Better degree of approximation by modified Bernstein-Durrmeyer type operators

Purshottam Narain Agrawal; Şule Yüksel Güngör; Abhishek Kumar

doi:10.3934/mfc.2021024

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2022, Volume 5, Issue 2: 75-92. Doi: 10.3934/mfc.2021024

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Better degree of approximation by modified Bernstein-Durrmeyer type operators

1.
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India
2.
Gazi University, Faculty of Science, Department Of Mathematics, 06500, Ankara, Turkey

^* Corresponding author: Ş. Y. Güngör
^* Corresponding author: Ş. Y. Güngör

Received: June 30, 2021

Revised: August 31, 2021

Early access: October 2021

Published: May 2022

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Abstract

In the present article we investigate a Durrmeyer variant of the generalized Bernstein-operators based on a function $ \tau(x), $ where $ \tau $ is infinitely differentiable function on $ [0, 1], \; \tau(0) = 0, \tau(1) = 1 $ and $ \tau^{\prime }(x)>0, \;\forall\;\; x\in[0, 1]. $ We study the degree of approximation by means of the modulus of continuity and the Ditzian-Totik modulus of smoothness. A Voronovskaja type asymptotic theorem and the approximation of functions with derivatives of bounded variation are also studied. By means of a numerical example, finally we illustrate the convergence of these operators to certain functions through graphs and show a careful choice of the function $ \tau(x) $ leads to a better approximation than the generalized Bernstein-Durrmeyer type operators considered by Kajla and Acar [11].

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Mathematics Subject Classification: Primary: 41A25; Secondary: 41A36.

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Figure 2. The convergence of $ M_{n}^{(\alpha )}(f;x) $ and $ M_{n, \tau }^{(\alpha )}(f;x) $ operators to the function $ f(x) $ for $ n = 10 $

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Figure 3. The convergence of $ M_{n}^{(\alpha )}(f;x) $ and $ M_{n, \tau }^{(\alpha )}(f;x) $ operators to the function $ f(x) $ for $ n = 20 $

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References

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