Degree of Approximation by Generalized Boolean Sum of $$\lambda $$-Bernstein Operators

Abstract

The purpose of the present paper is to investigate the degree of approximation of the \(\lambda \)-Bernstein operators introduced by Cai et al. (J Inequal Appl 61:1–11, 2018 [9]) by means of the Steklov mean, the Ditizian–Totik modulus of smoothness and the approximation of functions with derivatives of bounded variation. We introduce the bivariate case of the above operators and investigate the rate of convergence with the aid of the total and partial modulus of continuity and the Peetre’s K-functional. Furthermore, we define the associated GBS (Generalized Boolean Sum) operator of the bivariate operators and establish the degree of approximation in terms of the mixed modulus of smoothness for Bogel continuous and Bogel differentiable functions.