@article{Zhou2024, 
author = {Guorong Zhou and Qing-Bo Cai},
title = {Bivariate    λ-Bernstein operators on triangular domain},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {6},
pages = {14405-14424},
keywords = {bivariate λ-Bernstein operators, modulus of continuity, Korovkin-type theorem, rate of convergence, local approximation, voronovskaja asymptotic formula},
url = {https://www.sciopen.com/article/10.3934/math.2024700},
doi = {10.3934/math.2024700},
abstract = {This paper introduced a novel class of bivariate    λ-Bernstein operators defined on triangular domain, denoted as        B          m                      λ        1            ,              λ        2              (  f  ;  x  ,  y  ). These operators leverage a new class of bivariate Bézier basis functions defined on triangular domain with shape parameters        λ    1   and        λ    2  . A Korovkin-type approximation theorem for        B          m                      λ        1            ,              λ        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                      λ        1            ,              λ        2              (  f  ;  x  ,  y  ). Finally, the convergence of        B          m                      λ        1            ,              λ        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  ).}
}