The Average Errors for Linear Combinations of Bernstein Operators on the Wiener Space*

(1)

http://dx.doi.org/10.4236/jdaip.2013.14009

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: #[email protected]

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 determined on the Wiener space.

Keywords: Linear Combinations; Bernstein Operators; Weighted Lp-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 T F: X such that

 

X f  f T f is a measurable mapping is called an approximation operator. The p-average error of T is defined as



, , ,



:

 

d 1 .

p p

p X X

F

e T F   _ f T f  f





  

Let

 

 





0: 0,1 0

F  f C f 0 .

For every f F0 set

 

0 1

: max .

C _t

f f t

  

Then



F0,C



becomes a separable Banach space. Denote by B

 

F0 the Borel class of



F0, C



and by

0

 the Wiener measure on B

 

F0 (see [1]). From [1,

p. 70] we know

     

 





 

0 0

d

1

min , , , 0,1 .

2

F f s f t f

s t s t s t s t



      



(1)

The Bernstein operator on C

 

0,1 defined by





,

 

0

, : n

n n

k k

B f x f p x

n 

   _{ }  



k

, .



,

where

 





, 1 , 0,1,

n k k

n k

n

p x x x k n

k



 

_{ }  

  

This operator turned out to be a very interesting operator, 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

n m i n

i

L f x C n B f x







,

(2)

where ni and C ni

 

satisfy the following conditions:

 

0 1

1

0 1

0

( ) ;

( ) 1;

( ) 0, 1, 2, , 1.

m m

i i

m

i i m

i i

i

a n n n Cn

b C n C

c C n

d C n n   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).

(2)

independent of n and x, which may be a different constant in different cases.

For

 

1 0,1 0,1

L p

 ,  

   ,

the weighted Lp-norm of f C

 

0,1 is defined by

 





1

0

: 1 p d .

,

p p

f 



f t  t 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 approximafunc-tion by linear combinafunc-tions 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),

1  p L_{n m}_,



f x,

 



 

x 0

L₁ 0,1  and

 , 

 

x is continuous on

. Then we have

(0,1)





   













 

1

4

, 0 , 0

1 2

1 1

0 0

1 1

4 0

, , ,

2

1 d ,

p n m p

j i m m

j i

i j

i j i

i j

p _p

p

e L F

n n

C n C n

n _n _n

x x x x n

 

 

 

 





  

   

  

_ _  __

 _ _

  



 _ 

 

 

_  _ 

 

 





 

where

2

1

e d

2

x p

p x x.

  _ 





Here and in the following the notation an 

 

bn for

sequences

 

an and



bn



means that

lim_n_a b_n _n0.

3. Proof of Theorem 1

To prove Theorem 1 we need the following two lemmas. Lemma 1([5, p. 15]). If

1 0

2



  , then

 

,

k n

k n k

x n

p x Cn

 

  





for each k0, the constant C depending only on 

and k.

Lemma 2 ([5, p. 15]). For fixed 1

0 1,

3

x 

   ,

the asymptotic relation

 













_

_

 

1 2

,

2

,

1

2 1 exp

2 1

n k k

n k

n k n

p x x x

k

n k

x x n x

x x n

P x



 

_{ } 

 

 

   

    _ _  _ 

 

 



holds uniformly for all values of satisfying the ine-quality

k

. k

x n

n





 

(4) In other words,

 

, ,

lim _{n k} _{n k} 1

np x P x 

uniformly for all k satisfying (4).

Proof of Theorem 1. From [1, p.107] we have





 





 

2

 

0

, 0 , 0

1 2

, 0

0 .

, , ,

, d d

p

p n m p

p n m

F

L F

f x L f x f x x

 

 



 



 

 

 

p

e

v



(5)

By (2),

 





 

   

 

0

2

, 0

1 2

0 ,

0 0

1 1

0

0 0

, , 0

0 0

1 2 3

, d

d 2

d

2 .

i i

j i

i j

n m F

n m

i n k

F

i k

m m

i j

F

i j

i

n n

n k n s _F

k s i j

f x L f x f

f x f C n p x

k

f x f f C n C n

n

k s

p x p x f f f

n n

A x A x A x





 

 

 



 

 

 _{ } 

 

 

 _{   } 

   

  















(6)

On using (1), we obtain

 

   

0

2

1 _F 0 d .

