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Approximation Properties ◽  
Statistical Approximation ◽  
Type Approximation ◽  
Bivariate Case ◽  
Estimate Rate ◽  
Kantorovich Operators

In this study, we consider statistical approximation properties of univariate and bivariate ?-Kantorovich operators. We estimate rate of weighted A-statistical convergence and prove a Voronovskajatype approximation theorem by a family of linear operators using the notion of weighted A-statistical convergence. We give some estimates for differences of ?-Bernstein and ?-Durrmeyer, and ?-Bernstein and ?-Kantorovich operators. We establish a Voronovskaja-type approximation theorem by weighted A-statistical convergence for the bivariate case.

scholarly journals Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 448 ◽  
Author(s):  
H. Srivastava ◽  
B. Jena ◽  
S. Paikray ◽  
U. Misra
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Double Sequence ◽  
Modular Space ◽  
Double Sequences ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Korovkin Type Approximation Theorems ◽  
Kantorovich Operators ◽  
Sequence Of Functions

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. . The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

scholarly journals Generalized Baskakov Kantorovich operators

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6131-6151
Author(s):  
P.N. Agrawal ◽  
Meenu Goyal
Keyword(s):  
Rate Of Convergence ◽  
Bounded Variation ◽  
Statistical Convergence ◽  
Simultaneous Approximation ◽  
Weighted Approximation ◽  
Function Of Bounded Variation ◽  
Convergence Properties ◽  
Almost Everywhere ◽  
Direct Results ◽  
Kantorovich Operators

In this paper, we construct generalized Baskakov Kantorovich operators. We establish some direct results and then study weighted approximation, simultaneous approximation and statistical convergence properties for these operators. Finally, we obtain the rate of convergence for functions having a derivative coinciding almost everywhere with a function of bounded variation for these operators.

scholarly journals A NEW KIND OF THE VARIANT OF THE MODIFIED BERNSTEIN-KANTOROVICH OPERATORS DEFINED BY ¨OZARSLAN AND DUMAN

2021 ◽  
Author(s):  
Abhishek Kumar
Keyword(s):  
Rate Of Convergence ◽  
Present Article ◽  
Bounded Variation ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Type Theorem ◽  
Careful Choice ◽  
Derivatives Of ◽  
Kantorovich Operators ◽  
Infinitely Differentiable Function

In the present article, we dene a new kind of the modified Bernstein-Kantorovich operators defined by ¨ Ozarslan (https://doi.org/10.1080/01630563.2015.1079219) i.e. we introduce a new function ς(x) in the modified Bernstein-Kantorovich operators defined by Ozarslan with the property ({) is an infinitely differentiable function on [0; 1]; ς(0) = 0; ς(1) = 1 and ς’(x) > 0 for all x∈ [0; 1]. We substantiate an approximation theorem by using of the Bohman-Korovkins type theorem and scrutinize the rate of convergence with the aid of modulus of continuity, Lipschitz type functions for the our operators and the rate of convergence of functions by means of derivatives of bounded variation are also studied. We study an approximation theorem with the help of Bohman-Korovkins type theorem in A-Statistical convergence. Lastly, by means of a numerical example, we illustrate the convergence of these operators to certain functions through graphs with the help of MATHEMATICA and show that a careful choice of the function ς(x) leads to a better approximation results as compared to the modified Bernstein-Kantorovich operators defined by Ozarslan (https://doi.org/10.1080/01630563.2015.1079219).

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.

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