On the Average Errors of Multivariate Lagrange Interpolation

(1)

http://dx.doi.org

ABSTRAC

In this paper, first kind. The Keywords: M

1. Introduc

Let F be a r a probability Xbe another embedded in Any T F: 

measurable m The average e ( ,

e T F

For d1,

0,d

C 

The space

The classic is Gaussian w

( ,

d w

R s

For more d we refer to [1] In this pape

*_{Supported by Na}

no. 10871132 an higher school scie

#

Corresponding a

g/10.4236/jamp.

On

Department o

CT

we discuss the e average error Multivariate La

ction

real separable measure  o normed space

X. By || || _X X

 such th mapping is calle error of T is d ,|| || ) :X ||

F

F    f





let

{ [0,1] | whenever

d i

f C f

x 



0,d

C equipped [ || || su

t f





al Wiener she with mean zero

0,

1 , ) ( )

min{ ,

d C d

i i

t f s f

s







detailed discu ].

er, we let

ational Natural Sc nd 11271263) an ence and technolo author.

.2013.16001

the Ave

Lag

of Mathematics

e average error rs of the interp agrange Interpo

Banach space on the Borel e such that F

we denote th hat f || f 

ed an approxim defined as

2 ( ) ||X (

f T f 

1

( , , ) 0 for some 1

d

f x x 





d with the sup 0,1]

up | ( ) |

d

f t .

eet measure

w

and covarianc ( ) ( )

, }, ,

d

i f t w df

t s t

ussion and pro

cience Foundation nd by a grant fro ogy research (Z201

erage E

grange

Zengbo Zh

and Physics, No Email: #j Rece

rs of multivari olation sequen olation; Averag

e equipped wit sets of F. L

is continuousl he norm in X

( ) ||_X T f

 is

mation operato 1/2

(df) .



0, }.

i d

 

norm

d

w

on B(C0,d ce kernel

[0,1] .d (

operties of wd

n of China (Proje om Hebei provinc

10160).

Errors o

Interpo

hang, Yanjie

orth China Elect [email protected] ived July 2013

iate Lagrange i nce are determ

ge Error; Cheb

th Let

ly

X. a or.

)

d

1)

d,

For the mea

Let norm fo

Let

is the z Chebys well-kn based o

where

ect ce

of Multi

olation

*

Jiang#

tric Power Univ om

interpolation b mined on the mu

byshev Polyno

1 { (2

d

F f

f x

   

every measur asure of A by

1

( ) :

(2

d A f x

  

1 ( ,x ,x_d)

  

or f Fd is

2,

||f ||: || f ||

,

: :

k k n

  

zeros of T x_n( ) shev polynomi nown Lagrang

on





1, , d

i i

 

1 ,

, , 1

( , )

(

d

n d n i i L f x

f 







,j( )

j i j k

l x 

ivariate

*

versity, Baoding,

based on the C ultivariate Wie

omial; Wiener

[ 1,1] | ( 1, , 2 1)

d

C g

x

    

rable subset A

1

{ ( , ,

1, , 2 1

d d w g x

x



 

 



2

1 1

d i

i x







defined as

[ 1,1]

: | ( )

d

f t 



  







2 1

cos ,

2

k

n 

 

cosn (x

 

ial of the first ge interpolatio



1, ,d 1 n

ii is give

1, d)1,1( )1

i i l i x

 

1( )

,

j j

n

j k k i

i k x 

 

 

 



e

, China

Chebyshev nod ener space.

Sheet Measure

1

0, ( , , ) ) }.

d d

x x

C





( _d) ABF , w

)

1) }.

d x

A





1

, the weigh

1/ 2 2

) | ( )t dt .

   

1, ,

k  n

cos ) , the nth kind. For f on polynomia

en by

,

) ( ),

d

d i d

l x



, j1,, .d

des of the

e

we define

(2)

hted L₂-

th degree

d F

 , the al of f

(2)

2. Main Result

Since the polynomial interpolation operators are impor- tant approximation tool in the continuous functions space, there are a number of papers studying the convergence for interpolation polynomial, especially the interpolation polynomial based on roots of orthogonal polynomials. Xu Guiqiao [2] studied the average errors of univariate Lagrange interpolation based on the Chebyshev nodes on the Wiener space. Motivated by [2], we consider the average errors of multivariate Lagrange interpolation. We first study the bivariate Lagrange interpolation, then the general multivariate Lagrange interpolation. Our main results are the following:

