On the rate of convergence of two generalized Bernstein type operators
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On the rate of convergence of two generalized Bernstein type operators LIAN Bo-yong1
CAI Qing-bo2,∗
Abstract. In this paper, we introduce the B´ezier variant of two new families of generalized Bernstein type operators. We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity. By means of construction of suitable functions and the method of Bojanic and Cheng, we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.
§1
Introduction
In the year 1912, Bernstein [1] introduced a sequence of positive linear operators for f ∈ C[0, 1], as n ∑ k Bn (f, x) = f ( )pn,k (x), x ∈ [0, 1], n k=0
where pn,k (x) = (nk )xk (1 − x)n−k . Then many scholars have done a lot of relevant research. Lorentz [2] gave an exhaustive exposition of main facts about the Bernstein polynomials and discussed some of their applications in analysis. Cheng [3] obtained an estimate for the rate of convergence of Bn for functions of bounded variation in terms of the arithmetic means of the sequence of total variations and proved that the estimate was essentially the best possible at points of continuity. Bojanic [4] investigated the asymptotic behavior of Bn for some absolutely continuous functions having a derivative equivalent to a bounded variation function. King [5] defined a new type of Bernstein operators which preserve x2 . Quantitative estimates were given and compared with estimates of approximation by the class Bernstein polynomials Bn in [5]. Received: 2018-04-05. Revised: 2019-12-12. MR Subject Classification: 41A10, 41A25, 41A36. Keywords: Bernstein operators, modulus of smoothness, rate of convergence, bounded variation. Digital Object Identifier(DOI): https://doi.org/10.1007/s11766-020-3610-8. This work is supported by the National Natural Science Foundation of China(11601266), the Natural Science Foundation of Fujian Province of China (2016J05017), the Program for New Century Excellent Talents in Fujian Province University and the Program for Outstanding Youth Scientific Research Talents in Fujian Province University. We also thank Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing of Fujian Province University. *Corresponding author.
322
Appl. Math. J. Chinese Univ.
Vol. 35, No. 3
Very recently, Chen et al. [6] introduced a new family of generalized Bernstein operators based on a non-negative parameter α(0 ≤ α ≤ 1) as follows: n ∑ k (α) Tn,α (f, x) = f ( )pn,k (x), x ∈ [0, 1], (1) n k=0
where (α)
p1,0 (x) = (α)
pn,k (x) =
(α)
1 − x, p1,1 (x) = x, [ n−2 ] k−1 n ( k )(1 − α)x + (n−2 (1 − x)n−k−1 k−2 )(1 − α)(1 − x) + (k )αx(1 − x) x
for n ≥ 2 and (nk ) = 0(k > n). When α = 1, the operators Tn,α reduces to the Bernstein operators Bn . In [6], the authors studied many approximaiton properties of Tn,α such as uniform convergence, rate of convergence in terms of m
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