A x 



f x  f x

(7)

Note that

 





, ,

0 0

2 2 2

, 0

1, ,

1 ,

n n

n k n k

k k

n n k k

p x kp x nx

k p x n x nx x

 



 

  



(8)

(3)

 

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

( ) d

1 2 1 . 2 i i i i i i n m

i n k _F

i k i

n m

i n k

i k i i

n m

i n k

i k i

k 0

A x C n p x f x f f

n

k k

C n p x x x

n n

k

x C n p x x

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







 (9)

From [4,(3.24)], we know

 





 

1 2 , 0 2 1 . i i n

n k i

k i i

x x

k

p x x n

n n       





Combining(3) and (9) we get

 





 





 

_{ }

1 2 1 2 1 2 0 1 0 2 1 1 2 1 . 2 m i i i _i m i i _i x x

A x x C n n

n

x x C n

x n n  _   _   _                



  (10)

Now, we estimate the term A x3

 

. From (2) and (8),

 

0 1 1 3 ,

0 0 0 0

0

1 1

, ,

0 0 0 0

1 1

,

0 0 0

, 0 d 1 2 1 2 j i i j j i i j i i j j n n m m

i j n k n s

i j k s

F

i j

n n

m m

i j n k n s

i j k s

i j i j

n

m m

i j n k

i j k

n

n s s

,

A x C n C n p x p

k s

f f f

n n

C n C n p x p x

k s k s

n n n n

x C n C n p x

k p x                          _{   }            _    _     







x . i j s

n n

(11)

Using Lemma 1 and (3), we have

 

5 12 2 , . j j j n s

s _{x n}

n

p x Cn

    



Note that 0 , i j k s n n

 1,

we get

 

5 12 2 , . j j j n s i j

s _{x n} n

k s

p x Cn

n n



 

 



(12)

By (8) and (12), we obtain

 

5 12

5 12 5 12

23 12 5 12 , , 0 0 , , 0 , , , , , , j i i j i i j s j n j i j

k _i s

ni nj j

i j

k s

i

ni nj j

i j

k s

i

ni nj

n n

n k n s

k s i j

n

n k n s

k _{x n} i j

n k n s

i j

x n x n

n k n s

i j

x n x n

n k n s

x n

k s

p x p x

n n

k s

p x p x

n n

k s

p x p x

n n

k s

p x p x

n n

n p x p x

         _{ }                     



 5 12 .

j i j

x n k s n n   



(13) For 5 12 i i

k _x _n

n



  and 5 12

j j

s _x _n

n



  ,

by Lemma 2,

 













, , 2 2 1 1 2 1 exp .

2 1 2 1

i j

n k n s

i j i j i j j i i j k s

p x p x

n n

k s

n n

x x n n

n

n k s

x x

x x n x x n

       _ _ _ _ _ _      _     _ _  _           (14) Set



 



_

_

2

_

_

2

1, 2 : 1 2 exp 1 2

2 1 2 1

j

i n

n

.

F u u u u u u

x x x x

        _  _       For



1 2



1 1 1 1

, , ,

i i j j

k k s s

v v

n n n n

 

     

   

 

   

by the differential mean value theorem we have











 





 





 











2 2 1 2 2 2 1 2

1 1 2 2

exp 2 1

2 1

exp

2 1 2 1

, , i i j j j j i i j n k s

x x x

n n x x n

n _s

x x v x v

x x n

n n

x v x v

x x x x

k s

F v F v

(4)

By a simple computation we know where









1 , , 2 , .

F  C F   C





1 1 1

, , ,

i i j

k k s s

x x x x

n n n n

       _{ }     

  

1

j 

.

From (15),











 



_

_{ }



_

_{ }



2 2

11

2 2 ₁₂

1 2 1 2

exp

2 1 2 1

exp .

