On a New Kind of Λ-Bernstein Operators for Univariate and Bivariate Functions

Abstract

This paper presents a novel class of lambda-Bernstein operators, wherein the parameter lambda is an element of [-1, 1]. An approximation theorem of the Korovkin type is explored, a local approximation theorem is established and an asymptotic formula of the Voronovskaja type is derived. In addition, the bivariate tensor product operators are built, some approximation properties are discussed, including an asymptotic theorem of the Voronovskaja type and the order of convergence in relation to Peetre's K-functional. Finally, for certain continuous functions, numerical examples and plots to demonstrate our newly defined operators' convergence behavior are provided and there are also provided in comparison with the classical Kantorovich operators in terms of the approximation error.