Journal of Applied Mathematics

Volume 2013 (2013), Article ID 542653, 15 pages

http://dx.doi.org/10.1155/2013/542653

Research Article

Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

Hee Sun Jung¹ and Ryozi Sakai²

¹Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea

²Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 5 July 2013; Accepted 22 August 2013

Academic Editor: Qiankun Song

Copyright © 2013 Hee Sun Jung and Ryozi Sakai. 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.

Let and be the ultraspherical polynomials with respect to . Then, we denote the Stieltjes polynomials with respect to satisfying . In this paper, we consider the higher-order Hermite-Fejér interpolation operator based on the zeros of and the higher order extended Hermite-Fejér interpolation operator based on the zeros of . When is even, we show that Lebesgue constants of these interpolation operators are and , respectively; that is, and . In the case of the Hermite-Fejér interpolation polynomials for , we can prove the weighted uniform convergence. In addition, when is odd, we will show that these interpolations diverge for a certain continuous function on , proving that Lebesgue constants of these interpolation operators are similar or greater than log .

1. Introduction

Let and

For any real-valued function on and an integer , we recall that there exist unique Hermite and Hermite-Fejér interpolatory polynomials of higher order denoted by , and of degree , defined as follows:

We note that, by definition, is the Lagrange, is the Hermite-Fejér, and is the Krylov-Stayermann interpolatory polynomial. By (2), we may write

The polynomials

are unique, of degree exactly and satisfy the relations

where for nonnegative integers and

Here, are the well-known fundamental Lagrange polynomials of degree given by

and the coefficients may be obtained from the relations

If , then the Hermite interpolation polynomial of degree with respect to is defined by

We may express as

where for

is the unique polynomial of degree satisfying

Then, we easily see from the relations (5) and (12) that , , and for and (see [1]). Now, we have for any polynomial of degree ,

In what follows, we abbreviate several notations as , , and if there is no confusion. Here, we are interested in Hermite-Fejér and Hermite interpolations with respect to whose elements are the zeros of a sequence of Stieltjes polynomials and the product polynomials of Stieltjes polynomials and the ultraspherical polynomials, respectively. To be precise, we first consider the generalized Stieltjes polynomials defined (up to a multiplicative constant) by

where , , and is the th ultraspherical polynomial for the weight function . In 1935, Szegö [2] showed that the zeros of the generalized Stieltjes polynomials are real and inside and interlace with the zeros of whenever .

For the properties of interpolation operators based at the zeros of and the zeros of , Ehrich and Mastroianni [3, 4] proved that Lagrange interpolation operators based on the zeros of and extended Lagrange interpolation operators based on the zeros of have Lebesgue constants and of optimal order, that is, . For the Hermite-Fejér interpolation operator based on the zeros of and the extended Hermite-Fejér interpolation operator based on the zeros of , it is proved that Lebesgue constants and are of optimal order, that is, , in [5]. In this paper, we consider the higher-order Hermite-Fejér interpolation operator based on the zeros of and the higher-order extended Hermite-Fejér interpolation operator based on the zeros of . When is even, we show that Lebesgue constants of these interpolation operators are and , respectively; that is,    and   . In the case of the Hermite-Fejér interpolation polynomials for , we can prove the weighted uniform convergence. In addition, when is odd, we will show that these interpolations diverge for a certain continuous function on , proving that Lebesgue constants of these interpolation operators are similar or greater than .

This paper is organized as follows. In Section 2, we will introduce the main results. In Section 3, we will show the auxiliary propositions and estimate the coefficients of Hermite-Fejér interpolation polynomials in order to prove the main results. Finally, we will prove the results in Section 4.

2. Main Results

We first introduce some notations, which we use in the following. For the ultraspherical polynomials , , we use the normalization . We denote the zeros of by , , and the zeros of Stieltjes polynomials by , . We denote the zeros of by , . All nodes are ordered by increasing magnitude. We set , and for any two sequences and of nonzero real numbers (or functions), we write if and . We denote the space of polynomials of degree at most by .

