Approximation by the modified $ \lambda $-Bernstein-polynomial in terms of basis function

Mohammad Ayman-Mursaleen; Md. Nasiruzzaman; Nadeem Rao; Mohammad Dilshad; Kottakkaran Sooppy Nisar; Mohammad Ayman-Mursaleen; Md. Nasiruzzaman; Nadeem Rao; Mohammad Dilshad; Kottakkaran Sooppy Nisar

doi:10.3934/math.2024217

AIMS Mathematics

2024, Volume 9, Issue 2: 4409-4426. doi: 10.3934/math.2024217

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Research article

Approximation by the modified $ \lambda $-Bernstein-polynomial in terms of basis function

1.
School of Information and Physical Sciences, The University of Newcastle, University Drive, Callaghan, New South Wales 2308, Australia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 4279, Tabuk 71491, Saudi Arabia
3.
Department of Mathematics, University Centre for Research and Development, Chandigarh University, Mohali-140413, Punjab, India
4.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Received: 29 August 2023 Revised: 04 January 2024 Accepted: 08 January 2024 Published: 18 January 2024
MSC : 33C45, 41A25, 41A36

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.
- Bernstein-polynomial,
- $ \lambda $-Bernstein-polynomial,
- Bézier basis function,
- shifted knots,
- modulus of continuity,
- Lipschitz maximal functions,
- Peetre's $ K $-functional
Citation: Mohammad Ayman-Mursaleen, Md. Nasiruzzaman, Nadeem Rao, Mohammad Dilshad, Kottakkaran Sooppy Nisar. Approximation by the modified $ \lambda $-Bernstein-polynomial in terms of basis function[J]. AIMS Mathematics, 2024, 9(2): 4409-4426. doi: 10.3934/math.2024217

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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.

References

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