An Interpolation Process on the Roots of Hermite Polynomials on Infinite Interval

American Journal of Engineering Research (AJER)

2014

American Journal of Engineering Research (AJER) e-ISSN : 2320-0847 p-ISSN : 2320-0936 Volume-3, Issue-10, pp-75-83 www.ajer.org Research Paper

Open Access

An Interpolation Process on the Roots of Hermite Polynomials on Infinite Interval Rekha Srivastava Deptt. Of Mathematics And Astronomy Lucknow University, Lucknow, INDIA

ABSTRACT: For given arbitrary numbers

d k , k  1  1  n  1, d k , k  l  1  n  1 and d k , k  1  1  n , we *

**

seek to determine explicity polynomials R n  x  of degree atmost 3n-3 (n even), given by: n 1

Rn  x  

(1)



n 1

d kU

k

x 

k 1

n *

d kV k

x 

k 1

**

dk Wk

x,

k 1

Such that Rn '

Rn

 yk 

 d k , k  1  1  n  1,

 yk 

 d k , k  1 1  n  1 *

and Rn  xk



 d k , k  1 1  n , **

n 1

where  x k  k  1 are the zeros of n Hermite polynomial H n  x  and  y k  k  1 are the zeros of H n  x  .

Let the interpolated function f be continuously differentiable satisfying the conditions: n

th

L im

x   x

2v

f

x  x 

'

0 ,   0 ,1, 2 ., .....

and L im

x   x

2v

  x  f '  x   0 , w h e re   x   e

2

 x /2

,

further in (1) d x  f  y k  , k   1  n  1, 1    y2k * d k  f ' yk  ,    e  ( f ',  , k  1  1  n  1,     1 n   d k  f '  x k  , k  1  1  n , then for the sequence of inter polynomials R n  n  2 , 4 , ...  , we have the **

estimate e

 vx

2

f

x

Rn



f , x   0 (  ( f ';

1 n

) lo g n ),

v 

3 2

Which holds the whole real line, O does not depend on n a n d x a n d  is the modulus of continuity of introduce by G. Freud.

Keywords: Hermite, Interpolation

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f '

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I.

2014

INTRODUCTION

Earlier Pal [8] proved that when function values are prescribed on one set of n points and derivative values on another set of n  1 points, then there exists no unique polynomial of degree  2 n  1, but prescribed function value at one more point not belonging to the former set of n points there exists a unique polynomial of degree  2 n  1 . Eneduanya [2] proved its convergence on the roots of  n  x  .    x n , n  x n  1  ...  x 2 , n  x 1 , n   *

Let

*

be a given system of  2 n  1  distinct points. L  Szili [11] determined a unique polynomial R n lowest possible degree 2n-1 (for n even) given by: n 1

n

Rn  x  

Y

Ai ,n  x  

i ,n

i 1

Y

* 1, n

B i,n  x  ,

i 1

satisfies the conditions:

x   Y

Rn

i ,n

X

'

Rn

* v ,n

, i  1, 2 , ... n

i ,n

Y

' v ,n

, v  1, 2 , ... n  1

and Rn 0   0

If the interpolated function f is continuously differential f

 0   0 and

L im

x   f ' x  e

L im

x  e 2

 x /2

2  x /2

x

2v

x 

f

v  0 ,1, 2 , ...

0,

 0 , th e n th e s e q u e n c e e

 vx

2

x

f

 R  x  n

satisfies the relation

R n  x   0 (  ( f ',

1

) lo g n ),

v 1

2

which holds on the whole real line and 0 does not depend or n and x.

Further K.K. Mathur and R.B. Saxena [6] extended the results of L. Szili to the case of weighted (0,1,3,)-interpolation on Infinite interval.

In this paper, we consider a special problem of mixed type, (0,1;0)-interpolation on the zeros of Hermite polynomial (1.1)

Let

 X k  k 1 n

n 1

 y k  k 1

and

b e th e z e ro s o f H

n

x

a n d H n '  x  , w h e re

The fundamental polynomials of Lagrange interpolation are given by

(1.2)

1k

x

H

 H

n

x

x x n  k  '

xk



, k  1 1  n .

and

(1.3)

L

k

x

H

 H

' n

' yk

' n

x x 

yk



, k  1 1  n  1

In this paper, study the following:

 0 ,1 : 0  

In te r p o la tio n o n In f in ite In te r v a l.