For the Chebyshev polynomial , note that for and

In this paper, we let . In these cases, the results are well known or can easily be deduced. Therefore, we will consider the cases for and let in the following.

Let be the Hermite-Fejér interpolation polynomials of with respect to the zeros of . Also let be the Hermite-Fejér interpolation polynomials of with respect to the zeros of . The fundamental Lagrange interpolation polynomials and with respect to and , respectively, are given by

We let and be the Lebesgue constants based on the zeros of and , respectively. That is, the Lebesgue constants and are defined as follows:

and for a nonnegative real function ,

where and are the coefficients of the higher-order Hermite-Fejér interpolation polynomials defined in (4), with respect to and , respectively.

2.1. Uniform Convergence of Hermite-Fejér Interpolation Polynomials of Higher Order

Theorem 1. Let and .

(a) 

(b)Suppose that . Then, for a continuous function on one has uniformly for ,

Theorem 2. Let and .

(a) Then one has for

 and for

(b) For a continuous function on , if , then one has uniformly for ,

 and if , then

2.2. Divergence of Hermite-Fejér Interpolation Polynomials of Higher Order

Theorem 3. Let and . Then,

Theorem 4. Let and . Let . Suppose that

If , then

Theorem 5. Let and . Then,

Theorem 6. Let and . Let . Suppose that

If , then

If the Lebesgue constant is not bounded, then we know from Helley’s theorem that Hermite-Fejér interpolation does not converge for a certain continuous function on .

3. Estimation of the Coefficients of Higher-Order Hermite-Fejér Interpolation Polynomials

Proposition 7. Let .

(1)See [4, Theorem 2.1], for ,

Furthermore, .(2)See [4, Theorem  2.1] and [6,  ], for ,

Proposition 8 (see [4, Lemma  5.5]). Let . Then, for ,

and for ,

Proposition 9 (see [7, Theorem  2.1]). Let and a positive integer. Then, for all ,

Moreover, one has for ,

Proposition 10 (see [7, Theorem  2.2]). Let and a positive integer. Then, for all ,

Moreover, one has for ,

Proposition 11 (see [7, Theorems  2.3, 2.4]). Let and an even integer.

(a)For ,

(b)For ,

Proposition 12 (see [7, Lemma  4.9]). For , one has

Proposition 13 (see [7, Theorems  2.6, 2.7]). Let and .

(a)For and a positive integer , one has

(b)For and a positive integer , one has

where .

Theorem 14. Let , and .

(a) Uniformly for

and if is odd,

(b) Uniformly for ,

and if is odd,

Theorem 15. Let , and .

(a) Uniformly for ,

and if is odd, one has

(b) Uniformly for ,

and if is odd,

Theorem 16. Let and . Suppose that . Then, there exists a constant depending only on and such that

Theorem 17. Let and . Suppose that . Then, there exists a constant with depending only on and such that

Theorem 18. Let and . Suppose that . Then, there exists a constant depending only on and such that

Theorem 19. Let and . Suppose that . Then, there exists a constant with depending only on and such that

Proof of Theorem 14. We prove by induction on . Since

we know that

So, it holds for by (34) and (36). Now, assume that it holds for . Then, using Leibnitz’s rule for differentiation, we obtain

Suppose that is odd. Then,

Since is odd for an even and is even for an odd , we have by the mathematical induction on , (34), (36), (40), and (58),

These complete the proofs of (45) and (46). To prove (47) and (48), we proceed by induction on . Firstly, for , (47) is trivial since . For , we have by [8, ] and [8, ]

so that

Thus, if we assume that (47) holds for , , then by (45), we have

Suppose that is odd. Then,

Since is odd for an even and is even for an odd , we have by the mathematical induction, (45), (46), and (47)

These complete the proofs of (47) and (48).

Proof of Theorem 15. Using (35), (38), and (41), this is proved by the same method as the proof of Theorem 14.

For define and for

We rewrite the relation (67) in the form for ,

and for ,  ,

Now, for every , we will introduce an auxiliary polynomial determined by as the following lemma.

