Bernstein-type operators on elliptic domain and their interpolation properties

Abstract

Abstract The aim of this article is to construct univariate Bernstein-type operators ( ℬ m x G ) ( x , z ) \left({{\mathcal{ {\mathcal B} }}}_{m}^{x}G)\left(x,z) and ( ℬ n z G ) ( x , z ) , \left({{\mathcal{ {\mathcal B} }}}_{n}^{z}G)\left(x,z), their products ( P m n G ) ( x , z ) \left({{\mathcal{P}}}_{mn}G)\left(x,z) , ( Q n m G ) ( x , z ) \left({{\mathcal{Q}}}_{nm}G)\left(x,z) , and their Boolean sums ( S m n G ) ( x , z ) \left({{\mathcal{S}}}_{mn}G)\left(x,z) , ( T n m G ) ( x , z ) \left({{\mathcal{T}}}_{nm}G)\left(x,z) on elliptic region, which interpolate the given real valued function G G defined on elliptic region on its boundary. The bound of the remainders of each approximation formula of corresponding operators are computed with the help of Peano’s theorem and modulus of continuity, and the rate of convergence for functions of Lipschitz class is computed.