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

Purshottam Narain Agrawal; Jitendra Kumar Singh

doi:10.3934/mfc.2021040

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2023, Volume 6, Issue 2: 108-122. Doi: 10.3934/mfc.2021040

This issue Previous Article Next Article On Szász-Durrmeyer type modification using Gould Hopper polynomials

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

Purshottam Narain Agrawal and
Jitendra Kumar Singh^,

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India-247667

^*Corresponding author: Jitendra Kumar Singh
^*Corresponding author: Jitendra Kumar Singh

Received: October 2021

Revised: November 2021

Early access: January 2022

Published: May 2023

The second author is supported by Council of Scientific and Industrial Research, New Delhi, India, grant no.- 09/143(0914)/2018-EMR-I

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Abstract

The aim of this paper is to study some approximation properties of the Durrmeyer variant of $ \alpha $-Baskakov operators $ M_{n,\alpha} $ proposed by Aral and Erbay [3]. 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$ \ddot{u} $ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions $ e_0 $ and $ e_2 $ 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 $ M_{n,\alpha} $ and show the comparison of its rate of approximation vis-a-vis the modified operators.

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Mathematics Subject Classification: 26A15, 41A25, 41A35.

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