Abstract
In the present paper we define a q-analogue of the modified Bernstein-Kantorovich operators introduced by Ozarslan 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 rate of convergence by means of Lipschitz class and Peetre's K-functional and degree of approximation with the aid of a smoothing process e.g Steklov mean. Further, we introduce the bivariate case of modified q-Bernstein-Kantorovich operators and study the degree of approximation in terms of the partial and total modulus of continuity and Peetre's K-functional. Finally, we introduce the associated GBS (Generalized Boolean Sum) operators and investigate the approximation of the Bogel continuous and Bogel differentiable functions by using the mixed modulus of smoothness and Lipschitz class.
Highlights
Let : I ! R be an integrable function, I being [0,1] and pn;k(x) denote the usual Bernstein function given by pn;k(x) =n xk(1 k x)n k; 0 x1; k = 0; 1; 2; :::n: the classical Bernstein-Kantorovich operator is de...ned by Xn Z k+1 n+1Kn( ; x) = (n + 1) pn;k(x) k k=0 n+1(s)ds: The above operator may be expressed as follows: Z1Kn( ; x) = pn;k(x) k=0 k + s ds: n+1 (1.1)Received by the editors: March 27, 2019; Accepted: May 31, 2019. 2010 Mathematics Subject Classi...cation. 41A25, 41A36, 41A63, 41A10
In the present paper we de...ne a q-analogue of the modi...ed BernsteinKantorovich operators introduced by Özarslan and Duman(Numer
We introduce the bivariate case of modi...ed q-Bernstein-Kantorovich operators and study the degree of approximation in terms of the partial and total modulus of continuity and Peetre’s K-functional
Summary
1; k = 0; 1; 2; :::n: the classical Bernstein-Kantorovich operator is de...ned by. Kantorovich operators for functions of one and two variables and established some approximation properties. Several researchers have studied the q-analogues of positive linear operators and established many interesting approximation properties. Researchers introduced similar q-analogues of several positive linear operators and established many interesting approximation properties. The aim of the present paper is to establish the shape preserving properties and obtain the degree of approximation for the operators (1.4) by means of the Lipschitz class and the Peetre’s K-functional. We study the GBS operators of the modi...ed q-Bernstein-Kantorovich type and study the approximation of Bögel continuous and Bögel di¤erentiable functions
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