Bivariate $ \lambda $-Bernstein operators on triangular domain

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) $.

Keywords

• bivariate $ \lambda $-bernstein operators
• modulus of continuity
• korovkin-type theorem
• rate of convergence
• local approximation
• voronovskaja asymptotic formula