Approximation properties of a new generalized Bernstein-Kantorovich operators

By means of construction of suitable functions and the method of BojanicCheng, the author gives the rate of convergence of a new generalized BernsteinKantorovich operators for some absolutely continuous functions.


Introduction
For a function f (x) defined on the closed interval [0, 1], the expression is called the Bernstein polynomial of order n, where p n,k (x) = ( n k )x k (1 − x) n−k .The polynomials B n ( f, x) were introduced by S.Bernstein [1] in order to give an especially simple proof of the Weierstrass approximation theorem.Then many scholars have done a lot of relevant research work.Bernstein operators became popular for several reasons: (1) they are given explicitly and depend only on the values of a function for rational values of the variable.(2) they have various shape-preserving properties and provide the simplest means for the study of some problems.(3) they are easy to handle in computer algebra systems when the evaluation of f is difficult and time-consuming.
Lorentz [2] gave an exhaustive exposition of main facts about the Bernstein polynomials.He also discussed some of their applications in analysis.
Based on the arithmetic mean of the total variation sequence, the estimation of the conver-gence rate of the B n for the bounded variation function was obtained by Cheng [3].It is proved that the estimation is essentially the best possibility of the continuous point.
Bojanic [4] investigated the asymptotic behavior of B n for some absolutely continuous functions whose derivatives are bounded variation functions.
King [5] defined a new type of Bernstein operators which preserve x 2 .Quantitative estimates were compared with estimates of approximation by the class Bernstein polynomials B n in [5].
In the field of approximation theory, the applications of q-calculus are new area in the last 30 years.The first q-analogue of the well-known Bernstein polynomials was introduced by Lupas in the year 1987.In 1997 Phillips considered another q-analogue of the classical Bernstein polynomials.Next, the q-operators have become the research object of many scholars [6].
Recently, Chen et al. [7] introduced a new family of generalized Bernstein operators based on a non-negative parameter α(0 ≤ α ≤ 1) as follows: In [7], the authors studied many approximaiton properties of T n,α such as uniform convergence, rate of convergence in terms of modulus of continuity, voronovskaya-type asymptotic formula, and shape preserving properties.
To approximate Lebesgue integrable functions, Mohiuddine et al. [8] introduced the following integral modification of the operators T n,α : In [8], the uniform convergence of the operators and rate of convergence in local and global sense are studied.
The rate of approximation for some absolutely continuous functions whose derivatives are bounded variation functions is an intersting topic.This is mainly originated from Bojaniccheng [4] , then many scholars have done a lot of research in this field [9][10][11][12][13][14][15][16].
Base on this, this article studies the approximation of K n,α for some absolutely continuous functions f ∈ DBV[0, 1], where where χ k (t) is the characteristic function of the interval [ k n+1 , k+1 n+1 ] with respect to I = [0, 1].By the Lebesgue-Stieltjes integral representations, we have (3)

Some lemmas
The proof of our result are based on the following lemmas.Lemma 2.1 ( [8]) For e i = t i , i = 0, 1, 2, we have Remark 2.1 By simple applications of Lemma 2.1, we get Lemma 2.2 When n sufficient large, we have Proof By Cauchy-Schwarz inequality, we have The last inequality is obtained by Lemma 2.1 and Remark 2.1.
(ii) Using a similar method, we have

Main results
Theorem Let f ∈ DBV[0, 1].If h(x+) and h(x−) exist at a fixed point x ∈ (0, 1), then we have where Proof Let f satisfy the conditions of Theorem, by using Bojanic-Cheng's method [4] , we have where and From ( 7), ( 8), and noting To complete the proof, we must estimate the term K n,α ( t x ϕ x (u)du, x).