Statistical Blending-Type Approximation by a Class of Operators That Includes Shape Parameters <i>λ</i> and <i>α</i>

Abstract

This paper is devoted to studying the statistical approximation properties of a sequence of univariate and bivariate blending-type Bernstein operators that includes shape parameters α and λ and a positive integer. An estimate of the corresponding rates was obtained, and a Voronovskaja-type theorem is given by a weighted A-statistical convergence. A Korovkin-type theorem is provided for the univariate and bivariate cases of the blending-type operators. Moreover, the convergence behavior of the univariate and bivariate new blending basis and new blending operators are exhaustively demonstrated by computer graphics. The studied univariate and bivariate blending-type operators reduce to the well-known Bernstein operators in the literature for the special cases of shape parameters α and λ, and they propose better approximation results.

Keywords

• Voronovskaja-type theorem
• blending-type operators
• <i>α</i>-Bernstein operators
• <i>λ</i>-Bernstein operators
• shape parameters
• statistical convergence