Journal of Data Analysis and Information Processing, 2013, 1, 85-89
Published Online November 2013 (http://www.scirp.org/journal/jdaip)
http://dx.doi.org/10.4236/jdaip.2013.14009
Open Access JDAIP
The Average Errors for Linear Combinations of Bernstein
Operators on the Wiener Space *
Yanjie Jiang#, Ziqing Zhang
Department of Mathematics and Physics, North China Electric Power University, Baoding, China
Email: #jiangyj@126.com
Received September 18, 2013; revised October 20, 2013; accepted November 4, 2013
Copyright © 2013 Yanjie Jiang, Ziqing Zhang. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, we discuss the average errors of function approximation by linear combinations of Bernstein operators.
The strongly asymptotic orders for the average errors of the combinations of Bernstein operators sequence are deter-
mined on the Wiener space.
Keywords: Linear Combinations; Bernstein Operators; Weighted
p
L-Norm; Average Error; Wiener Space
1. Introduction
Let
F
be a real separable Banach space equipped with
a probability measure
on the Borel sets of
F
. Let
X
be another normed space such that
F
is continu-
ously embedded in
X
. By X
we denote the norm in
X
. Any such that :TFX

X
f
fTf is a
measurable mapping is called an approximation operator.
The p-ave rage error of is defined as
T

 
1
,,, :d.
p
p
pXX
F
eTFfTf f


Let



0:0,10FfC f 0.
For every 0
f
F set

01
:max .
Ct
f
ft

Then
0,C
F
becomes a separable Banach space.
Denote by

0
F
B the Borel class of
0,
FC and by
0
the Wiener measure on

0
F
B (see [1]). From [1,
p. 70] we know
 



00d
1
min,,,0,1 .
2
Ffsft f
sts ts tst

(1)
The Bernstein operato r on
0, 1C defined by
 
,
0
,:n
nn
k
k
Bfxfp x
n



k
,.
,
where
 
,1,0,1,
nk
k
nk
n
pxxx kn
k




This operator turned out to be a very interesting ope-
rator, easy to deal with and having many applications in
approximation theory and practice.
Since Bernstein operators cannot be used in the inves-
tigation of higher orders of smoothness, Butzer [2] in-
troduced combinations of Bernstein operators. Ditzian
and Totik [3, p. 116] extended this method and defined
the combinations as
 
1
,0
,: ,
i
m
nmi n
i
Lfx CnBfx
, (2)
where and
i
n
i
Cn
satisfy the following conditions:



01
1
0
1
0
1
0
() ;
() ;
() 1;
( )0,1,2,,1.
m
m
i
i
m
i
i
m
ii
i
annn Cn
bCnC
cCn
dCnn m
 
 
(3)
*Supported by National Natural Science Foundation of China (Project no
10871132 and 11271263) and by a grant from Hebei province higher
school science and technology research (Z2010160).
#Corres
p
ondin
g
autho
r
. Throughout this paper, denotes a positive constant C
Y. J. JIANG, Z. Q. ZHANG
86
independent of and
n
x
, which may be a different
constant in different cases.
For
10,1 0,1Lp,


,
the weighted
p
L-norm of
0,1fC is de fined by
 
1
0
:d.
p
1
,
p
p
f
ftt t

2. Main Result
Recently G. Q. Xu [4] studied the average errors of Bern-
stein operators approximation on the Wiener space. Mo-
tivated by [4], we considered the average errors of func-
tion approximation by linear combinations of Bernstein
operators. The strongly asymptotic orders for the average
errors of the linear combinations of Bernstein operators
sequence are determined on the Wiener space. We ob-
tain:
Theorem 1. Let , be given by
(2), 1p
,,
nm
Lfx

0x

1 and 0, 1L
,

x
is continuous on
. Then we have
(0,1)

 




1
4
,0 0
,
1
2
11
00
1
14
0
,, ,
2
2
1d ,
pnm p
j
i
mm ji
ij
ij iij
pp
p
eL F
n
n
nn
CnCn nnn
xx xxn


















 



where
2
2
1ed
2
x
p
p.
x
x


Here and in the following the notation
nn
ab for
sequences and
means that

n
a
n
b
lim 0
nnn
ab
 .
3. Proof of Theorem 1
To prove Theorem 1 we need the following two lemmas.
Lemma 1([5, p. 15]). If
1
02
,
then

,
k
n
k
nk
xn
pxCn

for each , the constant depending only on
0kC
and .
k
Lemma 2 ([5, p. 15]). Fo r fix ed
1
01,
3
x

