Applied Mathematics, 2011, 2, 389-397
doi:10.4236/am.2011.24047 Published Online April 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Pointwise Approximation Theorems for Combinations of
Bernstein Polynomials with Inner Singularities
Wenming Lu1, Lin Zhang2
1Department of Mathematics, Hangzhou Dianzi University, Hangzhou, China
2Department of Mathematics, Zhejiang University, Hangzhou, China
E-mail: lu_wenming@163.com, linyz@zju.edu.cn, godyalin@163.com
Received October 8, 2010; revised January 14, 2011; accepted January 17, 2011
Abstract
It is well-known that Bernstein polynomials are very important in studying the characters of smoothness in
theory of approximation. A new type of combinations of Bernstein operators are given in [1]. In this paper,
we give the Bernstein-Markov inequalities with step-weight functions
wx for combinations of Bernstein
polynomials with inner singularities as well as direct and inverse theorems.
Keywords: Bernstein Polynomials, Inner Singularities, Pointwise Approximation, Bernstein-Markov
Inequalities, Direct and Inverse Theorems
1. Introduction
The set of all continuous functions, defined on the inter-
val
I
, is denoted by

CI . For any

0,1fC, the
corresponding Bernstein operators are defined as fol-
lows:


,
0
,:
n
nk
k
Bpx
k
fxf n



,
where
 
,:C1 nk
kk
nk n
pxx x
, 0,1,,kn,
0,1x.
Approximation properties of Bernstein operators have
been studied very well (see [2-7], for example). In order
to approximate the functions with singularities, Della
Vecchia et al. [8] introduced some kinds of modified
Bernstein operators. Throughout the paper, C denotes
a positive constant independent of n and
x
, which
may be different in different cases. Ditzian and Totik
extended the method of combinations and defined the
following combinations of Bernstein operators:
 
1
,
0
,: ,
i
r
nri n
i
Bfx CnBfx
,
with the conditions:
a) 01 1r
nn nnCn
 ,
b) 1
0()
r
i
iCn C
,
c) 1
0() 1
r
i
iCn
,
d)

1
00
rk
ii
iCnn
, for 1, ,1kr.
For any positive integer r, we consider the determi-
nant
 

11 11
212 22341
221 2121 2223441
:
22132242322 41
r
rr rr
rrr rrrrr
A
rrr rr
 
 
 
 
 
We obtain 2
2!0
r
rj
Aj
 . Thus, there is a unique solution for the system of nonhomogeneous linear equations:
W. M. LU ET AL.
Copyright © 2011 SciRes. AM
390

 
 
12 21
12 21
12 21
12 21
1,
21 22410,
212 22214410,
21!3 2122410.
r
r
r
r
aa a
rar ara
rrarrarra
ra rarra
 
 

 

(1.1)
Let

21 2241
12 21
1, 01,
:0, 0,
1, 1.
rr r
r
axaxa xx
xx
x
 
 

with the coefficients 12 21
,,,r
aa a
satisfying (1.1).
From (1.1), we see that (2 )
(,)
() r
xC
 
,
01x

for 01
x
. Moreover, it holds that

11
,


00
i
, 0,1,, 2i
r
and


10
i
, 1,,2ir.
Let



1
1
,:
r
ii
i
H
fxf x lx
,
and



1
1,
1
1,
:
r
j
ji j
ir
j
ji ij
x
x
lx
x
x




,


12
i
nr i
xn

 

, 1,2,,1ir.
Further, let
1
2
,
nn
xn
2,
nn
xn


3,
nn
xn
4
2nn
xn


and

1
1
21
x
x
x
x
x




,

3
2
43
x
x
x
x
x





.
Set
 

 


12
12
,: 1
1.
n
F
fx Fx fxxx
xxHx


 

We have


 

 

 

 
52 32
11 5232
32 12
22 1232
,0,,1,
1,,,
,
,,,
1,,.
rr
rr
n
rr
rr
fxx xx
fxxxHxx xx
Ffx
Hxx xx
Hxxxfxxxx






