Pointwise Approximation Theorems for Combinations of Bernstein Polynomials with Inner Singularities

(1)

doi:10.4236/am.2011.24047 Published Online April 2011 (http://www.SciRP.org/journal/am)

Pointwise Approximation Theorems for Combinations of

Bernstein Polynomials with Inner Singularities

Wenming Lu1, Lin Zhang2

1_{Department of Mathematics}_,_{Hangzhou Dianzi University}_,_Hangzhou_,_China 2_{Department of Mathematics}_,_{Zhejiang University}_,_Hangzhou_,_China E-mail: lu_wenming@163.com, [email protected], 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 w x

 

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 C I

 

. For any f C



 

0,1



, the corresponding Bernstein operators are defined as fol-lows:





,

 

0

, : n _{n k}

k

B f x f k p x

n

      



_

,

where

 





, : C 1

n k k k

n k n

p x  x x  , k0,1, , n, x

 

0,1 .

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

, : r _i ,

n r i n

i

B f x  C n B f x





,

with the conditions:

a) n n 0n1nr1Cn,

b) ri 1₀C ni( ) C 

 



,

c) ri ₀1C ni( ) 1 

 



,

d) r1₀

 

k 0

i i

i C n n

 

 



, for k1, , r1.

For any positive integer r, we consider the determi-nant

 

 



 





 

















 



1 1 1 1

2 1 2 2 2 3 4 1

2 2 1 2 1 2 1 2 2 2 3 4 4 1 :

2 2 1 3 2 2 4 2 3 2 2 4 1

r

r r r r

r r r r r r r r

A

r r r r r

   

     



    

  

    

    

We obtain 2

2 ! 0

r

r j

(2)















 







 













 



1 2 2 1

1,

2 1 2 2 4 1 0,

2 1 2 2 2 2 1 4 4 1 0,

2 1 ! 3 2 1 2 2 4 1 0.

r

a a a

r a r a r a

r r a r r a r r a

r a r a r r a



   



 _ _ _ _{ } _ _

 _ _ _ _ _{ } _ _

  

       



 

  

(1.1)

Let

 

2 1 2 2 4 1

1 2 2 1 1, 0 1,

: 0, 0,

1, 1.

r r r

r

a x a x a x x

x x

x



  



      



_ 

 _





with the coefficients a a1, , ,2 a2r1 satisfying (1.1).

From (1.1), we see that (2 ) ( , )

( )_x _C r

  _{ } , 0

 

x 1 for 0 x 1. Moreover, it holds that 

 

1 1,

 i

_{ }

₀ ₀

  , i0,1, , 2 r and _ i

 

1 _0_, _i__{1, , 2}_ _r_. Let





1

   

1

, : r i i i

H f x  f x l x





,

and

 

11,

_





_

1 1,

:

r

j j i j

i _r

j j i i j

x x l x

x x

     

 



  ,







1 2



i

n r i

x

n



    

 

 , 1, 2, ,i  r1.

Further, let

1

2 ,

n n

x

n



  

 

  x₂ n n ,

n



  

 

 

3 ,

n n

x

n



  

 

  4

2

n n

x

n



  

 

 

and

 

1

2 1

x x x

x x

   __ _

 ,

 

3 2

4 3

x x x

x x

   __ _  . Set





 



 



 



 



 

1 2

, : 1

1 .

n

F f x F x f x x x

x x H x

 

   

 

We have





 



 



   

 



 



   

5 2 3 2

1 1 5 2 3 2

3 2 1 2

2 2 1 2 3 2

, 0, ,1 ,

1 , , ,

,

, , ,

1 , , .

r r

n

r r

f x x x x

f x x x H x x x x

F f x

H x x x x

H x x x f x x x x

 

 

 

 

 _  _{ } _



    

  

 

 



 _ _



 

  

 _ _



Obviously, F f x_n



,



is linear, reproduces polynomials of degree r, and



_,



 2r



 

_0,1



n

F f x C , provided

that _f__C 2r



 

_0,1



_{. Now, we can define our new com-} binations of Bernstein operators as follows:









1

 





, ,

0

, , r _i ,

n r n r n i n n

i

B f x B F x  C n B F x



 



(1.2)

where C ni

 

satisfy the conditions (a)-(d).