,
the asymptotic relation
 




1
2
,
2
,
1
21 exp21
nk
k
nk
nk
n
pxx x
k
nk
xxn x
xxn
Px







 





holds uniformly for all values of satisfying the ine-
quality k
.
k
x
n
n
 (4)
In other words,

,,
lim 1
nk nk
npxPx

uniformly for all satisfying (4). k
Proof of Theorem 1. From [1, p.107] we have
  
2
0
,0 0
,
12
,0
0.
,, ,
,d d
p
pn
mp
pnm
F
LF
fxLfxfx x






p
e
v
(5)
By (2),
 
 
  
 
 
0
0
0
0
2
,0
1
20,
00
11
000
,, 0
00
123
,d
d2
d
d
2.
i
i
j
i
ij
nm
F
n
m
ink
Fik
mm
ij
Fij
i
n
n
nkn sF
ks ij
fx Lfxf
fxfCn p x
k
fxffCnCn
n
ks
pxpxf ff
nn
AxAx Ax
















 



(6)
On using (1), we obtain

0
2
10
d
F.
A
xfxf
x
(7)
Note that
 
 
,,
00
222
,
0
1, ,
1,
nn
nk nk
kk
n
nk
k
px kpxnx
kpxnx nxx




(8)
by (1), we have
Open Access JDAIP
Y. J. JIANG, Z. Q. ZHANG 87
 
 
 
0
1
2,
00
1
,
00
1
,
00
() d
1
2
1.
2
i
i
i
i
i
i
n
m
ink
F
ik i
n
m
ink
ik ii
n
m
ink
ik i
k0
A
x Cnpxfxff
n
kk
Cnpxx x
nn
k
x Cnpxx
n








 



(9)
From [4,(3.24)], we know
 
1
2
,
0
21 .
i
i
n
nk i
kii
xx
k
pxxn
nn
 
Combining(3) and (9) we get
 
 
1
2
1
2
1
20
1
0
21
1
2
1.
2
m
ii
ii
mi
ii
xx
Ax xCnn
n
xx Cn
xn
n


 


 
(10)
Now, we estimate the term

3
A
x. From (2) and (8),


 
 

0
11
3,
000 0
0
11
,,
000 0
11
,
000
,
0
d
1
2
1
2
j
i
ij
j
i
ij
i
i
j
j
n
n
mm
ijnkns
ijk s
Fij
n
n
mm
ijnkns
ijk s
ijij
n
mm
ijnk
ijk
n
ns
s
,
A
xCnCnpxp
ks
ff f
nn
Cn Cnpxpx
ksks
nn nn
xCnCnpx
k
px

 

 















 
 

x
.
ij
s
nn
(11)
Using Lemma 1 and (3), we have
512
2
,.
j
j
j
ns
sxn
n
pxCn

Note that
0,
ij
ks
nn
1
,
we get

512
2
,.
j
j
j
ns ij
sxn
n
ks
px Cn
nn


(12)
By (8) and (12), we obtain
 
 
 
 
 
512
512512
512512
23
12
512
,,
00
,,
0
,,
,,
,,
j
i
ij
i
ij
sj
nj
ij
ks
i
nj
n
ij
ij
ks
i
nj
n
ij
ij
ks
i
nn
ij
n
n
nkn s
ks ij
n
nkn s
kij
xn
nkn sij
xn xn
nkn sij
xn xn
nkn s
xn
ks
pxpx
nn
ks
px px
nn
ks
px px
nn
ks
px px
nn
npxpx


 
 









512
.
jij
xn
ks
nn
(13)
For
512
i
i
kxn
n
 and 512
j
j
sxn
n
 ,
by Lemma 2,
 


 
,,
2
2
11
21
exp .
21 21
ij
nkn s
ij
ij
ij
j
i
ij
ks
pxpx
nn
ks
nn
xxnn
n
nks
xx
xx nxx n






 




(14)
Set
  
22
121212
,: exp
21 21
j
in
n.
F
uuuuuu
xx xx
 


For

12 11 11
,, ,
ii jj
kk ss
vv nn nn









by the differential mean value theorem we have

 


 

 
2
2
12
22
12
112 2
exp 21
21
exp 21 21
,,
i
ij
j
j
j
i
ij
n
ks
xx x
nn xxn
ns
xxvxv
xx n
n
nxv xv
xx xx
ks
Fv Fv
nn
 


 

 

 


 







,
i
k






 

 
 
(15)
Open Access JDAIP
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Open Access JDAIP
88
By a simple computation we know where

12
,, ,.
F
CF C
 


11 1
,, ,
ii j
kk ss
xx xx
nn nn



 