 

 

 




 

Obviously,

,
n
F
fx is linear, reproduces polynomi-
als of degree r, and




2
,0,1
r
n
Ffx C, provided
that



20, 1
r
fC. Now, we can define our new com-
binations of Bernstein operators as follows:




1
,,
0
,, ,
i
r
nrnr ninn
i
BfxBFx CnBFx

(1.2)
where

i
Cn satisfy the conditions (a)-(d).
2. The Main Results
Let
:0,1R
, 0
be an admissible step-weight
function of the Ditzian-Totik modulus of smoothness,
that is,
satisfies the following conditions:
1) For every proper subinterval
,0,1ab there
exists a constant
1,0CCab
such that
1
1
1
CxC
for
,
x
ab.
2) There are two numbers

00
and
10
for which




0
1
,as 0,
1,as 1.
xx
x
xx

(X~Y
means C-1Y X CY for some C).
Combining conditions (I) and (II) on
, we can de-
duce that

1
22
Cx xCx
 
 ,
0, 1x,
where




1
0
21xxx
.
W. M. LU ET AL.
Copyright © 2011 SciRes. AM
391
Let

wx x
 ,01

,0
and





0,1\: lim0
wx
CfC wfx
 
.
The norm in w
C is defined as

01
:sup
w
wC
x
f
wfwfx

 . Define





1
:: 0,1,
rr
rr
w
WfCf ACwf
 
,





1
,:: 0,1,
rr
rr
w
WfCfAC wf

 .
For w
fC, we define the weighted modulus of smoo-
thness by
 


001
,:supsup
rr
hx
whtx
Wftwx fx
 
,
where

0
1+
2
rk
rk
hr
k
r
f
xCfxkhx







,
 

0
1C +
rk
rk
hr
k
f
xfxrkh
 
.
Recently Felten showed the following two theorems in
[4]:
Theorem A. Let
 
=1
x
xx
and let
:0,1 R
, 0
be an admissible step-weight func-
tion of the Ditzian-Totik modulus of smoothness ([3])
such that 2
and 22
are concave. Then, for
0, 1fC and 02
,
 

212
,,
n
x
Bfx fxCfn
x





.
Theorem B. Let
 
=1
x
xx
and let
:0,1 R
, 0
be an admissible step-weight func-
tion of the Ditzian-Totik modulus of smoothness such
that 2
and 22
are concave. Then, for
0,1fC
and 02
,
 

12
,
n
x
Bfx fxCnx









implies


2,
f
tOt
.
Our main results are the following:
Theorem 2.1. For any 0
,
 

1
min0 ,1,
2w
f
C

, we have
 


2
,1 ,
r
r
r
nr
w xx BfxCnwf
. (2.1)
Theorem 2.2. For any 0
, r
f
W
,
we have
 



,1 ,
rr
rr
nr
wxxBfxw fC

. (2.2)
Theorem 2.3. For w
f
C
,
0
,
 

1
min0,12

,
00, r
,
 
1
nxx
n
 , we have

 

0
0
,1
1/2
,
,
nr
nr
w
wxf xBfx
x
OftOt
nx









(2.3)
3. Lemmas
Lemma 3.1. For any non-negative real u and v, we have
 
1
,
1
1
uv
nv
u
nk
k
nn
pxCx x
knk

 
 
 
. (3.1)
Lemma 3.2. If R
, then
 
2
,
0
n
nk
k
pxknx Cnx

. (3.2)
Lemma 3.3. For any r
f
W
, 0
, we have
 
rr
rr
n
wFCw f

. (3.3)
Proof. We first prove 5232
,
rr
xxx



(The same
as the others), we have
 

  


 


12
:
rr
rr
n
r
r
n
wx xFxwx xfx
wxxf xF
I
Ix


Obviously

1
r
r
Cw fI
For 2
I
, we have
 


  