Theorem 1. Let

2 1 2

( , ) [ 1,1]

x x x   ,

1 2

2 2

1 2

1 ( , )

1 1

x x

x x

 

  ,

and

, , 1, 1 2, 2

1 1

( , ) ( , ) ( ) ( ),

m n

mn i m j n i j

i j

L f x f   l x l x

 





where

1 , 1, 1

1( )

, ,

2 , 2, 2

1( )

, ,

( ) ,

( ) .

m

k m i

k k i

i m k m n

s n j

s s j

j n s n x

l x

x

l x

  

 

  

   



Then we have

2 2

2 2, 2, 2

2

2 2

2

( , ,|| || ) || ( ) ( , ) || ( )

sin cos sin (1 cos ) sin cos sin (1 cos )

2 2 2 2

2 _{(1 cos} ₎ _{(1 cos} ₎

sin cos sin (1 cos ) sin cos sin

1 ₂ ₂ ₂ ₂

2 _{(1 cos} ₎

mn mn

e L F f x L f x df

F

m m m m n n n n

m n

m m m m n n n

m

  

       



 

      

    

 _ _ _ _ 

 

   

 _ _ 

 

  

 

 2

(1 cos ) . (1 cos )

n



 



Theorem 2. Let 1

( , , d) [ 1,1]d

x x  x   ,





1

2 1 ( )x di 1 xi

     ,

and L_{n d}_, ( , )f x be defined by (3). Then we have 2

, 2,

2

, 2,

( , ,|| || )

|| ( ) ( , ) || ( )

n d d

Fd

e L F

f x L f x df



 

  _ 

1 1

1

2

1

( 1) 2

sin cos sin (1 cos )

2 2

(1 cos )

.

d _k

d k

k

d k d

k

n n n n

n

   

 

 





  

 





     

 

 

 

Remark 1. Let us recall some fundamental notions about the information-based complexity in the average case setting. Let F be a set with a probability measure

, and let G be a normed linear space with norm || || . Let S be a measurable mapping from F to G which is called a solution operator. Let N be a mea-

surable mapping from F into Rd, and let



be a measurable mapping from Rd into G which are called an information operator and an algorithm, respec- tively. The average error of the approximation N

with respect to the measure _{is defined by}





1

2 2

( , , ) : || ( ) ( ( )) || ( ) ,

F

e S N  



S f  N f  df

and the average radius of information N_{with respect to}

 is defined by

( , ) : inf ( , , ),

r S N e S N

 



where



ranges over the set of all possible algorithms that use information N. Furthermore, let _m be the class of all deterministic information operators N with cardinality m. Then, the mth minimal average radius is defined by

( , ) : inf ( , ).

m m

N

r S r S N



 

For FC0,d, w Sd, I (the identity mapping), and m consisting of function values taken at grid points, i e.,

1 1 1

1

( ) [ ( , , ), , ( , , ),

, (1 , ,1 )]

d d d

d

N f f h h f i h i h

f h h



 

  

(3)

for some h1,,hd, by [3,p.16] we know

1 2

1

( , _m) .

d

r S

m

 

From Theorem 2 we have

, 2, 1 2

1 ( _{n d}, _d,|| || ) .

e L F

n 

 

Note that d

mn , we can say that the average error of

,

n d

L is weakly equivalent to the corresponding nd th minimal average radius.

3. Proof of Theorem 1

Proof of Theorem 1. By a simple computation, we have





2 2 2

2

2 2

2

2 2, _{[ 1,1]} 2

2 2

2 [ 1,1]

2 2

2 2 2

[ 1,1] [ 1,1]

2

( , ,|| || ) | ( ) ( , ) | ( ) ( )

{ ( ) 2 ( ) ( , ) ( , ) ( )} ( )

{ ( ) ( )} ( ) 2 { ( ) ( , ) ( )} ( ) { ( , ) (

mn mn

F

mn mn

F

mn mn

F F

e L F f x L f x x dx df

f x f x L f x L f x df x dx

f x df x dx f x L f x df x dx L f x df

  

 

    



 

  

  

 



2

[ 1,1]

1 2 3

)} ( )

: 2 .

