Approximation degree of bivariate Kantorovich Stancu operators

Abstract

Abel et al. [U. Abel, M. Ivan, R. Păltănea, Appl. Math. Comput., 259 (2015), 116–123] introduced a Durrmeyer type integral variant of the Bernstein type operators based on two parameters defined by Stancu [D. D. Stancu, Calcolo, 35 (1998), 53–62]. Kajla [A. Kajla, Appl. Math. Comput., 316 (2018), 400–408] considered a Kantorovich modification of the Stancu operators wherein he studied some basic convergence theorems and also the rate of A-statistical convergence. In the present paper, we define a bivariate case of the operators proposed in [A. Kajla, Appl. Math. Comput., 316 (2018), 400–408] to study the degree of approximation for functions of two variables. We obtain the rate of convergence of these bivariate operators by means of the complete modulus of continuity, the partial moduli of continuity and the Peetre’s K-functional. Voronovskaya and Grüss Voronovskaya type theorems are also established. We introduce the associated GBS (Generalized Boolean Sum) operators of the bivariate operators and discuss the approximation degree of these operators with the aid of the mixed modulus of smoothness for Bögel continuous and Bögel differentiable functions.