Kantorovich variant of a new kind of q‐Bernstein–Schurer operators

Abstract

Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.

Theq-Chlodowsky andq-Szasz-Durrmeyer Hybrid Operators on Weighted Spaces

“…He obtained the rate of convergence on weighted spaces and a Voronovskaya-type theorem for these operators. Some other studies based on classical q− theory are [17][18][19][20][21][22][23][24].…”

Approximation by ( p , q ) $(p,q)$ -Lupaş–Schurer–Kantorovich operators

“…The development q -calculus applications established a precedent in the field of approximation theory. We may refer to some of them as Durrmeyer variant of q -Bernstein–Schurer operators [ 2 ], q -Bernstein–Schurer–Kantorovich type operators [ 3 ], q -Durrmeyer operators [ 8 ], q -Bernstein–Schurer–Durrmeyer type operators [ 12 ], q -Bernstein–Schurer operators [ 19 ], King’s type modified q -Bernstein–Kantorovich operators [ 20 ], q -Bernstein–Schurer–Kantorovich operators [ 23 ]. Lately, Mursaleen et al [ 17 ] pioneered the research of -analogue of Bernstein operators which is a generalization of q -Bernstein operators (Philips).…”