F

x dx

I I I

 

  



(4)

On using (1) and (2), we obtain

2 2

2 0 ,2

2 2 1 2

1 _{[ 1,1]} 2 _{[ 1,1]} 2

2

1 ₁ 1 ₂

1 2

1 2 1 2

1 2

1 1

{ ( ) ( )} ( ) { ( , ) ( )} ( )

2 2

1 1 1 1

.

2 ₁ 2 ₁ 4

F C

x x

I f x df x dx g w dg x dx

x x

dx dx

x x

  



 

 

 

 

 

 



(5)

From [2], we have

2

2 2

0 , 2

1 1

2 _{[ 1,1]} 2 ₁ ₁ 1 2 , , 2 1, 1 2, 2 1 2 1 2

1 1

1 1 ₁ ₂

1 1

, ,

2 1, 1 2, 2

{ ( ) ( , ) ( )} ( ) ( , ) ( , ) ( ) ( ) ( ) ( , )

1 1

( , )

2 2

1 1

( , ) ( ) ( ) ( )

2 2

m n

mn i m j n i j

i j

F F

m n

i j C

j n i m

i j

I f x L f x df x dx f x x f df l x l x x x dx dx

x x

g g w dg l x l x

     

 



  

 

   

  

 

    





  

  

















1 2 1 2

1 1 1 , 2 ,

1, 1 2, 2 1 2 1 2

1 1

1 1 , 1, 1 2 , 2, 2

1 2

1 2 2

1 1

1 2

2

( , )

1 min ,

( ) ( ) ( , )

2 2

1 min ,

1 min , ( ) ( )

2 ₁ 2 ₁

1 ₂ ₂

(

4 ₍₁

m n

j n i m

i j

m n

j n

i m i j

i j

x x dx dx

x x

l x l x x x dx dx x

x l x l x

dx dx

x x

m m m m

m

 



 

   



 

 



 

 

 

 

 

 

 

 

 







2

2 2

) ( ),

cos ) (1 cos )

n n n n

n

m n

   



 

 

 

 

(6)

and

0,2

2

3 _{[ 1,1]}2 2

2

, , , , 2 1, 1 1, 1 2, 2 2, 2 1 2 1 2

1 1 1 1 ₂

1 1 1 1

1 1

1 1 _, _,

1 1

{ ( , ) ( )} ( )

( , ) ( , ) ( ) ( ) ( ) ( ) ( ) ( , )

1 1

( , )

2 2

mn

m n m n

i m j n k m s n i k j s

i j k s

m m n

i j k s

F

F n

j m i m C

I L f x df x dx

f f df l x l x l x l x x x dx dx

g

 

     

 



   

 



  

 

 



  









, ,

2 1, 1, 2, 2, 1 2 1 2

1 , , 1, 1 1, 1 1 , , 2, 2 2, 2

1 2

2 2

1 1

1 1 ₁ 1 1 ₂

2

.

1 1

( , ) ( ) ( ) ( ) ( ) ( ) ( , )

2 2

1 min ,

1 min , ( ) ( ) ( ) ( )

2 ₁ 2 ₁

4

k m s n

i k j s

m m n n

j n s n

i m k m i k j s

i k j s

g w dg l x l x l y l y x x dx dx

l x l x l x l x

dx dx

x x

  _

   



 

   

  _

 

 

 











(4)

On combining (4)-(7), we obtain

2

2 2

2 2,

2

2 sin cos sin (1 cos )

2 2

(1 cos )

2 2

( 2

(1 cos )

1 2

2 2

( , ,|| || )

4 (1 cos )

2 2 ₎

(1 cos ) 1

( ) ( )

4 2

mn

m m m m

m

m m m m

m

n n n n

e L F

n

n n n n

n

n 

   



   



   

 

   



_

 

_



 





 





 





     

2 2

sin cos sin (1 cos ) sin cos sin (1 cos )

2 2 2 2

(1 cos ) (1 cos )

,

m m m m n n n n

m n

       

 

   



 

we complete the proof of Theorem 1.