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Let n be even, then for given arbitrary sequence of numbers  d k  k  1 ,  d k  n 1

unique polynomial R n  x 

*

n 1 k 1

and

d  **

k

n k 1

, there exists a

o f d e g r e e  3n  3 , such that

 R n  y k   d k , k  1 1  n  1  ' *  R n  y k   d k , k  1 1  n  1  and **   , R n  y k   d k , k  1 1  n

(1.4)

For n odd, R n  x  does not exist uniquely. Precisely we shall prove the following: Theorem 1: For n even, n 1

Rn  x  

(1.5)

n 1



d kU

n

 x    d kV k *

k

k 1

x



k 1

**

dk Wk

x,

k 1

where U k  x  , k  1  1  n  1 a n d W k  x  , k  1  1  n are the fundamental polynomial of the first kind and Vk

x,

k  1 1  n  1

are the fundamental polynomials of the second king of mixed type  0 , 1 : 0 

interpolation. Each such fundamental polynomial is of degree at most 3 n  3, given by:

(1.6)

U

(1.7)

Vk

k

x 

x



 x  L k ( x ) 1  2 y k  x  2

H

n

H

H

n

y k  

 yk 

 x  H n  x  Lk  x 

, k  1 1  n  1

'

n

H

 yk  H n  yk  ''

n

, k  1 1  n  1

and

(1.8)

Wk

x



H

2

n

 x  1k  x 

H

n

'

'

2

H k 

, k  1 1  n .

where 1 k  x  and L k  x  are given by (1.2) and (1.3) respectively Theorem 2: Let the interpolated function f : R  R be continuous differentiable, such that  2k x    x f  x    x   0  k  0 ,1, ...    and  L im   x      x  f '  x   0 , w h e re   x  = e  L im

(1.9)

x 

2

,   < 1 .

Further, taking the numbers  k as:

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

k  0 e

(1.10)

2

 yk



2014

1   , k  1  1  n  1,     1, n 

 w  f '; 

where w is the modulus of continuity of f ', then n 1

(1.11)



Rn

f ,x 

n 1



n

 y k  U k  x     kV k  x   

f

k 1

k 1

f

 xk W k  x 

k 1

satisfies the relation:

e

 vx

2

x

f

 1  3  R n  x   0  lo g n w  f '; , v  2. n   

which holds on the whole real line and 0 does not depend on n and x.

Remark. w



f ,

 denotes the special form of modulus of continuity introduced by G. Freud [3] given by:

(1.12)

w



f ,



x  t

 Sup W

f

X

 tW

x

f

x

 T  x W

x

f C

R

t

where

T

and



L im

x  W

denotes

x f x 

x 

  x , fo r x  1 .  1, fo r x  1  

the

sup-norm L im

0 , th e n    w



f ,

in



C

 R .

If

and

0.

II. PRELIMINARIES. In this section, we shall give some well known result which we shall use in the sequel.

The differential equation satisfied by H n  x  is given by: ''

(2.1)

H

n

(2.2)

H

n

'

x

x

2 xH

 2nH

' n

x

n 1

2nH

n

x

 0

 x .

From (1.2), we have

(2.3)

1K

x  j

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0     1

j  k fo r

,

k  1 1  n

j  k

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(2.4)

   H   xk

x 

'

1k

j

x  , x x  x  '

H

' n

2014

n

j

k

j

j  k

k

j  k.

,

Form (1.3), one has

(2.5)

(2.6)

y 

Lk

j

(2.7)

H

n

(2.8)

H

n

'

n 1

(2.9)



fo r

k  1 1 n  1

, j  k

2

k

2

xk 

j  k

0     1

n



x

 0 n

x  H

c2

1/ 4

2 1

n

n 2 

H



i

n

x

2/2

 , x  R

 y  H n 1  x  

2 i!

i0

xe

  x k2 !e , 0    1.  

 yHi x

i



2 n! 1 3

2

n

H

n 1

 n  1 ! y

yH n x

 x

From (1.2) and (2.9) at y  x k , we have v

(2.10)

x

1k



2



e

n!

