Journal of Applied Mathematics

Volume 2012 (2012), Article ID 482935, 25 pages

doi:10.1155/2012/482935

Research Article

Asymptotic Properties of Derivatives of the 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 16 March 2012; Accepted 24 May 2012

Academic Editor: Jin L. Kuang

Copyright © 2012 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 by satisfying , , , . In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials and the product . Especially, we estimate the even-order derivative values of and at the zeros of and the product , respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of and at the zeros of and on a closed subset of , respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-Fejér interpolation polynomials.

1. Introduction

Consider the generalized Stieltjes polynomials defined (up to a multiplicative constant) by

where , , and is the th ultraspherical polynomial for the weight function .

The polynomials , introduced by Stieltjes and studied by Szegö, have been used in numerical integration, whereas the polynomials have been used in extended Lagrange interpolation. In this paper, we will prove pointwise and asymptotic estimates for the higher-order derivatives of and . It is well known that these kind of estimates are useful for studying interpolation processes with multiple nodes.

In 1934, G. Szegö [1] showed that the zeros of the generalized Stieltjes polynomials are real and inside and interlace with the zeros of whenever . Recently, several authors [2–8] studied further interesting properties for these Stieltjes polynomials. Ehrich and Mastroianni [3, 4] gave accurate pointwise bounds of and the product on , and they estimated asymptotic representations for and at the zeros of and , respectively. In [6], pointwise upper bounds of , , , and are obtained using the asymptotic differential relations of the first and the second order for the Stieltjes polynomials and . Also the values of and at the zeros of and are estimated in [6]. Moreover, using the results of [6], the Lebesgue constants of Hermite-Fejér interpolatory process are estimated in [7].

In this paper, we find pointwise upper bounds of and for two cases of an odd order and of even order. Using these relations, we investigate asymptotic properties of derivatives of the Stieltjes polynomials and and we also estimate the values of and at the zeros of and , respectively. Especially, for the value of at the zeros of , we will estimate and for an odd at the zeros of and , respectively. Finally, we investigate asymptotic representations for the values of and at the zeros of and on a closed subset of , respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-Fejér interpolation polynomials.

This paper is organized as follows. In Section 2, we will introduce the main results. In Section 3, we will introduce the known results 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 and then we know that . 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 there exists a constant , independent of (and ) such that for large enough and write if and . We denote by the space of polynomials of degree at most .

For the Chebyshev polynomial , note that for and

Therefore, we will consider for .

Theorem 2.1. Let and be a positive integer. Then, for all ,

Moreover, one has

and especially one has, for ,

Theorem 2.2. Let and be a positive integer. Then, for all ,

Moreover, one has

and especially, for ,

In the following, we also estimate the values of and , at the zeros of and the zeros of , respectively.

Theorem 2.3. Let and be an even integer. For , one has

Theorem 2.4. Let and be an even integer. For , one has

Finally, we obtain the asymptotic representations for the values of and at the zeros of and on a closed subset of , respectively.

Theorem 2.5. Let and . Suppose . Then,

(a)

(b)

In addition,

Theorem 2.6. Let and . Suppose . Then,

(a)

(b)

In addition,

Theorem 2.7. Let and . Suppose . Then, one has, for a positive integer ,

where .

3. The Known Results

In this section, we will introduce the known results in [4, 6, 9] to prove main results.

Proposition 3.1. (a) Let . Then, satisfies the second-order differential equation as follows:

(b) Let . Then,

(c) Let . Then, for ,

(d) Let . Then, for ,

(e) Let . Then, for and ,

(f) Let . Then, for ,

(g) Let and . Then, satisfies the higher-order differential equation as follows:

Proof. (a) It is from [9, (4.2.1)]. (b) It is from [9, (4.7.14)]. (c) It is from [6, Lemma 3.4]. (d) It is from [9, (8.9.7)]. (e) For , it follows from [9, (7.33.5)], and, for , it comes from (b) and the case of . (f) It is from [6, Lemma 3.3 (3.23)]. (g) Equation (3.7) comes from (a).