2. The Main Results

Let : 0,1

 

R, 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

   

a b,  0,1 there exists a constant C C a b1

 

, 0 such that

 

1

1 1

C  x C

for x

 

a b, .

2) There are two numbers 

 

0 0 and 

 

1 0 for which

 





 

0

1

, as 0 , 1 , as 1 .

x x

x

x x



 



 _

 

 



～

(X~Y  means C-1_Y_≤_X_≤_CY_{for some}_C_).

Combining conditions (I) and (II) on , we can de-duce that

 

1

2 2

C x  x C x , x

 

0,1 ,

where 2

 

x x  0



1 x



 1

 

(3)

W. M. LU ET AL. 391

Let w x

 

 x ,0  1, 0 and

 

 





  



0,1 \ : lim 0



w

x

C f C wf x

 



   .

The norm in Cw is defined as

  

0 1

: sup

w

w C

x

f wf wf x

 

  . Define

 

_

_{ }

_

 



1



: : r 0,1 , r

r r

w

W_  f C f  AC w f   ,

 

_

_{ }

_

 



1



, : : 0,1 ,

r r

w

W_{ } _ f _C f  _AC w f _{ } _.

For f Cw, we define the weighted modulus of smoo-

thness by

 

0 0 1

, : sup sup

r r

h x w

h t x

W f t w x  f x

   

  ,

where

 

0

1 +

2

r _k

r k

h r

k

r

f x C f x k h x

 



   

   _ _  _ _

 



,

 





0

1 C +

r k

h r

k

f x f x r k h



 



  .

Recently Felten showed the following two theorems in [4]:

Theorem A. Let 

 

x = x



1x



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 _{ }2 2 _{are concave. Then, for}

 

0,1

fC and 0  2

,



_,



 

2 _, 1 2

_{ }

 

n

x

B f x f x C f n

x 

 





 

  _ _

 .

Theorem B. Let 

 

x = x



1x



and let

 

: 0,1 R

  , 0 be an admissible step-weight func-tion of the Ditzian-Totik modulus of smoothness such that _2_and_{ }2 2_{are concave. Then, for} _f__C

 

_0,1

and 0  2,



_,



 

1 2

_{ }

 

n

x

B f x f x C n

x    

_ _ 

 

  _ _

_ _ 

 

implies 2

 

_{f t}_, _{O t}

 

 

  .

Our main results are the following:

Theorem 2.1. For any  0,

   





1

min 0 , 1 , 2 f Cw

    , we have

   

 

_

_

₂

, 1 ,

r r

r n r

w x  x B  f x Cn wf . (2.1)

Theorem 2.2. For any  0, f Wr, we have

   

 

_

_

 

, 1 ,

r r

n r

w x  x B  f x 

C

w f . (2.2)

Theorem 2.3. For f Cw, 0,



   



1 min 0 , 1

2

   ,

 

0 0,r

  , n

 

x

 

x 1

n



   , we have

   





 

0

0 , 1

1/ 2

,

n r

n r

w

w x f x B f x

x

O f t O t

n x



 

 

 

_ _ _ 

 

 __ _ _ 

 

(2.3)

3. Lemmas

Lemma 3.1. For any non-negative real u and v, we have

 





1 , 1

1

u v

n

v u

n k k

n n

p x Cx x

k n k

 

 _





   _{ } _

   _ 

   



. (3.1)

Lemma 3.2. If R, then

 

2

 

, 0

n n k k

p x k nx Cn x



 _



 



. (3.2)

Lemma 3.3. For any fWr,  0, we have  r  r

r r

n

w F C w f . (3.3)

Proof. We first prove x xr5 2,xr3 2 (The same

as the others), we have

   

 

_{ }

_{   }

 

_{ }

     



 



  : 1 2

r r

n

r r

n

w x x F x w x x f x

w x x f x F x I I

 