H

n

(2.11)

0 1  2

n

 xk

0



' 2 n



n ,

ne

1

2

x

2

2

 xk

 , v1  1

 xk 

w h e re

 0

k 1

n

(2.12)

2



e

 xk

 H  x  '

n

2

 0 2

k

n 1

n !

1

, 0  1

k 1

and  n   2    :  2  

2

n

(2.13)

 n  1 :

 n

1/ 2

,

n = 1 ,2 ,...

III. PROOF OF THEOREM 1.

Using the results given in preliminaries and a little computation, one can easily see that the polynomials given (1.6), (1.7) and (1.8) satisfy the conditions: For k  1  1  n  1

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(3.1)

  U y  k  i    and  U    

k



j  k

0    1 

x  

fo r

j  1  1  n  1, U

,

' k

y  j

0 , j  1 1  n  1

j  k

j  1 1  n

0,

j

2014

For k  1  1  n  1

(3.2)

  V y  k  j    and

  0,

j  1  1  n  1, V k

'

y  j

 V  k   

 x   0,

j  1 1  n

W k    W k  

 y   0,

j  1  1  n  1, W k

j

j  k

0    1 

, j  1 1  n  1

fo r j  k

For k  1  1  n

(3.3)

j

x  j

'

 y   0, j

j  1 1  n  1

j  k

0    1 

, j  1 1  n

fo r j  k

IV. TO PROVE THEOREM 2, WE NEED Lemma 4.1 For k  1  1  n  1 a n d x     ,   , we have, 1 2   x  y2k  n 2  2 n !e  0  2 n H n  yk    v

Lk

x

  , v1  1 a n d k  1 1    

where L k  x  is given by (1.3).

Proof.

From (2.9) at y  y k and using (1.3) and (2.2), we get L k

x



2

n

H

n 2 n

 1!

 yk 

n 1

 i0

1 i

2 i!

H

i

x

H

i

 yk 

,

which on using (2.7) leads the lemma.

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V. Estimation of the fundamental polynomials Lemma 5.1: For k  1  1  n  1 a n d x     ,   n 1



2

e

 yk

U

x

k

 0





n e

2

vn

3

,v 

a n d 0    1,

2

k 1

where U k  x  is given by (1.6).

Proof.

From (1.6), we have when x  y k | n

1/ 2

n 1



e

2

n 1

2

 yk

U

x

k



k 1

e



 yk

H

k 1

2

n 1



x

2

Lk

2e



 yk

H

x

n

 yk 

n

.

xk x  yk Lk 2

H

k 1

xH n x

 yk 

n

 I1  I 2

(5.1)

Using (2.7), (2.13) and lemma 4.1, we get (5.2)

I1  0





n e

x

2

3

,v 

2

Similarly, owing to (2.6), (2.7), (2.13) and lemma 4.1, we have (5.3)

I2  0





n e

x

2

3

,v 

2

On combining (5.2) and (5.3), we get the lemma.

When x  y k  n

1/ 2

, using (1.3), we have

n 1



n 1

2

e

 yk

U

k

x



k 1

2

e

 yk

H H

k 1

n

n

x

2

Lk

 yk 

2

n 1



 k 1

2e

 yk

x

yk H H

n

n

x

 yk 

H '

H 'n

' n

x

Lk

x

 yk 

 I3  I4

From lemma 4.1,(2.7) and (2.13) we get

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(5.4)





n e

vx

2

2014

3

, v 

2

Similarly, using (2.6), (2.7), (2.13), lemma 4.1, (2.1) at x  y k and (2.2) we get I4  0

(5.5)





n e

vx

2

3

, v 

2

Owing to (5.4) and (5.5), we get the lemma.

Lemma 5.2 For k  1  1  n  1 a n d x     ,   , we have 2

n 1



e

n 1

2 yk



Vk  k





k 1



e

 yk

n

x

n

 yk 

H H

k 1

H

' n

x

H

'' n

Lk

x

 yk 

Using (2.1) at x  y k , (2.2), (2.7), (2.13) and lemma 4.1, we get the required lemma.