Proposition 3.2 (see [4]). Let . Let , and , . Then, for and ,

where and .

Proposition 3.3 ([6, Proposition  2.3]). Let . Then, for all ,

where

Then is a polynomial of degree satisfying

Proposition 3.4 ([4, Theorem  2.1]). Let . Then, for ,

Furthermore, .

Proposition 3.5 ([6, Theorem  2.5]). Let .

(a)For all ,

Moreover, one has, for ,

(b)For all ,

Moreover, one has, for ,

Proposition 3.6 ([6, Corollary  2.6]). Let . Then, for all ,

Here, is a polynomial of degree of defined in (4.37) such that, for ,

and, for ,

Proposition 3.7 ([6, Corollary  2.7]). Let .

(a)For all ,

Moreover, one has, for ,

(b)For all ,

Moreover, one has, for ,

Proposition 3.8 ([4, Lemma  5.5]). Let . Then, for ,

and, for ,

We now estimate the second derivatives at the zeros of and .

Proposition 3.9 ([6, Theorem  2.9]). Let . Then, for ,

and, for ,

4. The Proofs of Main Results

In this section, we let and . A representation of Stieltjes polynomials is (cf. [1, 10])

where

In the following, we state the asymptotic differential relation of the higher order of .

Lemma 4.1. Let . Then, for all and ,

and, for ,

Here, is a polynomial of degree defined in (3.10);

such that

Proof. For , (4.3) is obtained by times differentiation of (3.9). Equation (4.6) follows by (3.11) and the use of Markov-Bernstein inequality. Now, we prove (4.4). We know that the Chebyshev polynomial satisfies the second-order differential equation

so we have, by times differentiation of (4.7),

Let for a nonnegative integer ,

Observe that in the view of Szegö’s result (cf. [1])

Then, since (note (4.10)), we will prove that, for and ,

instead of (4.4). Since, from the proof of [6, Proposition  2.3],

and, for ,

we obtain that and . Using (4.8), we have, for ,

Therefore, (4.11) is proved by the mathematical induction on . Consequently, we have (4.4).

We obtain pointwise upper bounds of for two cases of an odd order and an even order in the following.

Lemma 4.2. Let and . Let . If is even, then one has

and, if is odd, then one has

Proof. Let . From (4.3) and (4.4), we have, for ,

and especially

that is, we have (4.15) for . From Proposition 3.2, we see , , so we have, for ,

Then, from (4.17) with and (4.18), we know that

that is, we have (4.16) for . Assume that (4.15) and (4.16) hold for times differentiation. Let be an even number. Then, we have, from (4.17), (4.19), and the assumptions for and ,

that is, we have (4.15). Similarly, we also have (4.16) for an odd .

Lemma 4.3. Let and

Then, is a polynomial of degree of and has distinct real zeros in . Moreover, if one lets be the zeros of , then is interlaced with the zeros of that is, , .

Proof. Since the sign of is , it is proved.

Lemma 4.4. Let be a nonnegative integer. Then, has distinct real zeros on . If one lets be the zeros of the polynomial with

then one has, for and ,

Proof. From Lemma 4.3, we know that has distinct real zeros on . By the interlaced zeros property of Lemma 4.3, we see that, for ,

Thus, (4.24) is proved.

Proof of Theorem 2.1. Let . Equation (2.2) comes from (4.15), (4.16), (3.12), and (3.13). From Propositions 3.4, 3.5, and (4.19), we have

Hence, using the Markov-Bernstein inequality, we have (2.3). To prove (2.4), we will use the mathematical induction. We use (3.14). The formula (2.3) holds for from (3.14). We suppose that, for and ,

Then, by Lemma 4.4 and (3.8), we have, for and ,

Here, the last inequality is obtained by Proposition 3.2, that is,

Therefore, we have, for and ,

Hence, from (2.3), we have (2.4). For , the proof is similar.

