APPROXIMATION PROPERTIES OF MODIFIED q-BERNSTEIN-KANTOROVICH OPERATORS

Abstract

In the present paper we deÖne a q-analogue of the modiÖed BernsteinKantorovich operators introduced by ÷zarslan and Duman(Numer. Funct.Anal. Optim. 37:92-105,2016). We establish the shape preserving properties of these operators e.g. monotonicity and convexity and study the rateof convergence by means of Lipschitz class and Peetreís K-functional and degree of approximation with the aid of a smoothing process e.g Steklov mean.Further, we introduce the bivariate case of modiÖed q-Bernstein-Kantorovichoperators and study the degree of approximation in terms of the partial andtotal modulus of continuity and Peetreís K-functional. Finally, we introducethe associated GBS (Generalized Boolean Sum) operators and investigate theapproximation of the Bˆgel continuous and Bˆgel di§erentiable functions byusing the mixed modulus of smoothness and Lipschitz class.