2 1 2 1

j i

i j i j

j i

n n

k s k s

x x

n n x x n x x n

n n

x v x v x v x v n

x x x x

 

      

 

  _     _ _  _   

   

 



    

 

     _   _  _{ }  _

   

 

(16)

Integrating two side of (16) about



v v1, 2



in

1 1

, ,

i i j j

k k s s

n n n n

 

 _  _   

 

   

we get











 



_

_{ }



_

_{ }



2 2

1 11

1

2 2 ₁₂

1 1 2 1 2 2

exp

2 1 2 1

d exp d

2 1 2 1

j i

i j

j i

i j i j

s k

n j

n i

i j k s

n n

k s k s

x x

n n x x n x x n

n n

n n v x v x v x v x v v n

x x x x



 _

       

 

  _     _ _  _   

   

 



    

 

            _ _

 

   

 



.

(17)

From (14)-(17), we have

 











 



_

_{ }



_

_{ }



5 12 5 12

, ,

1 1

1

2 2

1 2 1 2

1 1

d

2 1

exp

2 1 2 1

i j

j i

i j

n k n s

i j

k x n s x n

n n

n k n s

i j

k _{x n} s _{x n}

n n

s k

i j n n

k s

n n

k x n s x n

n n

j i

k s

p x p x

n n

k s

p x p x

n n

n n v

x x

n n

x v x v x v x v

x x x x

 

   

     



 

 

 



  



    _    _

 









 







 





 





 



 







 



23 12

5 12 5 12

5 12 5 12 1 5 12 5 12

2

1 1

2 2

1 1 2 1 2

1

1 1 2

d

d exp

2 1 2 1 2 1

exp

2 1

j i

i j

i j _{n j}

i

i j

s k

i j n n i j

k s

n n

k x n s x n

n n

x n x n

i j n i

x n x n

v n

n n n n

v x v x v x v x v v

x x x x x x

n

n n _n

dv x v x v

x x

 

 

    



   

 

 _ 



  

 

 _ _     _   _ 

 

 





   

 











2

d







2

_

_{ }



2

 

1

1 2 d 2 .

2 1 2 1

j

n

x v x v v n

x x x x

 

 

 _ _ _  _

 _ _ 

 

  

(18) Let



 





 



1 1 , 2

2 1 2 1

j

i n

n

w x v w

x x x x

   

  x v2

,

(5)

 





   

 







5 12 5 12

1 12 1 12

1 12 1 2 1 12 1 2

, ,

2 1 2 1 1 2 2 2 1

1 1 2

2 1 2 1

2 1

d exp d

i j

j i

i i j j

n k n s

i j

k x n s x n

n n

x x x x

n n n n

i j

x x x x

k s

p x p x

n n

x x w w

w w w

n n

 

   

  

   

 



   







 w₂ n



.

(19)

By (3), suppose that

2 2

,

i j

n _c n _c

n  n 

,

from (19) and the convergence of the improper integral



2 2



1 1 2 1 2

dw c wi c wj exp w w w

 

    



d 2

,

we have

 









 





_

_





1 2

5 12 5 12

1

2 2

1 2

, , 1 1 2 2

2 2

1 2 2 1

1 1 2 2 1

2 1

d exp d

2 1 2 1

d exp d 2 d

exp

i j

j i

n k n s

i j

k x n s x n i j

n n

n w

i j n j i

x x w w

k s

p x p x w w w w n

n n n n

x x _w _w x x _w _w

w w w w w

n n n n

w w

 

  

 

   

   

  



      



   

 

   

       

   

 _ _  _ _

  











 









_{ }





_{ }









 

1 2

1 1

2 2

2

2 2

1 2

d

1 1

2 1 _e 2 1 _e

d d

2 1 ₁ ₁ 2 1

.

j i

i j

j i

j j

i i

i j i j j i

j i _i _j

w n

n n

w w

n n

x x x x

w w n

n n

x x n n x x n n

n n

n n n n n n

n n _n _n



  

 

 



   

 _ _  _ _

     

  

 

   

 _   _  _ _

    _  _

 

 

 _ _ _ _ _ _





 

(20)

Combining (11) and (20), we obtain

 





_{ }

1 2

1 1

3 , ,

0 0 0 0

1 1

0 0

1 2

2 1

. 2

j i

i j

n n

m m

i j n k n s

i j k s i j

m m

i j j i

i j

i j _i _j

k s

A x x C n C n p x p x

n n

n n n n

x x

x C n C n

n n

 

   

  _

 

   

 

   

 



 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