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Adv. Inequal. Appl. 2014, 2014:13 ISSN: 2050-7461

A NEW KIND OF HERMITE INTERPOLATION

S. BAHADUR∗, M. SHUKLA2

Department of Mathematics and Astronomy, University of Lucknow, Lucknow 226007, India

Copyright c2014 Bahadur and Shukla. 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 study the convergence of Hermite interpolation polynomials on the nodes obtained by projecting vertically the zeros of 1−x2Pn(α,β)(x), wherePn(α,β)(x)stands for the Jacobi polynomial.

Keywords: Jacobi polynomial, explicit representation, convergence.

2000 AMS Subject Classification:41A05

1. Introduction

In a paper, Goodman and Sharma [3] considered convergence and divergence behaviour of Hermite interpolation in the circle of radiusρ

3

2. In [4], Goodman, Ivanov and Sharma

consid-ered the behaviour of the Hermite interpolation in the roots of unity. In [1], Bahadur and Math-ur proved the convergence of quasi-Hermite interpolation on the nodes obtained by projecting vertically the zeros of 1−x2Pn (x) on the unit circle, where Pn(x) stands fornth Legendre polynomial. Later on convergence of Hermite interpolation was considered in [8] on the same set of nodes. Recently, Berriochoaa, Cachafeiros and Breyb [2] studied the convergence of the

Corresponding author

E-mail address: swarnimabahadur@ymail.com (S. Bahadur); manishashukla2626@gmail.com (M. Shukla) Received August 28, 2013

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Hermite -Fej´er and the Hermite interpolation polynomials, which are constructed by taking e-qually spaced nodes on the unit circle. As a consequence, they achieved some improvements on Hermite interpolation problems on the real line.

In [7], Mathur and Saxena Investigated the convergence of Quasi-Hermite -Fej´er interpola-tion. In [11], Xie considered the regularity of(0,1,2, ...,r−2,r)*-interpolation on the nodes obtained by projecting vertically the zeros of 1−x2Pn(α,β)(x) onto the unit circle, where

Pn(α,β)(x) stands for the Jacobi Polynomial. In [9], Szabo gave a generalization of P´al-type interpolation for the zeros of Jacobi Polynomial. In [6], Lenard studied the convergence of the modified(0,2)- interpolation procedure if the inner nodal points are the roots of the Ultraspher-ical polynomials with odd integer parameter.

In this paper, we consider the zeros of 1−x2Pn(α,β)(x), which are projected vertically onto the unit circle. In Section 2, we give some preliminaries. In Section 3, we describe the problem and obtain the existence of interpolatory polynomials. In Section 4, explicit formulae of inter-polatory polynomials are given. In Section 5 and Section 6, the estimation and convergence of interpolatory polynomials are considered, respectively.

2. Preliminaries

In this section, we give some well known results. The differential equation satisfied byPn(α,β)(x)is

(2.1) (1−x2)Pn(α,β)00(x) + β−α−(α+β+2)xP(α,β)

0

n (x) +n(n+α+β+1)Pn(α,β)(x) =0,

(2.2) W(z) =

2n

k=1

(z−zk) =KnPn(α,β)

1+z2

2z

zn,

(2.3) R(z) = z2−1W(z).

We shall require the fundamental polynomials of Lagrange interpolation based on the nodes as zeros of R(z)is given by

(2.4) Lk (z) = R(z)

R0(zk) (z−zk), k =0(1)2n+1

We will also use the following results

(2.5) (−1)nW0(zn+k) =W0(zk)

=−12KnPn(α,β)0(xk) 1−z2k

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(2.6) (−1)n−1W00(zn+k) =W00(zk)

=−12KnPn(α,β)0(xk)

2(β−α)zk−(α+β+2) 1+z2k

+2n 1−z2k−2znk−3, k=1(1)n

(2.7) R0(zk) = z2k−1W0(zk) (2.8) R00(zk) =W0(zk)

4z2k+2(β−α)zk−(α+β+2) 1+z2k

−2n 1−z2k+2z−k1.