  

Obviously

 

1

r r

C w f

I  

For I₂, we have

     



 



 

   



 



 

0 2

2

r r

n i

r _{r i}

r

n i

w x x f x F x

w x x n f x F x

I 

 

 





By [3], we have

 





 

5 2 3 2

5 2 3 2 _{5 2} _{3 2}

,

2 2

, _, .

r r

r r _r _r

r i n

x x

r i i

r

x x _x _x

f x F x

C n f H n f

 

  _ _



 

 

 _

  _ _

  _ _



 

 _   _

 

(4)

   

 

5 2 3 2

5 2, 3 2 1

2 2 ,

2

:

r r

x x r

r

x x

Cn w x x f H

Cw x x f T

I

T



 

 

 

 

 



 

 

By Taylor expansion, we have

 





 

_{ }



 



 

1

0

1

! 1

d . 1 !

i u r

u i

i u

x r r

i x

x x

f x f x

u

x s f s s

r 





 

 







(3.4)

It follows from (3.4) and the identity

 

1

r

v v

i i i

x l x Cx





, v0,1, , r. we have









 

_{   }





 





 

_{ }





 





 

1 0

1 1

1 0 1

1 1

,

! 1

d 1 !

1

d , 1 !

i

u r r

u i

i i u

r _x

r r i _x i

i

r u r

u v

u v v

u i i

u v i

r _x

r r i _x i

i

x x

H f x f x l x

u

l x x s f s s

r

f x C f x C x x l x

l x x s f s s

r

 







  



  

 



 

  _  _

 

 













which implies that

     







    

1

 





1  

 

,

1 _{d ,}

1 !

i r

r _x

r r

r

i _x i

i

w x x f x H f x

w x x l x x s f s s

r



 

 

  







Since

 

i

l x C, for x_{ }x_r__{5 2},x_r__{3 2}_, i1, , r.

It follows from

 

1 1

r r

i i

x s x x

w s w x

 

 ,

s between xi and x, then

     





   





 

_{ }

 

1 1

/ 2

,

d

.

i

r

r _x

r r r

i x i r r

r

w x x f x H f x

Cw x x x s f s s

C

n w f









 





So

 

2

r r

C w f

I   .

Then, the lemma is proved.

According to methods of Lemma 3.3, we can easily get:

Lemma 3.4. If f W_r,  0, then

   





 

,

r r

n x r

w x g x H g x C w g

n x 

  

  _ _

  (3.5)

Lemma 3.5. For any 0, f Cw, we have

 

, 1

n r

wB  f C wf . (3.6)

Proof. By (1.2), we have

 





 





 

   

 

   

 

, 1 , 1 1

1

, 0 1

1

,0 0

1

, 0

1 2 3

, ,

.

0

1

:

i

i i

n r n r n

n r

i n n k

i k i

r

i n n

i r

i n n n

i

w x B f x w x B F x

k

w x C n F p x

n

w x C n F p x

w x C n F p

I I

x

I

 

 



 



 

 _{ }

 



  

 



Now, the theorem can be proved easily.

Lemma 3.6. Let min



   

0 , 1



1 2

   , then for rN,

1 0

8

t r

  and 1

2 2

rt _{  }_x rt_{, we have}

 

2 2

1 1

2 2

d d

t t _r

r r r

k r

t t

k

x u u u Ct x

 

  _

 _  _

 













 (3.7)

Lemma 3.7. Let n

 

:

 

n k,

 

k n n

A x w x p x

  





, then

 

/ 2

n

A x Cn for 0  1 and  0.

Proof. If x 3

n



  , then the statement is trivial.

Hence assume 0 x 3 n



   ( the case 3 x 1

n

   

can be treated similarly). Then for a fixed x the maxi-mum of p_{n k}_,

 

x is attained for kkn:n n. By using Stirling’s formula, we get

 





,

1 n

n

n n n

n

n k k

n k k n k

n n

n

nx x

e

p x C

k _k n k _{n k}

e e



  _

    



    _

   

   

1 1 .

n n

k n k

n n

k nx k nx

C

k n k

n



     

 _  _{ }  _



(5)

W. M. LU ET AL. 393

Now from the inequalities









1 1 , 2 n

k nx n n nx n x

n n x

      _  _       and 2 1 2

1 u e u u , 1_{ }_{u e}u_, _u_₀_.

