b: Department of Computer Science, University of Kentucky, Lexington, Kentucky, ... for both cases and the estimates are the best possible for continuous points.
Journal of Approximation Theory 109, 242256 (2001) doi:10.1006jath.2000.3538, available online at http:www.idealibrary.com on
On the Rates of Approximation of Bernstein Type Operators Xiao-Ming Zeng 1 Department of Mathematics, Xiamen University, Xiamen 361005, People's Republic of China E-mail: xmzengjingxian.xmu.edu.cn
and Fuhua (Frank) Cheng Department of Computer Science, University of Kentucky, Lexington, Kentucky 40506-0046 E-mail: chengcs.engr.uky.edu Communicated by Ranko Bojanic Received October 17, 1999; accepted in revised form September 12, 2000; published online February 5, 2001
Asymptotic behavior of two Bernstein-type operators is studied in this paper. In the first case, the rate of convergence of a Bernstein operator for a bounded function f is studied at points x where f (x+) and f (x&) exist. In the second case, the rate of convergence of a Szasz operator for a function f whose derivative is of bounded variation is studied at points x where f (x+) and f (x&) exist. Estimates of the rate of convergence are obtained for both cases and the estimates are the best possible for continuous points. 2001 Academic Press
1. INTRODUCTION For a function f defined on [0, 1] the Bernstein operator B n is defined by n
B n( f, x)= : f k=0
k p nk(x), n
\+
p nk(x)=
n k x (1&x) n&k. k
\+
(1)
For a function f defined on [0, ) the Szasz operator S n is defined by
Sn( f, x)= : f k=0
1
k q (x), n nk
\+
q nk(x)=e &nx
(nx) k . k!
Supported by NSFC 19871068 and Fujian Provincial Science Foundation of China.
242 0021-904501 35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.
(2)
243
BERNSTEIN TYPE OPERATORS
In 1983 Cheng [1] proved that B n(sgn(t&x), x)=O(n &16(x(1&x)) &52 ),
x # (0, 1),
(3)
where
{
t>0
1,
t=0 t