Lemma 5.3 For k  1  1  n a n d x     ,   n



2

e

 xk

Wk

x



 0 e

vx

2

,v 

3

and 0    1.

2

k 1

Where W k  x  is given by (1.8).

Proof.

From (1.8), we have 2

n



e



n

2 xk

Wk

x

k 1





e

 xk

H

' 2 n

H

k 1

 x  1k  x 

' 2 n

 xk 

Using (2.8), (2.10), (2.12) and (2.13), we get the lemma.

VI. IN THIS SECTION, WE MENTION CERTAIN THEOREMS OF G. FREUD AND L. SZILI REQUIRED IN THE PROOF OF THEOREM 2.

Theorem (G. Freud, Theorem 4[4] and theorem 1[3]) Let f : R  R be continuously differentiable. Further, let L im

x   x

2k

 x f

 x  0,

k  0 ,1, 2 , .... .

and L im

x   x

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2k

  x  f ' x   0,

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American Journal of Engineering Research (AJER)

2014

then there exist polynomials Q n  x  of degree  n , such that (6.1)

 x f

1   1    0    f '; , x  R, n   n  

 x   Qn  x 

where  stands for modulus of continuity defined by (1.12) and   x  the weight function.

Szili ([11] lemma 4, theorem 4) established the follow (6.2)

r

  x  Qn

x

 0  1  , r  0 ,1 : x  R

VII. PROOF OF THE MAIN THEOREM 2. n 1

(7.1)

Qn  x  



n 1

Qn



yk U

k

x



k 1

n



'

Qn

yk Vk

x

k 1



Q n  xk



.

Wk

x

k 1

From (7.1) and (1.11), we have Rn e

 vx

x 2

f

x

Rn  x   f

 Rn



x

 e

 vx

 e

 vx

f  Qn 2

x

Qn  x  f n 1

2

 Qn  x  f



2

e

  yk

f

Y 

x  yk   Qn  yk 

U

k

2

x

e

  yk

k 1 n 1

 e

 vx

2



 k Vk  x 

k 1 n 1

 e

 vx

2



2

e

  yk

'

Qn

 yk 

Vk

x

2

e

  yk

k 1 n

 e

 vx

2



2

e

  yk

f

 xk 

 Qn Wk

x

2

e

  yk

k 1

Owing to (6.1), (6.2), (1.10) and lemmas 5.1-5.3, theorem follows

Acknowledgement In this Paper I have got good cooperation from Prof. K.K. Mathur.

REFERENCES [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11]

Blazs, J.: Sulyosott (0,2)-interrolacio ultraszferibus polinomk gyokein, MTA III; Oszt. Kozl, 11(1961), pp. 305-338.

Enduanya, S.A. : On the convergence of interpolation polynomials, Analysis, Maths., 11(1985), pp. 12-22.

Freud, G.: On polynomials approximation with the weight exp  1 ibid, 24 (1973), pp. 367-371.  x   2 

2 k

 , 

Freud, G.: On two polynomial inequalities I, Acta Math, Acad. Sci. Hung., 22 (1971), pp. 109-116.

Freud, G.: On two polynomial inequalities II, ibid., 23 (1972) pp. 137-145.

Mathur, K.K. and Sexena, R.B.: On weighted (0,1,3)-interpolation on the abscissas as zeros of Hermite polynomials. Acta. Math Hung., 62 (1-2) (1993), pp. 31-47.

Mathur, K.K. and Srivastava Rekha.: Pal-type Hermite Interpolation on Infinite- interval, Journal of Mathematical Analysis and Applications 192, 346-359 (1995).

Pal L.G.: A new modification of the Hermite Fejer interpolation, Analysis Math. 1 (1975), pp. 197-205.

Szego, G.: Orthogonal polynomials Amer. Math. Soc. Coll. Publ. New York, 1959.

Szili, L.: A convergence theorem for the Pal method of interpolation on the roots of Hermite polynomials, Anal. Math. 11(1985) pp. 75-84.

Szili, L.: Wieghted (0,2)-interpolation on the roots of Hermite polynomials. Annales Univ. Sci. Budapest. Eotvos Sect. Math. 27 (1984), pp. 153-166.

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