0
2
2
r
r
n
i
rri
r
n
i
wxxf xFx
wxxnfxFx
I

By [3], we have
 



52 32
52 3252 32
,
22
,,
.
rr
rr rr
ri
n
xx
ri i
r
xx xx
fxFx
Cnf Hnf

 


 
 

 



So
W. M. LU ET AL.
Copyright © 2011 SciRes. AM
392
 
 

52 32
52 32
1
,
2
2
,
2
:
rr
rr
r
r
xx
r
r
xx
Cn w xxfH
Cw xxTf
I
T







By Taylor expansion, we have
 






1
0
1
!
1d.
1!
i
u
ru
i
i
u
xrr
i
x
xx
fxf x
u
x
sfss
r

(3.4)
It follows from (3.4) and the identity

1
rvv
ii
i
x
lx Cx
, 0,1, ,vr.
we have




 





 
 



10
1
1
10 1
1
1
,!
1
d
1!
1
d,
1!
i
i
u
rr u
i
i
iu
rxrr
ii
x
i
ru r
uv
uvv
uii
uv i
rxrr
ii
x
i
xx
Hfxf xlx
u
lxxsfs s
r
f
xC fxCxxlx
lxxsfs s
r

 


 





which implies that
 

 



1
1
,
1d,
1!
i
r
rxrr
r
ii
x
i
wxxf xHfx
wxxl xxsfss
r

Since
i
lx C, for 5232
,
rr
xxx



, 1, ,ir.
It follows from
 
11rr
ii
x
sxx
ws wx


,
s between i
x
and
x
, then
 
 




1
1
/2
,
d
.
i
r
rxrr
r
i
x
i
r
r
r
wxxf xHfx
Cw xxxsfss
C
nwf

So

2
r
r
Cw fI
.
Then, the lemma is proved.
According to methods of Lemma 3.3, we can easily
get:
Lemma 3.4. If r
f
W
, 0
, then
 


,
r
r
nr
x
wxgxH gxCwg
nx





(3.5)
Lemma 3.5. For any 0
, w
fC, we have
,1nr
wBfC wf
. (3.6)
Proof. By (1.2), we have
  

 
  
  
,1 ,1
1
1
,
01
1
,0
0
1
,
0
123
,,
.
0
1
:
i
i
i
ii
nrnr n
n
r
in nk
ik i
r
inn
i
r
innn
i
wxBfxwxBF x
k
wxCn Fpx
n
wxC n Fpx
wxC n Fp
II
x
I







Now, the theorem can be proved easily.
Lemma 3.6. Let
 

1
min0 ,12

, then for rN
,
1
08
tr
 and 1
22
rt rt
x
 , we have

22
1
1
22
dd
tt r
rrr
kr
tt
k
x
uu uCtx








 (3.7)
Lemma 3.7. Let

,
:
nnk
knn
A
xwxpx

, then
/2
n
Ax Cn
for 01
and 0
.
Proof. If 3
xn
 , then the statement is trivial.
Hence assume 3
0xn
 ( the case 31x
n

can be treated similarly). Then for a fixed x the maxi-
mum of
,nk
px is attained for :
n
kk nn
 
.
By using Stirling’s formula, we get
 
,
1n
n
nnn
n
nk
k
nk knk
nn
nn
nnx x
e
pxC
knk
knk
ee



  
  
  
11 .
nn
knk
nn
nn
knxknx
C
knk
n


 


W. M. LU ET AL.
Copyright © 2011 SciRes. AM
393
Now from the inequalities


1
1,
2
n
knx nnnxnx
nnx


 


and
2
1
2
1uu
ue

 , 1u
ue, 0u.
We have that the second inequality is valid. To prove the
first one we consider the function

