Some approximation results for the new modification of Bernstein-Beta operators

Qing-Bo Cai; Melek Sofyalıoğlu; Kadir Kanat; Bayram Çekim; Qing-Bo Cai; Melek Sofyalıoğlu; Kadir Kanat; Bayram Çekim

doi:10.3934/math.2022105

AIMS Mathematics

2022, Volume 7, Issue 2: 1831-1844. doi: 10.3934/math.2022105

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Research article

Some approximation results for the new modification of Bernstein-Beta operators

1.
Fujian Provincial Key Laboratory of Data-Intensive Computing Key Laboratory of Intelligent Computing and Information Processing School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China
2.
Ankara Hacı Bayram Veli University, Polatlı Faculty of Science and Letters, Department of Mathematics, Ankara, Turkey
3.
Gazi University, Faculty of Science, Department of Mathematics, Ankara, Turkey

Received: 30 June 2021 Accepted: 27 October 2021 Published: 03 November 2021
MSC : 41A25, 41A36, 47A58

This paper deals with the newly modification of Beta-type Bernstein operators, preserving constant and Korovkin's other test functions $ e_i = t^i $, $ i = 1, 2 $ in limit case. Then the uniform convergence of the constructed operators is given. The rate of convergence is obtained in terms of modulus of continuity, Peetre-$ \mathcal{K} $ functionals and Lipschitz class functions. After that, the Voronovskaya-type asymptotic result for these operators is established. At last, the graphical results of the newly defined operators are discussed.
- Korovkin type approximation theorem,
- modulus of continuity,
- functions of Lipschitz class,
- Peetre's $ \mathcal{K} $-functionals
Citation: Qing-Bo Cai, Melek Sofyalıoğlu, Kadir Kanat, Bayram Çekim. Some approximation results for the new modification of Bernstein-Beta operators[J]. AIMS Mathematics, 2022, 7(2): 1831-1844. doi: 10.3934/math.2022105

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Abstract

This paper deals with the newly modification of Beta-type Bernstein operators, preserving constant and Korovkin's other test functions $ e_i = t^i $, $ i = 1, 2 $ in limit case. Then the uniform convergence of the constructed operators is given. The rate of convergence is obtained in terms of modulus of continuity, Peetre-$ \mathcal{K} $ functionals and Lipschitz class functions. After that, the Voronovskaya-type asymptotic result for these operators is established. At last, the graphical results of the newly defined operators are discussed.

References

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