Approximation properties of modified q-Bernstein-Kantorovich operators

Abstract

In the present paper we define a q-analogue of the modified Bernstein-Kantorovich operatorsintroduced by Ozarslan and Duman (Numer. Funct. Anal. Optim. 37:92-105,2016). We establishthe shape preserving properties of these operators e.g. monotonicity and convexity and study the rateof convergence by means of Lipschitz class and Peetre's K-functional and degree of approximation withthe aid of a smoothing process e.g Steklov mean. Further, we introduce the bivariate case of modifiedq-Bernstein-Kantorovich operators and study the degree of approximation in terms of the partial andtotal modulus of continuity and Peetre's K-functional. Finally, we introduce the associated GBS (GeneralizedBoolean Sum) operators and investigate the approximation of the Bogel continuous and Bogeldifferentiable functions by using the mixed modulus of smoothness and Lipschitz class.