Bivariate $ \lambda $-Bernstein operators on triangular domain

Guorong Zhou; Qing-Bo Cai; Guorong Zhou; Qing-Bo Cai

doi:10.3934/math.2024700

AIMS Mathematics

2024, Volume 9, Issue 6: 14405-14424. doi: 10.3934/math.2024700

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Bivariate $ \lambda $-Bernstein operators on triangular domain

Guorong Zhou ¹,
Qing-Bo Cai ^{2
,
,}

1.
School of Mathematics and Statistics, Xiamen University of Technology, Xiamen 361024, Fujian, China
2.
School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, Fujian, China

Received: 25 February 2024 Revised: 09 April 2024 Accepted: 15 April 2024 Published: 22 April 2024
MSC : 41A10, 41A25, 41A36

This paper introduced a novel class of bivariate $ \lambda $-Bernstein operators defined on triangular domain, denoted as $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $. These operators leverage a new class of bivariate Bézier basis functions defined on triangular domain with shape parameters $ \lambda_1 $ and $ \lambda_2 $. A Korovkin-type approximation theorem for $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $ was established, with the convergence rate being characterized by both the complete and partial moduli of continuity. Additionally, a local approximation theorem and a Voronovskaja-type asymptotic formula were derived for $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $. Finally, the convergence of $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $ to $ f(x, y) $ was illustrated through graphical representations and numerical examples, highlighting instances where they surpass the performance of standard bivariate Bernstein operators defined on triangular domain, $ B_{m}(f; x, y) $.
- bivariate $ \lambda $-Bernstein operators,
- modulus of continuity,
- Korovkin-type theorem,
- rate of convergence,
- local approximation,
- voronovskaja asymptotic formula
Citation: Guorong Zhou, Qing-Bo Cai. Bivariate $ \lambda $-Bernstein operators on triangular domain[J]. AIMS Mathematics, 2024, 9(6): 14405-14424. doi: 10.3934/math.2024700

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Abstract

This paper introduced a novel class of bivariate $ \lambda $-Bernstein operators defined on triangular domain, denoted as $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $. These operators leverage a new class of bivariate Bézier basis functions defined on triangular domain with shape parameters $ \lambda_1 $ and $ \lambda_2 $. A Korovkin-type approximation theorem for $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $ was established, with the convergence rate being characterized by both the complete and partial moduli of continuity. Additionally, a local approximation theorem and a Voronovskaja-type asymptotic formula were derived for $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $. Finally, the convergence of $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $ to $ f(x, y) $ was illustrated through graphical representations and numerical examples, highlighting instances where they surpass the performance of standard bivariate Bernstein operators defined on triangular domain, $ B_{m}(f; x, y) $.

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