Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators

<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id="M3">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type="bibr" rid="b3">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id="M4">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id="M5">\begin{document}$ e_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ e_2 $\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id="M7">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>

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Statistical approximation properties of modified Durrmeyer q-Baskakov type operators with two parameters α and β

Asian-European Journal of Mathematics ◽

10.1142/s1793557117500280 ◽

2016 ◽

Vol 10 (02) ◽

pp. 1750028

Author(s):

Vishnu Narayan Mishra ◽

Preeti Sharma

Keyword(s):

Modulus Of Continuity ◽

Maximal Function ◽

Statistical Convergence ◽

Approximation Theorem ◽

Rates Of Convergence ◽

Approximation Properties ◽

Statistical Approximation ◽

Baskakov Operators ◽

Two Parameters ◽

Lipschitz Type Maximal Function

The main aim of this study is to obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established. Our results show that rates of convergence of our operators are at least as fast as classical Durrmeyer type modified Baskakov operators.

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Stancu-variant of generalized Baskakov operators

Filomat ◽

10.2298/fil1709625r ◽

2017 ◽

Vol 31 (9) ◽

pp. 2625-2632 ◽

Cited By ~ 2

Author(s):

Nadeem Rao ◽

Abdul Wafi

Keyword(s):

Rate Of Convergence ◽

Modulus Of Continuity ◽

Type Theorem ◽

Modulus Of Smoothness ◽

Order Of Approximation ◽

Baskakov Operators ◽

Direct Estimate ◽

Voronovskaya Type Theorem

In the present paper, we introduce Stancu-variant of generalized Baskakov operators and study the rate of convergence using modulus of continuity, order of approximation for the derivative of function f . Direct estimate is proved using K-functional and Ditzian-Totik modulus of smoothness. In the last, we have proved Voronovskaya type theorem.

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On approximation properties of Baskakov-Schurer-Szász-Stancu operators based on q-integers

Filomat ◽

10.2298/fil1804359m ◽

2018 ◽

Vol 32 (4) ◽

pp. 1359-1378 ◽

Cited By ~ 4

Author(s):

M. Mursaleen ◽

A.A.H. Al-Abied ◽

Khursheed Ansari

Keyword(s):

Rate Of Convergence ◽

Maximal Function ◽

Statistical Convergence ◽

Type Theorem ◽

Weighted Approximation ◽

Approximation Properties ◽

Stancu Operators ◽

Voronovskaya Type Theorem ◽

Lipschitz Type Maximal Function

In the present paper, we introduce Stancu type generalization of Baskakov-Schurer-Sz?sz operators based on the q-integers and investigate their approximation properties. We obtain rate of convergence, weighted approximation and Voronovskaya type theorem for new operators. Then we obtain a point-wise estimate using the Lipschitz type maximal function. Furthermore, we study A-statistical convergence of these operators and also, in order to obtain a better approximation.

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Rate of convergence of Szász-beta operators based on q-integers

Demonstratio Mathematica ◽

10.1515/dema-2017-0015 ◽

2017 ◽

Vol 50 (1) ◽

pp. 130-143 ◽

Cited By ~ 1

Author(s):

Pooja Gupta ◽

Purshottam Narain Agrawal

Keyword(s):

Rate Of Convergence ◽

Modulus Of Continuity ◽

Maximal Function ◽

Statistical Convergence ◽

Weighted Modulus Of Continuity ◽

Weighted Modulus ◽

Lipschitz Type Maximal Function

Abstract The purpose of this paper is to establish the rate of convergence in terms of the weighted modulus of continuity and Lipschitz type maximal function for the q-Szász-beta operators. We also study the rate of A-statistical convergence. Lastly, we modify these operators using King type approach to obtain better approximation.

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Approximation of Modified Baskakov Operators Based on Parameter s

Iraqi Journal of Science ◽

10.24996/ijs.2021.62.2.24 ◽

2021 ◽

pp. 588-593

Author(s):

Ali J. Mohammad ◽

S. A. Abdul-Hammed ◽

T. A. Abdul-Qader

Keyword(s):

Numerical Results ◽

Order Of Approximation ◽

Test Functions ◽

Baskakov Operators ◽

Type Formula ◽

Better Than

In this article, we define and study a family of modified Baskakov type operators based on a parameter . This family is a generalization of the classical Baskakov sequence. First, we prove that it converges to the function being approximated. Then, we find a Voronovsky-type formula and obtain that the order of approximation of this family is . This order is better than the order of the classical Baskakov sequence whenever . Finally, we apply our sequence to approximate two test functions and analyze the numerical results obtained.

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Approximation by modified Jain–Baskakov operators

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2008 ◽

2020 ◽

Vol 27 (3) ◽

pp. 403-412

Author(s):

Vishnu Narayan Mishra ◽

Preeti Sharma ◽

Marius Mihai Birou

Keyword(s):

Weighted Approximation ◽

Approximation Order ◽

Continuous Functions ◽

Basis Functions ◽

Test Functions ◽

Approximation Properties ◽

Baskakov Operators ◽

Closed Intervals ◽

Direct Results ◽

Modified Forms

AbstractIn the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions {e_{0}} and {e_{1}}. It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

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Approximation properties of modified Srivastava-Gupta operators based on certain parameter

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v38i1.36907 ◽

2018 ◽

Vol 38 (1) ◽

pp. 41-53 ◽

Cited By ~ 1

Author(s):

Alok Kumar ◽

Dr Vandana

Keyword(s):

Maximal Function ◽

Statistical Convergence ◽

Approximation Theorem ◽

Type Theorem ◽

Weighted Approximation ◽

Linear Functions ◽

Approximation Properties ◽

Voronovskaja Type Theorem ◽

Direct Approximation ◽

Lipschitz Type Maximal Function

In the present article, we give a modified form of generalized Srivastava-Gupta operators based on certain parameter which preserve the constant as well as linear functions. First, we estimate moments of the operators and then prove Voronovskaja type theorem. Next, direct approximation theorem, rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Then, we obtain point-wise estimate using the Lipschitz type maximal function. Finaly, we study the $A$-statistical convergence of these operators.

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Jakimovski–Leviatan operators of Kantorovich type involving multiple Appell polynomials

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2013 ◽

2019 ◽

Vol 28 (1) ◽

pp. 73-82 ◽

Cited By ~ 1

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).

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Blending type Approximations by Kantorovich variant of α-Schurer operators

10.22541/au.162526026.60514857/v1 ◽

2021 ◽

Author(s):

Nadeem Rao ◽

Pradeep Malik ◽

Mamta Rani

Keyword(s):

Numerical Analysis ◽

Modulus Of Smoothness ◽

Second Order ◽

Weight Functions ◽

Order Of Approximation ◽

Approximation Properties ◽

Global Approximation ◽

Measurable Functions ◽

Two Parameters ◽

Kantorovich Operators

In the present manuscript, we present a new sequence of operators, i:e:, -Bernstein-Schurer-Kantorovich operators depending on two parameters 2 [0; 1] and > 0 foe one and two variables to approximate measurable functions on [0:1+q]; q > 0. Next, we give basic results and discuss the rapidity of convergence and order of approximation for univariate and bivariate of these sequences in their respective sections . Further, Graphical and numerical analysis are presented. Moreover, local and global approximation properties are discussed in terms of rst and second order modulus of smoothness, Peetre’s K-functional and weight functions for these sequences in dierent spaces of functions.

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