2
1
21
uu
ueu


.
Here

00
,
 
2
1
2
11
uu
uue

 ,
00
,
 
2
1
2
20
uu
uuue

  
, whence

0u
for
0u.
Hence

2
,
1
exp 2
n
nn
nk nn
nn
knxknx
C
px kknx
kk
n
















2
2
exp 2
Cn x
n
n
knx
Ce
k
n







.
Thus
 

2
Cn x
n
AxCxe

 . An easy calcula-
tion shows that here the maximum is attained when
C
xn
 and the lemma follows.
Lemma 3.8. For 01

, ,
0
, we have
 
2
,nk
knn
wxpx knxCnx



(3.8)
Proof. By (3.2) and the lemma 3.7, we have
 

21
2
1
2,
1
2
22
,() .
n
n
nnk
knn
n
n
nk
kn n
wxwxp x
pxknxCn x








 



Lemma 3.9. For any 0
, r
f
W
,we have


,1
rr
nr
wBfCn wf
. (3.9)
Proof. We first prove 1
0,xn
(The same as
1
1,1xn



, now





 
 
 

,1
2
1,
00
2
,
000
2
,0
00
2
00
,
!
!
i
i
i
i
i
i
r
nr
nr
rr
i
innrk
ik
ii
n
nr
rr
rj
irn nrk
ikj i
rr
rj
irnnr
ij i
rr
rj i
irn
ij i
wxB fx
nk
wxC nFpx
nr n
kr j
Cw xnCFpx
n
rj
Cw xnCFpx
n
nj
Cw xnCFn
























,
1
2
,
010
123
:.
ii
i
i
nrnr
nr
rr
rj
irn nrk
ikji
px
kr j
Cw xnCFpx
n
HHH









We have
 

2
1,0
00
2
00
1
.
i
i
rr
r
in nr
ij i
rr nr
i
r
ij i
r
rj
HCwx nFpx
n
nx
Cn wfx
rjn
Cn wf












Similarly, we can get 2
r
H
Cn wf, and
3
r
H
Cn wf.
When 11
,1xnn

, according to [3], we have


 


 




 




23
,1 ,1
2
2
00
,
2
2
00
,12
,,
,
,
:
i
i
i
i
rr
nrnr n
rr
rj
jiii
ij
j
nnk
kn Aii
rr
rj
jiii
ij
j
nk
xknx ii
wxBfxwxBF x
wxxQxn Cnn
kk
xFpx
nn
wxxQxn Cnn
kk
xHpx
nn















where
23
:0, ,1Axx

, H is a linear function. If
i
k
A
n
, when

2
1ii
i
wx Cnknx
wkn

 



, we have
2
i
i
n
kn
 , also



2
,1
rj
ji i
Qxnnx x
 , and





22
2
,
rj
rj
jii i
xQxnnCnx

. By (3.2), then
W. M. LU ET AL.
Copyright © 2011 SciRes. AM
394
  
 
2
2
1 ,
2
000
2
2
2
,12
2
000
1:.
i
i
i
i
rj j
n
rr
i
nnk
ijk ii
rj j
n
rr
i
ii nk
ij ki
nkk
Cw xxFpx
nn
x
nk
Cwfnk nxxpxII
n
x
 
 
















 
 
By a simple calculation, we have 1
r
I
Cn wf. By (3.2), then
 
2
2
2
2 ,
2
00 0
i
i
rj
n
rr jjr
i
iink
ij k
n
I
CwfnknxpxCnwf
x




 





 .
We note that
 


14
max ,:
i
k
H
Hx HxHa
n
 


 .
if
14
,
x
xx

,we have
 
wx wa.So, if
14
,
x
xx
,
then
 
2
rr
Cn waHaCnwf
.
If
14
,
x
xx

, then

2
i
wa n
, by (3.8), we have
 

23
2
2
2
22
00
,.
i
i
rj
rr
i
i
ij
j
r
nk
xknx i
n
Cw aHawxnx
k
xpxCnwf
n







 

