scholarly journals A new kind of Bi-variate $ \lambda $ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Purshottam Narain Agrawal ◽  
Behar Baxhaku ◽  
Rahul Shukla
Keyword(s):  
Rate Of Convergence ◽  
Type Theorem ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Type Operator ◽  
Mixed Modulus Of Smoothness ◽  
Voronovskaja Type Theorem ◽  
Differentiable Functions ◽  
Mixed Modulus ◽  
Approximation Of A Function

<p style='text-indent:20px;'>In this paper, we introduce a bi-variate case of a new kind of <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich type operator with shifted knots defined by Rahman et al. [<xref ref-type="bibr" rid="b31">31</xref>]. The rate of convergence of the bi-variate operators is obtained in terms of the complete and partial moduli of continuity. Next, we give an error estimate in the approximation of a function in the Lipschitz class and establish a Voronovskaja type theorem. Also, we define the associated GBS(Generalized Boolean Sum) operators and study the degree of approximation of Bögel continuous and Bögel differentiable functions by these operators with the aid of the mixed modulus of smoothness. Finally, we show the rate of convergence of the bi-variate operators and their GBS case for certain functions by illustrative graphics and tables using MATLAB algorithms.</p>

scholarly journals Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
S. A. Mohiuddine
Keyword(s):  
Rate Of Convergence ◽  
Differentiable Function ◽  
Type Theorem ◽  
Modulus Of Smoothness ◽  
Order Of Approximation ◽  
Mixed Modulus Of Smoothness ◽  
Voronovskaja Type Theorem ◽  
Gbs Operators ◽  
Mixed Modulus ◽  
Boolean Sum

AbstractWe construct the bivariate form of Bernstein–Schurer operators based on parameter α. We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the help of Peetre’s K-functional of our newly defined operators. Moreover, we define the associated generalized Boolean sum (shortly, GBS) operators and estimate the rate of convergence by means of mixed modulus of smoothness. Finally, the order of approximation for Bögel differentiable function of our GBS operators is presented.

scholarly journals On the Approximation Properties of q − Analogue Bivariate λ -Bernstein Type Operators

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Edmond Aliaga ◽  
Behar Baxhaku
Keyword(s):  
Type Theorem ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Approximation Properties ◽  
Bernstein Type ◽  
Matlab Software ◽  
Type Space ◽  
Mixed Modulus Of Smoothness ◽  
Mixed Modulus ◽  
Boolean Sum

In this article, we establish an extension of the bivariate generalization of the q -Bernstein type operators involving parameter λ and extension of GBS (Generalized Boolean Sum) operators of bivariate q -Bernstein type. For the first operators, we state the Volkov-type theorem and we obtain a Voronovskaja type and investigate the degree of approximation by means of the Lipschitz type space. For the GBS type operators, we establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate q -Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MATLAB software.

scholarly journals Bivariate α,q-Bernstein–Kantorovich Operators and GBS Operators of Bivariate α,q-Bernstein–Kantorovich Type

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1161 ◽  
Author(s):  
Qing-Bo Cai ◽  
Wen-Tao Cheng ◽  
Bayram Çekim
Keyword(s):  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Moduli Of Continuity ◽  
Mixed Modulus Of Smoothness ◽  
Differentiable Functions ◽  
Gbs Operators ◽  
Mixed Modulus ◽  
Bivariate Operators ◽  
Kantorovich Operators ◽  
Boolean Sum

In this paper, we introduce a family of bivariate α , q -Bernstein–Kantorovich operators and a family of G B S (Generalized Boolean Sum) operators of bivariate α , q -Bernstein–Kantorovich type. For the former, we obtain the estimate of moments and central moments, investigate the degree of approximation for these bivariate operators in terms of the partial moduli of continuity and Peetre’s K-functional. For the latter, we estimate the rate of convergence of these G B S operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness.

Jakimovski–Leviatan operators of Kantorovich type involving multiple Appell polynomials

2019 ◽  
Vol 28 (1) ◽  
pp. 73-82 ◽  
Author(s):  
Pooja Gupta ◽  
Ana Maria Acu ◽  
Purshottam Narain Agrawal
Keyword(s):  
Rate Of Convergence ◽  
Bounded Variation ◽  
Maximal Function ◽  
Polynomial Growth ◽  
Weighted Space ◽  
Type Theorem ◽  
Degree Of Approximation ◽  
Type Operator ◽  
Appell Polynomials ◽  
Lipschitz Type Maximal Function

Abstract The purpose of the present paper is to obtain the degree of approximation in terms of a Lipschitz type maximal function for the Kantorovich type modification of Jakimovski–Leviatan operators based on multiple Appell polynomials. Also, we study the rate of approximation of these operators in a weighted space of polynomial growth and for functions having a derivative of bounded variation. A Voronvskaja type theorem is obtained. Further, we illustrate the convergence of these operators for certain functions through tables and figures using the Maple algorithm and, by a numerical example, we show that our Kantorovich type operator involving multiple Appell polynomials yields a better rate of convergence than the Durrmeyer type Jakimovski Leviatan operators based on Appell polynomials introduced by Karaisa (2016).

scholarly journals Bivariate-Schurer-Stancu operators based on (p,q)-integers

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1251-1258 ◽  
Author(s):  
Nadeem Rao ◽  
Abdul Wafi
Keyword(s):  
Rate Of Convergence ◽  
Uniform Approximation ◽  
Type Theorem ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Second Order ◽  
Korovkin Type Theorem ◽  
Stancu Operators