We have that the second inequality is valid. To prove the

first one we consider the function

 

2

1

2 ₁

u u

u e u

      .

Here 

 

0 0,

 





2 1 2

1 u u 1

u u e

       , 

 

0 0,

 

_u _{u u}



₂



_e u 12u2 ₀

      , whence 

 

u 0 for 0

u .

Hence

 

2 , 1 exp 2 n n n

n k n n

n n

k nx k nx

C

p x k k nx

k k n   _ _ _ _   _ _  _   _ _   _      _ _  





2  2

exp 2 Cn x n n k nx C e k n     _     _ _     .

Thus

 





Cn x2 n

A x _C _x e   _{. An easy calcula-} tion shows that here the maximum is attained when

C x

n

   and the lemma follows.

Lemma 3.8. For 0  1, ,  0, we have

 

2

 

,

n k k n n

w x p x k nx Cn x

   _       



(3.8)

Proof. By (3.2) and the lemma 3.7, we have

 

2 1 2 1 2 , 1 2 2 ₂ , ( ) . n n n n k k n n

n n n k

k n n

w x w x p x

p x k nx Cn x

    _                           



Lemma 3.9. For any  0, f W_r,we have  

_{ }

, 1

r r

n r

wB  f Cn wf . (3.9)

Proof. We first prove x 0,1 n

 

_{ }_ (The same as

1 1 ,1 x n    _ _

 , now

 

_

_

   

 

, 1 2 1 , 0 0 2 ,

0 0 0

2 ,0 0 0 2 0 0 , ! ! i i i i i i r n r n r r r i

i n n r k

i i k n i

n r

r r

r j

i r n n r k

i k j i

r r

r j

i r n n r

i j i

r r

r j i

i r n

i j i

w x B f x

n k

w x C n F p x

n r n

k r j

Cw x n C F p x

n

r j

Cw x n C F p x

n

n j

Cw x n C F

n                        _{ }  _{ }              _ _       _ _  



  

 



 

, 1 2 ,

0 1 0

1 2 3

: .

i i

i

i n r n r

n r

r r

r j

i r n n r k

i k j i

p x

k r j

Cw x n C F p x

n

H H H

              _ _     

  

We have

 





2 1 ,0 0 0 2 0 0 1 . i i r r r

i n n r

i j i

r r

n r i

r

i j i

r

r j

H Cw x n F p x

n

n x

Cn wf x

r j n Cn wf         _       _ _              

 



Similarly, we can get _H₂__{Cn wf}r _{, and}

3

r

H Cn wf .

When x 1,1 1

n n

 

_  _

 , according to [3], we have

 

_

_

_{ }

 





 



 





  

 



 





  

 

2 3

, 1 , 1

2 2 0 0 , 2 2 0 0

, 1 2

, , , , : i i i i r r

n r n r n

r r

r _j

j i i i

i j j

n n k

k n A i i

r r

r _j

j i i i

i j j

n k

x k n x i i

w x B f x w x B F x

w x x Q x n C n n

k k

x F p x

n n

w x x Q x n C n n

k k

x H p x

n n                         _  _ _{ }           _  _ _{ }      









where A: 0,



x2





 

x3,1 , H is a linear function. If

i

k _A

n  , when

 





1 i 2 i

i

w x

C n k n x

w k n



 

 

 _   _

 , we have

2

i i

n

k n   , also



,





1



2

r j

j i i

Q x n _ n x _x _  _ _{, and}

 



2





_,





2

 



2

r j

r _j

j i i i

x Q x n n C n x

(6)

 

_{ }

 

2 2

1 2 ,

0 0 0

2 2

2

, 1 2 2

0 0 0

1 : .

i

r j

j n

r r i

n n k

i j k i i

r j

j n

r r i

i i n k

i j k i

n k k

Cw x x F p x

n n

x

n k

C wf n k n x x p x I I

n x



 



 

  



 _

  

     

 _ _     

   

 

   

 _ _       

 

 









By a simple calculation, we have I1Cn wfr . By (3.2), then

 

2 2

2

2 2 ,

0 0 0

i

r j n

r r j _j

r i

i i n k

i j k

n

I C wf n k n x p x Cn wf

x 

 



 

 _  _ 

  

 

 _ _  

 





.