1
j
. From (15),
 
 
 

2
2
11
22
12
121 2
exp 2121
exp .
21 21
j
i
iji j
j
i
n
n
ksk s
xx
nnx xnx xn
n
n
xv xvxvxvn
xx xx






 







 







(16)
Integrating two side of (16) about in
12
,vv
11
,,
ii jj
kk ss
nn nn



we get
 
 
 

2
2
111
122
12
1121 22
exp 21 21
dexp d
21 21
j
i
ij
j
i
iji j
s
knj
ni
ijks
nn
n
n
ksk s
xx
nnx xnx xn
n
n
nnvx vx vx vx vvn
xx xx






 















 .
(17)
From (14)-(17), we have
 
 



 
 

512512
512512
512512
,,
,,
1
1
1
22
1212
11 d
21
exp 21 21
ij
ij
ij
ij
ij
ij
j
i
ij
ij
ij
nkn sij
ks
xn xn
nn
nkn s
ij
ks
xn xn
nn
s
k
ij n
n
ks
nn
ks
xn xn
nn
j
i
ks
px px
nn
ks
pxpx
nn
nn v
xx
n
n
xv xvxvxv
xx xx
 
 
 



 




 
  
 


 
23
12
512512
512512 1
512512
2
1
122
1121 2
1
1
112
d
dexp
2 12121
exp
21
j
i
ij
ij
ij
ijnj
i
ij
s
k
ij nj
ni
ks
nn
ks
xn xn
nn
xn xn
ij ni
xn xn
vn
nn n
n
vxvxvxv xvv
xxxx xx
n
nn n
dvxvxv
xx

 
 





 




 


2
d
 


22
1
122
d.
21 21
j
n
xvxvv n
xx xx


 




(18)
Let

 

112
,
21 21
j
in
n
wxvw
xx xx


2
xv
,
by (18), we get
Y. J. JIANG, Z. Q. ZHANG 89
 






512512
112
112
11212112 12
,,
21 2122 1
12
112
21 21
21 dexpd
ij
ij
ij
j
i
ii jj
nkn sij
ks
xn xn
nn
n
n
xx xx
nnnn ij
xx xx
ks
px px
nn
xx ww
www
nn

 

 



2
.
wn

(19)
By (3), suppose that
22
,
ij
ij
nn
cc
nn
,
from (19) and the convergence of the improper integral
22
112 12
dexp
ij
wcwcw www
 
 
 2
d,
we have
  




1
2
512512
1
22
12
,,1 122
22
12 21
11221
2
1
21 dexpd
21 21
dexpd2d
exp
ij
ij
ij
j
i
nkn sij
kij
s
xn xn
nn
nw
ij ji
n
xx ww
ks
pxpxwwww n
nn nn
xx xx
ww ww
wwwww
nn nn
ww
 
 
 
  
 
 
 

 

 

 

 
 

 



1
2
1
2
11
22
2
22
22
12
12
d
11
21 21
ee
dd
21 21
11 .
ji
ij
ji
jj
i i
ijijj i
ji ij
wn
nn
ww
nn
xx xx
wwn
nn
xx xx
nn
nn
nn
nnnnn n
nn nn

 

 
 
 

 











 



(20)
Combining (11) and (20), we obtain
  
  
1
2
11
3,,
00 00
11
00
1
2
21 .
2
j
i
ij
n
n
mm
ij nkns
ij ksij
mm ij ji
ij
ij ij
ks
Ax xCnCnpxpxnn
nnn n
xx
xCnCn
nn

 


 
 
 
 n
(21)
From (5)-(7), (10), and (21), we complete the proof of
Theorem 1.
REFERENCES
[1] K. Ritter, “Average-Case Analysis of Numerical Prob-
lems,” Springer-Verlag, Berlin, 2000.
[2] P. L. Butzer, “Linear Combinations of Bernstein Polyno-
mials,” Canadian Journal of Mathematics, Vol. 5, 1953,
pp. 559-567. http://dx.doi.org/10.4153/CJM-1953-063-7
[3] Z. Ditzian and V. Totik, “Moduli of Smoothness,” Sprin-
ger-Verlag, Berlin, 1987.
http://dx.doi.org/10.1007/978-1-4612-4778-4
[4] G. Q. Xu, “The Simultaneous Approximation Average
Errors for Bernstein Operators on the R-Fold Integrated
Wiener Space,” Numerical Mathematics Theory Methods
and Applications, Vol. 5, No. 3, 2012, pp. 403-422.
[5] G. G. Lornetz, “Bernstein Polynomials,” University of
Toronto, Toronto, 1953.
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