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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

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 haswhere .

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