We note that

 



1 4



 

max , :

i

k

H H x H x H a

n

  _ _

 

 

  .

if x



x x₁ , ₄



,we have w x

 

w a

 

.So, if x



x x₁ , ₄



, then _₂__{Cn w a H a}r

   

__{Cn wf}r _.

If x



x x1 , 4



, then w a

 

ni 2





 , by (3.8), we have

     

_{ }

 

2 3

2 2

2

2 2

0 0

, .

i i

r j

r r

i i i j j

r n k

x k n x i

n

Cw a H a w x n

x

k

x p x Cn wf

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 f Cw, min





   

0 , 1



1 2, we discuss it as

follows:

Case 1. If 0

 

x 1 n



  , by (3.9), we have

   





 

_{ }

 

_

_

( ) , 1

2 , 1

,

r r

n r

r r

n r r

w x x B f x

x

C x w x B f x Cn wf

x

  





   (4.1)

Case 2. If

 

x 1 n

  , we have

 

_

_

 

_

_

 







  

, 1 , 1

2 2

0 0

, ,

, .

r r

n r n r n

r r

r _j

j i i i

i j

B f x B F x

x Q x n C n n



 

  





 

, 0

i

i j

n

n n k

k i i

k k

x F p x

n n



   



   

   



,

where



_,





₁



2

r j

j i i

Q x n n x x



 

 

 

  , and

 



2





_,





2

 



2

r j

r _j

j i i i

x Q x n n C n x

    

So

   

 

_

_

_{   }

 

   

_{ }

 

   

_{ }

, 1

2 2

, 2

0 0 0

2 2

2

0 0

2 2

, 2

0 0

,

i

r

r r

n r r j

j n

r r

i

n n k

i j k i i

r j

j

r r

r i

n

i j k n A i i

r j

r r

r i

n k

i j

w x x B f x Cw x x

n k k

x F p x

n n

x

n k k

Cw x x x F

n n

x

n

p x Cw x x

x

k x

n

 









 

 

  

 

  

 

  

     

 _ _     

   

 

     

 _ _     

   

 

 

  _ _

 

 











 

2 3

, : 1 2.

i i

j

n k

x k n x i i

k

H p x

n  

 

   

 

   

   



(4.2) where A: 0,



x₂





 

x₃,1 , 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 _f _Wr



 , by [3], we have

 





2

_{ }

, 1 1 ,

0 0

, i _i

i n r r

r r r

n r n i i n n r k

i k n i

k

B F x C n n F p x

n

 

 

 

 _{ }

 

(7)

W. M. LU ET AL. 395

If 0  k ni r, we have

 

1

1 ₀i d

i

r r

r r _n

n i n

i i

n

k k

F Cn F u u

n n

 

   

 

   

 



 

 . (4.4)

If k0, we have

 

1  

_{ }

1 0 ₀i d

i

r

r _n r

n n

n

F C

_

u F u u

 . (4.5)

Similarly





 

_{ }

1

1 ₂

1 ₁ 1 d .

i i

r r

r i r

r

n i n

n i

n

n r

F Cn u F u u

n

     

 

 

 



 (4.6)

By (4.3), we have

   

 

_

_

   

 

   

 

   

 

   

 

, 1 2

1 ,

0 0 1 2

1 ,

0 1 2

1 ,0 0

2

1 ,

0

,

0

1 .

i

i i

i

i i

i i i

r r

n r

n r r

r r r

i n n r k

i k n i

n r r

r r r

i n n r k

i k n i

r

r r r

i n n r

i n

r

r r r

i n n r n r

i _n

w x x B f x

k

Cw x x n F p x

n

k

Cw x x n F p x

n

Cw x x n F p x





 



 

  



 



 



 



 

  _{ }

   

  _{ }

 

 

 





(4.7)

which combining with (4.4)-(4.6) give

   

 

_

_

 

, 1 ,

r r

n r

w x  x B _ f x C w f .

