Approximation on univariate and bivariate modified Bernstein-Kantorovich operators

Weiping Zhou; Yujie Liu; Wentao Cheng; Weiping Zhou; Yujie Liu; Wentao Cheng

doi:10.3934/math.2026858

AIMS Mathematics

2026, Volume 11, Issue 7: 21128-21152. doi: 10.3934/math.2026858

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Research article

Approximation on univariate and bivariate modified Bernstein-Kantorovich operators

1.
School of Mathematics and Statistics, Anqing Normal University, Anhui, Anqing 246133, China
2.
School of Mathematical Sciences, Guizhou Normal University, Guizhou, Guiyang 550001, China

Received: 08 April 2026 Revised: 29 June 2026 Accepted: 07 July 2026 Published: 16 July 2026
MSC : 26D20, 41A25, 41A35, 41A36

In this manuscript, we introduce a new $ \rho $-th-order Kantorovich-type generalization of the modified Bernstein operators. First, we establish Voronovskaya-type theorems for the proposed operators. Then, we derive rates of convergence and establish global approximation theorems by means of the modulus of continuity, the $ \mathbb{K} $-functional, the Ditzian-Totik weighted moduli of smoothness and the weighted $ \mathbb{K} $-functional. Also, several numerical examples are presented to illustrate the approximation behavior using MATLAB. Finally, we construct a bivariate analog of these operators and investigate its convergence properties.
- Bernstein operators,
- Voronovskaya-type theorems,
- modulus of continuity,
- uniform convergence,
- global approximation,
- bivariate operators
Citation: Weiping Zhou, Yujie Liu, Wentao Cheng. Approximation on univariate and bivariate modified Bernstein-Kantorovich operators[J]. AIMS Mathematics, 2026, 11(7): 21128-21152. doi: 10.3934/math.2026858

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Abstract

In this manuscript, we introduce a new $ \rho $-th-order Kantorovich-type generalization of the modified Bernstein operators. First, we establish Voronovskaya-type theorems for the proposed operators. Then, we derive rates of convergence and establish global approximation theorems by means of the modulus of continuity, the $ \mathbb{K} $-functional, the Ditzian-Totik weighted moduli of smoothness and the weighted $ \mathbb{K} $-functional. Also, several numerical examples are presented to illustrate the approximation behavior using MATLAB. Finally, we construct a bivariate analog of these operators and investigate its convergence properties.

References

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