We will also use the following well known inequalities(see [6], [10]) (2.9) 1−x2

1 2P(α,β)

n (x) = o nα−1

,

for α >0 ,x∈[−1,1] (2.10) 1−xk2 −1∼ kn−2

,

(2.11)

P

(α,β)0

n (xk)

∼ k

−α−3

2 nα+2.

3. THE PROBLEM AND REGULARITY

Let

(3.1) Zn=

 

z0=1,z2n+1=−1

zk=cosθk+isinθk,zn+k=−zk,k=1(1)n

be the vertical projections on the unit circle of the zeros of 1−x2Pn(α,β)(x), we determine the interpolatory polynomials Rn(z)of degree≤4n+3 satisfying the conditions:

(3.2)

 

Rn(zk) = αk ; k=0(1)2n+1

R0n(zk) = βk ; k=0(1)2n+1

where αk and βk are arbitrary complex numbers and establish the convergence theorem of

Rn(z).

Theorem 1. Hermite interpolation is regular on Zn.

Proof. It is sufficient if we show the unique solution of(3.2)is Rn(z)≡0, when all data αk

=βk=0.Clearly in this case we have Rn(z) =R(z)q(z),whereq(z)is a polynomial of degree ≤2n+1.

AsR0n(zk) =0 k=0(1)2n+1

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We haveq(zk) =0. It follows that

q(z) = (az+b)W(z)

Asq(±1) =0,we geta=b=0,which gives thatq(z)≡0 leading toRn(z)≡0.This completes the proof.

4. EXPLICIT REPRESENTATION OF INTERPOLATORY

POLYNO-MIALS

We shall write Rn(z)satisfying(3.2)as

(4.1) Rn(z) =

2n+1

k=0

αkAk(z) + 2n+1

k=0

βkBk(z),whereAk(z)andBk(z)are fundamental polynomials of the first and second type respectively , each of degree atmost 4n+3 satisfying the conditions:

For j, k=0(1)2n+1,

(4.2)

 

Ak zj=δjk,

A0k zj

=0

(4.3)

 

Bk zj

=0, B0k zj=δjk.

Theorem 2. For k=0(1)2n+1,we have (4.4) Bk(z) = R(z)Lk (z)

R0(zk)

.

Theorem 3.For k=0(1)2n+1,we have

(4.5) Ak(z) =L2k (z)−2L0k(zk)Bk(z).

One can prove theorems 2 and 3 owing to(4.3)and(4.2)respectively.

5. ESTIMATION OF FUNDAMENTAL POLYNOMIALS

Lemma 1.Let Lk(z)be given by(2.4). Then

(5.1) max

|z|=1 2n+1

k=0

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where c is a constant independent of n and z .

Proof.From maximal principal, we know

λn =max

|z|=1 λn(z)

λn = 2n+1

k=0

|Lk(z)|.

Letz=x+iyand|z|=1. Then we see, for 0≤argz<π andk=1,2, ...n.

| Lk(z)|=

z2−1W(z)

z2k−1W0(zk) (z−zk) .

Using(2.2)and (2.5), we get that

| Lk(z)|=

1−x2

1 2P(α,β)

n (x)

(1−xxk) + 1−x2

1

2 1x2

k

1

2

12

2 1−x2kPn(α,β)0(xk) (x−xk)

≤ 1−x 212

Pn(α,β)(x) (1−xxk)12

1−x2kPn(α,β)0(xk) (x−xk)

= Gk (x).

Also, we have| Ln+k(z)| ≤ Gk (x).

Similarly forπ≤argz<2π andk=1,2, ...n , we have

| Lk(z)| ≤ Gk (x), | Ln+k (z)| ≤ Gk (x).

For a fixedz=x+iy,|z|=1 and−1<x<1, we see that

λn(z) ≤ 2 n

k=1

Gk(x) +| L0(z)|+| L2n+1(z)|

= 2 ∑

|xk−x|≥12(1−x2k)

Gk(x) +2 ∑

|xk−x|<12(1−xk2)

Gk(x) + 2 Using(2.9),(2.10)and(2.11), we get the desired result.

