Bivariate $\lambda$-Bernstein operators on triangular domain

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, $B_m(f;x,y)$.