Combining with the theorem 2.1 and theorem 2.2, we can obtain

Corollary For any  0, 0  1, we have

 

_

_

   

_{ }





 

, 1

1 2 1

2

,

max , , ,

, .

{

r r

n r

r r

r

w r

r r

w

w x x B f x

Cn n x wf f C

C w f f W



 



 



 

 



 _

 

 

(4.8)

4.3. Proof of Theorem 2.3 4.3.1. The Direct Theorem

We know

 







1

 



1  

 

d

1 !

n n n

t r r

n x

F t F x F t x

t u F u u

r





   

 







(4.9)









, 1 , 0

k n r

B   x x  , k1, 2, , r1 (4.10)

According to the definition of _Wr

 , for any g W r, we have  , 1



,



 , 1



 

,



r r

n r n r n

B  g x B  G g x , and

   

n n r, 1



n,



 

n r, 1



r



n, , ,



w x G x B  G x w x B  R G t x x ,

there of



, ,



t





r1  r

 

d

r n _x n

R G t x 

_

t u  G u u, we have

   





 

_{ }

   

 

_{ }

 

, 1

1 , 1

1 1 2 , 1 2

,

d ,

d , .

n n r n

r t r

r

n n r x r

r t r

r

n n r x r

r t n r _x

w x G x B G x

t u

C w G w x B du x

w u u

t u

C w G w x B u x

u

t u

B u x

w u 





 

 _ 

 



 

 

  _ 

  



 

 _ _

 

  _ 

  



 

 _ _

 



(4.11)

also

 

1 1

2 d 2 , 2 d 2

r r r r

t t

r r

x x

t u t x t u t x

u C u C

u x w u w x

 

 

   

 



.

(4.12) By (3.2), (3.3) and (4.12), we have

   





 

_{ }





 

, 1

,

n n r n

r r

n n r

r

n r

n

w x G x B G x

C w G x B t x x

x

C w G

n x

 



 

 

  

 _ _

 

. (4.13)

By (3.3), (3.5) and (4.13), when _{g W}r

  , then

   





   





  







   





 

, 1

, ,

, , ,

δ δ

.

n r n

n n r

r r

n r n r

n

w x g x B g x w x g x G g x

w x G g x B g x w x g x H g x

x x

C w G C w g

n x  n x 



  

   

   

 _ _  _ _

   

(4.14) For f Cw, we choose proper g W r, by (3.6) and (4.14), then

   





   

 





   





 

, 1

, 1 , 1

,

, ,

δ

, .

n r

n r n r

n r

w

w x f x B f x w x f x g x

w x B f g x w x g x B g x

x

C f

n x

 

 

 

  

   

 

 _ _

 

4.3.2. The Inverse Theorem

(8)

 







 



, , inf :

r

r r r r

r _w _g

K  f t  w f g t w g g W  .

By [3], we have

 

1

,

, , ,

r r r

r

w w w

C f t K  f t C f t . (4.15)

Proof. Let  0, by (4.15), we choose proper g so

that





r



,



w

w f g C f  , w gr  r C rr



f,



w



   .