Lemma 20 (see [9, Lemma  11]).

(i)For , there exists a unique polynomial of degree such that

(ii) and , .

Lemma 21 (see [9, Lemma  13]). If , then for ,

Lemma 22 (see [9, Proof of Lemma  14]). For positive integers and ,

Proof of Theorem 16. Similarly to Theorem 14, we use mathematical induction with respect to . From (58), (43), and (34), we know that

Then, from the following relations:

we have the results by induction with respect to .

Proof of Theorem 17. We prove (54) by induction on . Since and , (54) holds for . From (63), we write in the form of

Then, by (46) and (48), is . For , we suppose (54). Then, since we know from (53)

we have for

Then, using Lemma 22 and , we have the following form:

Therefore, we proved the result.

Proof of Theorems 18 and 19. These theorems are proved by the same method as the above theorems.

4. The Proofs of Theorems

4.1. Proof of Theorems 1 and 2

From now on, we assume that . We first state some known results for the Stieltjes polynomials. Let , , , , and , . Ehrich and Mastroianni [4] proved that for and

where and . It implies that for and ,

where and .

Lemma 23. Let be a positive integer and . Then,

(a)for ,

(b)for and ,

(c)for ,

(d)for and ,

Proof. (a)

where and .

: Suppose that . Then, since , we have

Since we know for ,

we have

Therefore, we have for

: Since for ,

Therefore, we have

: Since for , we have for

Since for ,

with , we have

Therefore, we have the result (a).

(b) Similarly to (a), we let

Then, for , we have using (87)

by the use of . Therefore, we have for

: Since and for ,

Therefore,

: Since for , we have using (93)

and we know that

Therefore, we see that

Consequently, we have the result (b).

(c) and (d): Since , we know . Then, we have

because we see if , then

and if , then

that is, we have (c). Similarly,

Then, for

and for

On the other hand, by (97) and (99), we know that

Since for , we have

Therefore, we have

So, by (108) and (109) we have (d).

Lemma 24. Let . Let be a fixed, sufficiently small constant. Let for some . Then, one has uniformly for

Proof. For the simplicity, we denote

Then, we know that

Then, for some with , we have by (80)

Therefore, we have the result from Lemma 23 (a) with , .

For convenience, we let

then

Let

Lemma 25. Let be a positive integer. Then, one has for

Proof. By [10, (26)], it is sufficient to prove (120) for , where is a fixed, sufficiently small constant. In the following case, let for some . Then, we have from (32), (34), and (47)

The last inequality follows from Lemma 24. On the other hand, if , there exists between and such that we see from (34) and (36) with ,

Hence, from (47) and (80), we conclude that

Therefore, we have the result.

Lemma 26. Let be an even integer. Then,

Proof. First, we note that is odd. Let (, ) for some . Then, considering , as , , we have from (32), (34), (48), and Lemma 23(b)

where it is integrated under . Besides, we have, similarly to (123),

Therefore, we have the result.

Lemma 27. For any polynomial , one has

Proof. Let    for some .

Here, is the coefficient of the higher-order Hermite interpolation polynomial based on the zeros of , defined in (11). Since , we can see from (47) that uniformly for ,

Hence, using (32) and (34), we see for

Here, using Lemma 24 with as for , we have the right formula in the lemma. We also see that for

Consequently, we have

Proof of Theorem 1. (a) From Lemma 26, the result is trivially proved.

(b) Since is continuous on , for given , there exists a polynomial such that for

Then, one has from Lemmas 26 and 27

This implies (21).

Lemma 28. Let . Let be a fixed, sufficiently small constant.    for some . Then we have uniformly for

where

Proof. When , we can show the lemma as the proof of Lemma 24 (see (116)). If , we have from Lemma 23(c),

Let

then

Let

and for a nonnegative real function ,

Lemma 29. Let be a positive integer. If , then one has for

If , then one has for

Proof. Similarly to the proof of Lemma 25, using (51), we have from Proposition 7(2), Proposition 8 (35), and Lemma 28

where and are defined by (136) or (137). Besides, we easily know from (51), (35), (38), and (80)

Therefore, we have the result.

