Approximation by $\lambda$-Bernstein type Operators on Triangular Domain

Abstract

In this paper, a new type of $\lambda$-Bernstein operators $\big(B^{w}_{m,\lambda}g\big)(w,z)$ and $\big(B^{z}_{n,\lambda}g\big)(w,z)$ over triangular domain are constructed. Their Products $\big(P_{mn,\lambda}g\big)(w,z),$ $\big(Q_{nm,\lambda}g\big)(w,z)$ and their Boolean sums $\big(S_{mn,\lambda}g\big)(w,z)$, $\big(T_{nm,\lambda}g)(w,z) $ on triangle $\mathcal{R}_h,$ with parameter $\lambda \in [-1,1] $ are obtained. Convergence theorem for Lipschitz type continuous functions and a Voronovskaja-type asymptotic formula are studied for these introduced operators. Remainder terms for error evaluation by using modulus of continuity are discussed. Graphical representations are added to demonstrate consistency to theoretical findings for the operators approximating functions on triangular domain. Parameter $\lambda$ will provide flexibility in approximation and in some case approximation will be better than its classical analogue.