(4.16)

For r N , 0 1 8

t r

  and 1

2 2

rt_{  }_x rt_{, we have}

 



 





 





 





 

     

 

0

, 1 , 1 , 1

1

2 2 2

, 1 1

0 ₂ ₂ 1

, , ,

2

, d d

2

r r r r

h h n r h n r h n r

h x h x n

r r

r j

r h x h x n r k r

j k

h

w x f x w x f x B f x w x B f g x w x B g x

r

x j h x

C n w x B f g x u u u

r

x j h x

w x

   



 



 

  



 

 



        

 _  _ _  

  _ _  _ _

 

 

  _   _

 _ _ _ _ _ _

 

  _ _ 

 















 _{ }

     

 

2 2

, 1 1 1 2 3

1 2 2

, d d :

h x h x _r

r

n r k r

h x x

k

B g x u u u J J J

 

 

 _

 _  _ _ _

 











(4.17)

Obviously

 

0

1 1/ 2

n x

J C

n x





  

 _ _

  . (4.18)

By (3.9) and (4.16), we have





2 _{ } 2 _{ }

  



2 1

2 2

d d ,δ

h x h x

r r r r r

r

h x h x w

J Cn w f g u u Cn h x f

 



   

 

 

_



_

  (4.19)

By the first inequality of (4.8) and (4.16), we let  1, then





 

     

 

  



2 2

2 1

1 2 2

d d ,δ .

h x h x

r _r r

r r r r r

k r

h x h x w

k

J Cn w f g x u u u Cn h x x f

 



     

  

 

  _  _ 













 (4.20)

By (3.7) and (4.16), we have

 

_{ }

   





1

2 2

3 1

1 1

2 2

d d δ ,δ

h x h x _r _r

r

r r r r r

k k r

h x h x w

k k

J C w g w x w x u x u u u Ch f

 

  

   

   

   

 _  _ _  _ 





 











 (4.21)

Now, by (4.17)-(4.21), there exists a constant M 0 so that

 









 









0

2 1/ 2

1 2 1/ 2

min , , ,

, , .

r r n

r r r r r r r r

h r w w

r

n r r n r r r r

w w

x x

w x f x C n n x h f h f

n x x

x x

C h M n f h f

x

n x



  



 



  

 

 

 



 _

_ _ _ __ __ _



  __ _        _

 

    



_ _ _ _ _ _ 

 

 _ _  _ _    

_ _ _ _ 

 

When n2, we have

  



 

1 ₁ 1

2 ₂ 2

1

1 2

n n n

n  x  n   _ x  n  x ,

Choosing proper x, δ, n N , so that

 





 

1 1

1

2 n x 1 2 n x

n n

x x

 

  _{  } _    _,

Therefore

 

r

 



0 r r r



_,





h w

w x _f x C    h _ f 

which implies

 

_,



0



_,





r r r r

w w

f t C  h f

 

(9)

W. M. LU ET AL. 397

So, by Berens-Lorentz lemma in [3], we get

 

_, 0

r w

f t Ct



  .

5. References

[1] D. S. Yu, “Weighted Approximation of Functions with Singularities by Combinations of Bernstein Operators,”

Journal of Applied Mathematics and Computation, Vol. 206, No. 2, 2008, pp. 906-918.

[2] Z. Ditzian, “A Global Inverse Theorem for Combinations of Bernstein Polynomials,” Journal of Approximation Theory, Vol. 26, No. 3, 1979, pp. 277-292.

doi:10.1016/0021-9045(79)90065-0

[3] Z. Ditzian and V. Totik, “Moduli of Smoothness,” Springer- Verlag, Berlin, 1987.

[4] M. Felten, “Direct and Inverse Estimates for Bernstein

Polynomials,” Constructive Approximation, Vol. 14, No. 3, 1989, pp. 459-468. doi:10.1007/s003659900084

[5] S. S. Guo, C. X. Li and X. W. Liu, “Pointwise Approxi-mation for Linear Combinations of Bernstein Operators,”

Journal of Approximation Theory, Vol. 107, No. 1, 2000, pp. 109-120. doi:10.1006/jath.2000.3504

[6] G. G. Lorentz, “Bernstein Polynomial,” University of Toronto Press, Toronto, 1953.

[7] J. J. Zhang and Z. B. Xu, “Direct and Inverse Approxi-mation Theorems with Jacobi Weight for Combinations and Higher Derivatives of Baskakov Operators,” Journal of Systems Science and Mathematical Sciences, In Chi-nese, Vol. 28, No. 1, 2008, pp. 30-39.