It follows from combining the above inequalities that
the lemma is proved.
4. Proof of Theorems
4.1. Proof of Theorem 2.1
When w
fC,


min0,11 2

, we discuss it as
follows:
Case 1. If

1
0xn
, by (3.9), we have
 
 
 


()
,1
2
,1
,
,
rr
nr
rr
r
r
nr
r
wx xBfx
x
CxwxB fxCnwf
x
 (4.1)
Case 2. If

1
xn
, we have







,1 ,1
2
2
00
,,
,.
rr
nrnr n
rr
r
j
j
ii i
ij
BfxBFx
x
QxnCnn




,
0
i
i
j
n
nnk
kii
kk
x
Fpx
nn



,
where



2
,1
rj
ji i
Qxnnx x



 , and





22
2
,
rj
rj
jii i
xQxnnCnx

So
 

  
 
 
  
,1
2
2
,
2
000
2
2
2
00
2
2
,2
00
,
i
i
i
i
r
rr
nr
rj j
n
rr
i
nnk
ijk ii
rj j
rr
ri
n
ijknAii
rj
rr
ri
nk
ij
wx xBfxCwxx
nkk
xFpx
nn
x
nkk
Cw xxxF
nn
x
n
pxCwx xx
k
xn

 
 









 

 
 


 



 
 


23
,12
:.
i
i
j
nk
xknx ii
k
Hpx
n



 


(4.2)
where
23
:0, ,1Axx

, we can easily get
2
1
r
Cn wf
, and 2
2
r
Cn wf
. By bringing these
facts together, the theorem is proved.
4.2. Proof of Theorem 2.2
When r
f
W
, by [3], we have


 
2
,11 ,
00
,
i
i
i
nr
r
rrr
nrniinnrk
ik
i
n
k
BFxCnnFp x
n






. (4.3)
W. M. LU ET AL.
Copyright © 2011 SciRes. AM
395
If 0i
knr, we have

1
10d
i
i
r
r
rr
n
nin
ii
n
kk
F
CnFu u
nn





. (4.4)
If 0k, we have



1
10
0d
i
i
r
r
rr
n
nn
n
F
CuF uu
. (4.5)
Similarly



1
12
111d.
i
i
rr
rr
ir
ni n
n
i
n
nr
F
CnuFu u
n





(4.6)
By (4.3), we have
 


 
 
  
  
,1
2
1,
00
1
2
1,
01
2
1,0
0
2
1,
0
,
0
1.
i
i
i
i
i
i
i
i
ii
i
r
r
nr
nr
r
rrr
innrk
ik i
n
nr
r
rrr
innrk
ik i
n
r
rrr
innr
in
r
rrr
innrnr
in
wx xBfx
k
Cw xxnFpx
n
k
Cw xxnFpx
n
Cw xxnFpx
Cw xxnFpx
















(4.7)
which combining with (4.4)-(4.6) give
 



,1 ,
rr
rr
nr
wxxBfxCwf

.
Combining with the theorem 2.1 and theorem 2.2, we
can obtain
Corollary For any 0
, 01
, we have
 


 



,1
12 1
2
,
,
max ,,,
,.
{
r
r
nr
rr
r
w
r
rr
w
wxxBfx
Cnnx wff C
Cwff W


(4.8)
4.3. Proof of Theorem 2.3
4.3.1. The Direct Theorem
We know
 




1
1
d
1!
nnn
trr
n
x
FtFxFt x
tu F uu
r


(4.9)


,1 ,0
k
nr
Bxx
 , 1, 2,,1kr (4.10)
According to the definition of r
W
, for any r
g
W
,
we have





,1 ,1
,,
rr
nrnr n
BgxBGgx

, and
 


,1 ,1
,,,,
nnrn nrrn
w
xGxBGxwxBR Gtxx

 ,
there of



1
,, d
trr
rn n
x
RGtxtuGuu

, we have
 

  

 