Lemma 2.Let Bk(z)be given by(4.4). Then

(5.2) 2n

+1

k=0

|Bk(z)| ≤c logn n

where c is a constant independent of n and z .

Proof.In view of Lemma 1 and using(2.3),(2.7),(2.9),(2.10)and(2.11), we get the required result.

Lemma 3.Let Ak(z)be given by(4.5). Then (5.3) 2n∑+1

k=0

|Ak(z)| ≤clogn, where c is a constant independent of n and z .

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6.CONVERGENCE:

Let f(z) be analytic for|z| <1 and continuous for|z| ≤ 1 andω(f,δ)be the modulus of

continuity of f eix

THEOREM 4 :Let f(z)be continuous in|z| ≤1 and analytic in|z|<1. Let the arbitrary

numbersβk’s be such that

(6.1) |βk| =o nω f,n−1

, k=1(1)2n

Then Rnbe defined by

(6.2) Rn(z) =

2n+1

k=0

f(zk)Ak(z) + 2n+1

k=0

βkBk(z)

satisfies the relation

(6.3) |Rn(z)−f(z)|=o ω(f,n−1)logn,

whereω f,n−1is the modulus of continuity of f(z).

To prove theorem 4, we shall need the following:

Let f(z)be continuous in|z| ≤ 1 and analytic in |z| <1.Then there exists a polynomial of degree 2n−2 satisfying Jackson’s inequality

(6.4) |f(z)−Fn(z)| ≤ cω f,n−1 , z=eiθ(0<θ ≤2π)

and also an inequality due to O.Kiˇs[5] (6.5)

F

(m)

n (z)

≤ cn

mω f,n−1

form=1.

PROOF: SinceRn(z)be given by(6.2) is a uniquely determined polynomial of degree≤ 4n+3 , the polynomialFn(z)satisfying(6.4)and (6.5) can be expressed as

Fn(z) = 2n+1

k=0

F(zk)Ak(z) +2n

+1

k=0

Fn0(zk)Bk(z)

Then , |Rn(z) −f(z)| ≤ |Rn(z)−Fn(z)|+|Fn(z)−f(z)| ≤2n∑+1

k=0

| f(zk)−Fn(zk)| |Ak(z)|

+

2n+1

k=0

{ |βk|+Fn0(zk)} |Bk (z)|+|Fn(z) −f(z)|

Using z=eiθ (0<

θ ≤2π),(6.1),(6.4),(6.5)and Lemma 2 and 3 , we get(6.3) Conflict of Interests

The authors declare that there is no conflict of interests.

REFERENCES

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[2] E. Berriochoaa, A. Cachafeirob, B.M. Breyb, Some improvements to the Hermite - Fej´er interpolation on the unit circle and bounded interval, Comput. Math. Appl. 61 (2011) 1228-1240.

[3] T.N.T. Goodman, A. Sharma, A property of Hermite interpolation in the roots of unity, Ganita, 43 (1992) 171-180.

[4] T.N.T. Goodman, K.G. Ivanov, A. Sharma, Hermite interpolation in the roots of unity, J. Approx. Theory 84 (1996) 41-60.

[5] O. Kiˇs, Remarks on interpolation (Russian); Acta Math. Acad. Sci. Hungar 11 (1960) 49-64.

[6] M. Lenard, Modified (0,2)interpolation on the roots of Jacobi Polynomials II (convergence), Acta Math. Hungar, 89 (2000) 161-177.

[7] K.K. Mathur, R.B. Saxena, On the convergence of Quasi-Hermite-Fej´er interpolation, Pacific J. Math. 20 (1967) 245-259.

[8] P. Mathur, K.S. Ashwini, Hermite interpolation on unit circle, J. Ultra. Phys. Sci. 21 (2010).

[9] V.E.S. Szabo, A generalization of P´al-type interpolation IV The Jacobi case, Acta Math. Hungar, 74 (1997) 287-300.

[10] G. Szeg˝o, Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ. New York, (1959).

References

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