The aim of this article is to introduce a bivariate extension of Schurer-Stancu operators based on (p,q)-integers. We prove uniform approximation by means of Bohman-Korovkin type theorem, rate of convergence using total modulus of smoothness and degree of approximation via second order modulus of smoothness, Peetre?s K-functional, Lipschitz type class.

scholarly journals About Some Linear Operators

2007 ◽  
Vol 2007 ◽  
pp. 1-13
Author(s):  
Ovidiu T. Pop
Keyword(s):  
Rate Of Convergence ◽  
General Class ◽  
Type Theorem ◽  
Modulus Of Smoothness ◽  
Positive Operators ◽  
Linear Operators ◽  
Linear Positive Operators ◽  
Voronovskaja Type Theorem

Using the method of Jakimovski and Leviatan from their work in 1969, we construct a general class of linear positive operators. We study the convergence, the evaluation for the rate of convergence in terms of the first modulus of smoothness and we give a Voronovskaja-type theorem for these operators.

scholarly journals Better numerical approximation by λ-Durrmeyer-Bernstein type operators

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1405-1419
Author(s):  
Voichiţa Radu ◽  
Purshottam Agrawal ◽  
Jitendra Singh
Keyword(s):  
Numerical Approximation ◽  
Modulus Of Smoothness ◽  
Asymptotic Formulas ◽  
Bernstein Type ◽  
Mixed Modulus Of Smoothness ◽  
Differentiable Functions ◽  
Mixed Modulus ◽  
Bivariate Operators ◽  
Durrmeyer Variant ◽  
Theoretical Results

The main object of this paper is to construct a new Durrmeyer variant of the ?-Bernstein type operators which have better features than the classical one. Some results concerning the rate of convergence in terms of the first and second moduli of continuity and asymptotic formulas of these operators are given. Moreover, we define a bivariate case of these operators and investigate the approximation degree by means of the total and partial modulus of continuity and the Peetre?s K-functional. A Voronovskaja type asymptotic and Gr?ss-Voronovskaja theorem for the bivariate operators is also proven. Further, we introduce the associated GBS (Generalized Boolean Sum) operators and determine the order of convergence with the aid of the mixed modulus of smoothness for the B?gel continuous and B?gel differentiable functions. Finally the theoretical results are analyzed by numerical examples.

scholarly journals Bezier variant of genuine-Durrmeyer type operators based on Polya distribution

2017 ◽  
Vol 33 (1) ◽  
pp. 73-86
Author(s):  
TRAPTI NEER ◽  
◽  
ANA MARIA ACU ◽  
P. N. AGRAWAL ◽  
◽  
...  
Keyword(s):  
Rate Of Convergence ◽  
Approximation Theorem ◽  
Type Theorem ◽  
Modulus Of Smoothness ◽  
Global Approximation ◽  
Voronovskaja Type Theorem ◽  
Polya Distribution ◽  
Durrmeyer Type Operators ◽  
Direct Approximation ◽  
Bézier Variant

In this paper we introduce the Bezier variant of genuine-Durrmeyer type operators having Polya basis functions. We give a global approximation theorem in terms of second order modulus of continuity, a direct approximation theorem by means of the Ditzian-Totik modulus of smoothness and a Voronovskaja type theorem by using the Ditzian-Totik modulus of smoothness. The rate of convergence for functions whose derivatives are of bounded variation is obtained. Further, we show the rate of convergence of these operators to certain functions by illustrative graphics using the Maple algorithms.

scholarly journals Generalized p,q-Gamma-type operators

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Wen-Tao Cheng ◽  
Qing-Bo Cai
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Type Theorem ◽  
Weighted Approximation ◽  
Local Approximation ◽  
Pointwise Estimates ◽  
Approximation Properties ◽  
Central Moments ◽  
Voronovskaja Type Theorem

In the present paper, the generalized p,q-gamma-type operators based on p,q-calculus are introduced. The moments and central moments are obtained, and some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre K-functional. Also, the rate of convergence, weighted approximation, and pointwise estimates of these operators are studied. Finally, a Voronovskaja-type theorem is presented.

Bivariate q-Bernstein–Chlodowsky–Durrmeyer type operators and the associated GBS operators

2019 ◽  
Vol 13 (05) ◽  
pp. 2050091
Author(s):  
Tarul Garg ◽  
Nurhayat İspir ◽  
P. N. Agrawal
Keyword(s):  
Lipschitz Space ◽  
Continuous Functions ◽  
Modulus Of Smoothness ◽  
Approximation Properties ◽  
Space Of Continuous Functions ◽  
Mixed Modulus Of Smoothness ◽  
Gbs Operators ◽  
Durrmeyer Type Operators ◽  
Mixed Modulus ◽  
Bivariate Operators

This paper deals with the approximation properties of the [Formula: see text]-bivariate Bernstein–Chlodowsky operators of Durrmeyer type. We investigate the approximation degree of the [Formula: see text]-bivariate operators for continuous functions in Lipschitz space and also with the help of partial modulus of continuity. Further, the Generalized Boolean Sum (GBS) operator of these bivariate [Formula: see text]–Bernstein–Chlodowsky–Durrmeyer operators is introduced and the rate of convergence in the Bögel space of continuous functions by means of the Lipschitz class and the mixed modulus of smoothness is examined. Furthermore, the convergence and its comparisons are shown by illustrative graphics for the [Formula: see text]-bivariate operators and the associated GBS operators to certain functions using Maple algorithms.

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