,1
1
,1
1
12
,1 2
1
12
,1 2
,
,
d,
d,.
nnrn
r
t
r
r
nnr r
x
r
t
r
r
nnr
r
x
r
t
nr x
wx GxBGx
tu
Cw GwxBdux
wu u
tu
CwGwx Bux
u
tu
Bux
wu




















(4.11)
also
  
11
2222
d, d
rrrr
tt
rr
xx
tutx tutx
uC uC
uxwuwx





.
(4.12)
By (3.2), (3.3) and (4.12), we have
 







,1
,1
,
,
nnrn
r
r
rr
nnr
r
r
nr
n
wx GxBGx
Cw GxBtxx
x
CwG
nx






. (4.13)
By (3.3), (3.5) and (4.13), when r
g
W
, then
 
  






,1
,1
,,
,, ,
δδ
.
nr n
nnr
rr
rr
nn
rr
n
wx gxBgxwx gxGgx
wxGgx Bgxwxgx Hgx
xx
CwGCwg
nx nx




 

 
 
 
(4.14)
For ,
w
f
C
we choose proper r
g
W
, by (3.6) and
(4.14), then
 
  


,1
,1 ,1
,
,,
δ
,.
nr
nr nr
n
r
w
wxf xBfxwxfxgx
wx Bfgxwx gxBgx
x
Cf
nx







4.3.2. The Inverse Theorem
The weighted K-function is given by
W. M. LU ET AL.
Copyright © 2011 SciRes. AM
396




,,inf :
r
rrrr
rwg
KftwfgtwggW

.
By [3], we have



1
,
,, ,
rrr
r
ww
w
CftKftCft
 


. (4.15)
Proof. Let 0
, by (4.15), we choose proper
g
so
that
,
r
w
wf gCf

,


,
r
rrr
w
wg Cf

 .
(4.16)
For rN
, 1
08
tr
 and 1
22
rt rt
x , we have


  

 



 

0
,1 ,1,1
1
22
2
,1 1
0 1
22
,,,
2,dd
2
rrr r
hhnrh nrh nr
hx hx
n
r r
r
j
rnrk r
hx hx
j k
h
wxfxwxfxBfxwxBf gxwxBgx
r
xjhx
CnwxBfgxu uu
r
xjhx
wx
 



 

 
 






 







 













 
22
,11123
1
22
,dd:
hx hxr
r
nrk r
hx x
k
BgxuuuJJJ






(4.17)
Obviously


0
11/2
nx
JC
nx




. (4.18)
By (3.9) and (4.16), we have



 
22
21
22
dd ,δ
hx hx
rrrrr
r
hx hxw
JCnwfgu uCnh xf






 (4.19)
By the first inequality of (4.8) and (4.16), we let 1
, then





 
22
2 2
2 1
1
22
dd ,δ.
hx hx
r r
r
rrrrr
kr
hx hxw
k
JCnwfgxuu uCnhxxf










 (4.20)
By (3.7) and (4.16), we have







1
22
3 1
11
22
dd δ,δ
hx hxrr
r
rrrrr
kkr
hx hxw
kk
JCwgwxwxuxuuuChf



 
 






 (4.21)
Now, by (4.17)-(4.21), there exists a constant 0M so that
 


 



 
0
0
2
1/2
1
2
1/2
min ,,,
,,.
r
r
n
rrrrrrrr
hrww
r
nn
rrr rrr
ww
xx
wxf xCnnxhfhf
nxx
xx
ChMnfhf
x
nx





 

 




 
 

 
 




When 2n, we have
 
11
1
22
21
12
nnn
nxnx nx

 ,
Choosing proper x, δ, nN, so that

  

111
22
1
nn
x
x
nn
x
x


 ,
Therefore
 
0,
rrrr
hw
wxf xChf

 
which implies
 
0
,,
rrrr
ww
ft Chf


.
W. M. LU ET AL.
Copyright © 2011 SciRes. AM
397
So, by Berens-Lorentz lemma in [3], we get

0
,
r
w
f
tCt
.
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