Abstract
In this paper, a kind of Schurer type q-Bernstein-Kantorovich operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of these operators are also studied by means of the modulus of continuity and the help of functions of the Lipschitz class. Then, the global approximation property is given for these operators.
Highlights
In, Phillips [ ] introduced and studied q analogue of Bernstein polynomials
The book Convergence Estimates in Approximation Theory written by Gupta and Agarwal introduced some approximation properties of certain complex q-operators in compact disks
The goal of this paper is to introduce a kind of Schurer type q-Bernstein-Kantorovich operators and to study the approximation properties of these operators with the help of the Korovkin type approximation theorem
Summary
In , Phillips [ ] introduced and studied q analogue of Bernstein polynomials. During the last decade, the applications of q-calculus in the approximation theory have become one of the main areas of research, q-calculus has been extensively used for constructing various generalizations of many classical approximation processes. The goal of this paper is to introduce a kind of Schurer type q-Bernstein-Kantorovich operators and to study the approximation properties of these operators with the help of the Korovkin type approximation theorem. In , Muraru [ ] introduced and studied the following q-Bernstein-Schurer operators for any fixed p ∈ N ∪ { }: Bn,p(f ; q; x) = Pn+p,k(q; x)f [k]q/[n]q , k=. In , Özarslan and Vedi [ ] introduced the q-Bernstein-Schurer-Kantorovich operators Knp. Comparing the results of our present paper with [ ], we find that the literature [ ] only estimated the rate of convergence in the pointwise sense for these operators Knp. In the present paper, we estimate the rate of convergence in the pointwise sense, and give the global approximation for these operators Sn,p defined by ( ), and about the estimate of the rate of convergence in the pointwise sense for these operators Sn,p, we get some new results, which are different from those in [ ]. – f (x) + f x – hφ(x) be the second order Ditzian-Totik modulus of smoothness, and let
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