Bernstein-type operators on elliptic domain and their interpolation properties

Abstract

The aim of this article is to construct univariate Bernstein-type operators (ℬmxG)(x,z)\left({{\mathcal{ {\mathcal B} }}}_{m}^{x}G)\left(x,z) and (ℬnzG)(x,z),\left({{\mathcal{ {\mathcal B} }}}_{n}^{z}G)\left(x,z), their products (PmnG)(x,z)\left({{\mathcal{P}}}_{mn}G)\left(x,z), (QnmG)(x,z)\left({{\mathcal{Q}}}_{nm}G)\left(x,z), and their Boolean sums (SmnG)(x,z)\left({{\mathcal{S}}}_{mn}G)\left(x,z), (TnmG)(x,z)\left({{\mathcal{T}}}_{nm}G)\left(x,z) on elliptic region, which interpolate the given real valued function GG 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.

Keywords

• bernstein-type operators
• elliptic domain
• interpolation
• product operators
• boolean sum operators
• modulus of continuity
• peano’s theorem
• 41a35
• 41a36
• 41a80