4. Proof of Theorem 2

Proof of Theorem 2. Similar to the proof of Theorem 1,

2 2

, 2, , 2,

2 2

, ,

[ 1,1]

2 2

, ,

[ 1,1] [ 1,1]

( , ,|| || ) || ( ) ( , ) || ( )

{ ( ( ) 2 ( ) ( , ) ( , )) ( )} ( )

{ ( ) ( )} ( ) 2 { ( ) ( , ) ( )} ( ) { ( , ) ( )

d d

n d d n d d

F_d

n d n d d

F

d n d d n d d

F F

e L F f x L f x df

f x f x L f x L f x df x dx

f x df x dx f x L f x df x dx L f x df

 

 

    



 

   

  



[ 1,1]

1 2 3

} ( )

: 2 .

d d F

x dx

J J J





  



(8)

Form (1) and (2),

0,

2 2 1

1 _{[ 1,1]} _{[ 1,1]}

1

1 2

1

1 1

{ ( ) ( )} ( ) { ( , , ) ( )} ( )

2 2

1 1

.

2 ₁ 2

d d

d

d d

F C

d d

k

k k

k

x x

J f x df x dx g w df x dx

x

dx x

  



 

 

 

 

  

 _{  }

  







(9)

By a simple computation similar to (6)-(7) we obtain

1 1

1

1 0,

2 _{[ 1,1]} ,

1 1, 1 , 1 1

[ 1,1] , . 1

1 [ 1,1]

, . 1

{ ( ) ( , ) ( )} ( )

{ ( , , ) ( , , ) ( )} ( ) ( ) ( , , ) 1

1 1

1

{ ( , , ) ( , , ) ( )

2 2 2 2

d d

d _d _d

d d

n d d

F n

d i i d i d i d d d

i i F

n

i i

d

i i C

J f x L f x df x dx

f x x f df l x l x x x dx dx

x x

g g w dg

 

   

 



 



  

 

  



 



 





    

 









1

1 1

1 ,

1 2 ₂

1 1

1, 1 , 1 1

[ 1,1]

, . 1 1

1 min , ( ) 1 ₂ ₂

(

2 ₁ 2

(1 cos )

( ) ( ) ( , , ) 1 min ,

( ) ( ) ( , , ) 2

k k

k

d

k

d _d

d

d n

k i k i k

k d

i k

k

i d i d d d

d n

k i

i d i d d d

i i k

x l x _n _n _n _n

dx

x _n

n

l x l x x x dx dx

x

l x l x x x dx dx

   



 _







  



 

 



  

 _



 



 









  

)d,

(5)

and

1 0 ,

1 1

2

3 _{[ 1,1]} ,

, , 1 1

[ 1,1]

, , 1 , , 1 1 1

[ 1,1]

, , , , 1

1 1

( , , )

2 2

{ ( , ) ( )} ( )

{ ( , , ) ( , , ) ( )} ( ) ( ) ( , , )

{ d

d

d d

d _d _d _k _s

d d d

i i

d C d

n d d

F

d d

n n

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

i i j j F k s

n

i j j

g

J L f x df x dx

f f df l x l x x x dx dx

 

 

     



   

 

 



  



_



 





 

 



   





1

, , 1 1

1 1

1 , ,

1 2

1 1

( , , ) ( ) ( ) ( ) ( , , )

2 2

1 min , ( ) ( )

2 ₁ 2 .

d

k s

k k k k

k k

d d

j j

d k i k s j s d d

k s

d

d n n

i j k i k k j k

k

i j

k

n i

g w dg l x l x x x dx dx

l x l x

dx x

 



  

 

   



 

 



 



     



 







   (11)

On combining (8)-(11), we obtain

2

, 2, 1

2

1 1

2 1

( , ,|| || )

1 ₂ ₂

( )

2 2 _{(1 cos} ₎ 2

1 ₂ ₂

( 1) ( ) .

2 _{(1 cos} ₎

n d d

d d

d

k k d k

d k

e L F n n n n

n

d _n _n _n _n

k _n

n



   

 _ 



   

 



 

 



 

   

_{ }   _{ }

  _  

 

   _{ }  

  _



We complete the proof of Theorem 2.

REFERENCES

[1] K. Ritter, “Average-Case Analysis of Numerical Prob- lems,” Springer-Verlag Berlin Heidelberg, New York, 2000.

[2] G. Q. Xu, “The Average Errors for Lagrange Interpola-

tion and Hermite-Feje’r Interpolation on the Wiener Space (in Chinese),” Acta Mathematica Sinica, Vol. 50, No. 6, 2007, pp. 1281-1296.

[3] A. Papageorgiou and G. W. Wasilkowski, “On the Aver- age Complexity of Multivariate Problems,” Journal of Complexity, Vol. 6, No. 1, 1990, pp. 1-23.