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Research Article | Open Access

Approximation by the modified λ-Bernstein-polynomial in terms of basis function

Mohammad Ayman-Mursaleen1, Md. Nasiruzzaman2, Nadeem Rao3, Mohammad Dilshad2, Kottakkaran Sooppy Nisar4( )
School of Information and Physical Sciences, The University of Newcastle, University Drive, Callaghan, New South Wales 2308, Australia
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 4279, Tabuk 71491, Saudi Arabia
Department of Mathematics, University Centre for Research and Development, Chandigarh University, Mohali-140413, Punjab, India
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
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Abstract

In this article by means of shifted knots properties, we introduce a new type of coupled Bernstein operators for Bézier basis functions. First, we construct the operators based on shifted knots properties of Bézier basis functions then investigate the Korovkin's theorem, establish a local approximation theorem, and provide a convergence theorem for Lipschitz continuous functions and Peetre's K-functional. In addition, we also obtain an asymptotic formula of the type Voronovskaja.

Keywords

Bernstein-polynomialλ-Bernstein-polynomialBézier basis functionshifted knotsmodulus of continuityLipschitz maximal functionsPeetre's K-functional
CLC number: 33C45, 41A25, 41A36

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AIMS Mathematics
Volume 9 Issue 2,
February 2024
Pages 4409-4426
DOI: 10.3934/math.2024217

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Ayman-Mursaleen M, Nasiruzzaman M, Rao N, et al. Approximation by the modified λ-Bernstein-polynomial in terms of basis function. AIMS Mathematics, 2024, 9(2): 4409-4426. https://doi.org/10.3934/math.2024217

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Received: 29 August 2023
Revised: 04 January 2024
Accepted: 08 January 2024
Published: 15 February 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)