Let

then

Let

Lemma 30. Let be an even integer. Then, one has for

and for

Proof. First, we note that is odd. Similarly to the proof of Lemma 26, using Proposition 7(2), Proposition 8 (35), and (51), we have

where . If , considering and as and , respectively, we have from Lemma 23(b)

and if , considering and as and , respectively, we have from Lemma 23 (d)

Hence, we conclude that

Besides, we easily know that

Therefore, we have the result.

Lemma 31. For any polynomial , one has

Proof. We have using Lemma 28

Proof of Theorem 2. Using Lemmas 29, 30, and 31, it is similar to the proof of Theorem 1.

4.2. Proof of Theorems 3 and 4

Proof of Theorems 3 and 4. Suppose that , and let

From (17), we know that

Let be the least positive zero of . Then, we have

Since we know from (42), (34), and (47)

Thus, we have

Therefore, we have the result from Lemma 29.

Remark 32. Similarly, if we let be the least zero of with and if we consider for , then we have for

Proof of Theorem 4. Suppose that . Let be the least positive zero of . Then, similarly to the proof of Theorem 3, we have using the assumption (27)

Here, we used the followings:

because . Thus, we have , and it implies (28) from (20).

Proposition 33 (see [6, ]). For ,

Proposition 34 (see [11, Theorem]). For ,

Lemma 35. For ,

Proof. It is proved from (167) and (168).

Proof of Theorem 5. Suppose that , and let

From (18), we know that

Let . Then, we know from (169). Then, similarly to the proof of Theorem 3, we have from (35), (56), and (51), for ,

Therefore, we have the result from Lemma 29.

Proof of Theorem 6. Suppose that . Let . Then, since we know that as we see above, similarly to the proof of Theorem 4, we have using the assumption (30)

Here, we used the following:

because . Therefore, we have the result.

References

P. Vértesi, “Hermite-Fejér interpolations of higher order. II,” Acta Mathematica Hungarica, vol. 54, no. 3-4, pp. 369–380, 1990.

G. Szegö, “Über gewisse orthogonale polynome, die zu einer oszillierenden Belegungsfunktion gehören,” Mathematische Annalen, vol. 110, no. 1, pp. 501–513, 1935.

S. Ehrich and G. Mastroianni, “Stieltjes polynomials and lagrange interpolation,” Mathematics of Computation, vol. 66, no. 217, pp. 311–331, 1997.

S. Ehrich and G. Mastroianni, “On the generalized Stieltjes polynomials and Lagrange interpolation,” in Approximation Theory and Function Series, vol. 5 of Bolyai Society Mathematical Studies, pp. 187–203, 1996.

H. S. Jung, “Hermite and Hermite-Fejér interpolation for Stieltjes polynomials,” Mathematics of Computation, vol. 75, no. 254, pp. 743–766, 2006.

G. Szego, Orthogonal Polynomials, vol. 23, American Mathematical Society. Colloquium Publications, Providence, RI, USA, 1975.

H. S. Jung and R. Sakai, “Asymptotic properties of derivatives of the Stieltjes Polynomials,” Journal of Applied Mathematics, vol. 2012, Article ID 482935, 25 pages, 2012.

P. Vértesi, “Hermite-Fejér interpolations of higher order. I,” Acta Mathematica Hungarica, vol. 54, no. 1-2, pp. 135–152, 1989.

Y. Kanjin and R. Sakai, “Pointwise convergence of Hermite-Fejér interpolation of higher order for Freud weights,” Tohoku Mathematical Journal, vol. 46, no. 2, pp. 181–206, 1994.

P. Nevai, “Mean convergence of lagrange interpolation, III,” Transactions of the American Mathematical Society, vol. 282, pp. 669–698, 1984.

S. Ehrich, “Asymptotic properties of Stieltjes polynomials and Gauss-Kronrod quadrature formulas,” Journal of Approximation Theory, vol. 82, no. 2, pp. 287–303, 1995.