Lemma 4.5. Let be a nonnegative integer and . Then,

Proof. (a) When , it is obvious from (3.5) and (3.12). Now, suppose . From (3.12), (2.2), and (3.5), we have

(b) From (4.4) and (3.5), we have

(c) Similarly to the proof of (a), we have, from (2.2) and (3.5),

Lemma 4.6. Let . Then, for all and ,

and, for ,

Here, is a polynomial of degree of defined as follows:

Furthermore, one has

Proof. Similarly to the proof of Lemma 4.1, (4.35) is obtained by times differentiation of the second-order differential relation with respect to , that is, (3.17). So it is sufficient to prove (4.36) and (4.38). From (4.37), we know that

By Lemma 4.5 (a) and (b), we have, for ,

From (4.19) and Lemma 4.5 (c), we have, for and ,

When , we can similarly obtain that

Therefore, we have (4.36). On the other hand, from (2.4), (4.6), and (3.2), we know that, for a nonnegative integer ,

Then, similarly to the proof of Lemma 4.5, we obtain that

Therefore, we have (4.38).

Lemma 4.7. Let . Then, for , if is even, one has, for ,

and, if is odd, one has

Proof. Using (4.35) and (4.36), we obtain the result similarly to the proof of Lemma 4.2.

Proof of Theorem 2.2. Equation (2.5) comes from Lemma 4.7 and Proposition 3.7. We will show (2.6) and (2.7). From Proposition 3.7, we see

Hence, using Markov-Bernstein inequality for , we have (2.6). Now, we show (2.7). By Proposition 3.7 (a), it is true for . We suppose that, for ,

As the proof of Theorem 2.1, we have, for and for ,

Therefore, we see that by induction with Proposition 3.7, (2.7) holds for every .

Corollary 4.8. Let and . Then, for ,

Proof. Corollary 4.8 comes from (4.35), (2.7), and (3.8).

Proof of Theorem 2.3. Equation (2.8) comes from (4.15) and (3.24).

Lemma 4.9. For and ,

Proof. Since we know from (3.24) and (3.25) that

(4.51) is obviously proved. Similarly, since we have, from (3.4) and (3.25),

(4.52) is obtained.

Lemma 4.10. Let and . Let be an odd integer.

(a)For ,

(b)For ,

Proof. (a) We know, from (3.3),

So (4.55) holds for . Assume that, for ,

Then, we have from (3.5), (3.7), and (4.58) that

Therefore, we have the result using the mathematical induction.

(b) For an odd integer , we have from (3.6), (4.16), and (4.52)

Lemma 4.11. Let . If is even, then

Proof. It is easily proved from (3.4) and (3.7).

Lemma 4.12. Let be a positive integer. Then, one has, for ,

Proof. From (4.37), we know that

From Lemma 4.5 (a) and (b), we know that for ()

On the other hand, we estimate splitting into three terms as follows:

Here, from Lemma 4.5 (c), we have for

For the first term, we also split into two terms as follows:

From (2.2) and (4.55), we know that for even and

Also, we know from (4.56) and (3.5) that for even and

Then, we have

Similarly, for odd and , we have, by (2.8) and (3.5),

and, by (2.2) and (4.61),

Thus, we have

Therefore, we have the result.

Proof of Theorem 2.4. When , (2.9) holds from (3.27). Let even , and suppose that (2.9) holds for . Since we know by (4.35), (2.5), (4.62), and (4.19)

we obtain, using mathematical induction,

Therefore, (2.9) is proved.

Proof of Theorem 2.5. (a) From (4.3), we know that

Therefore, we have, by (2.2), (3.24), and (3.26),

Suppose that, for an integer ,

Then, from (4.3), we obtain

Therefore, by (2.2), (2.8), and (4.4), we have

(b) Similarly to the proof of (a), by (4.51), (3.3), and (3.7), we can obtain

In addition, we see that, from (4.51),

Proof of Theorem 2.6. (a) From (3.9), we know that

Therefore, by (4.4) and (4.56), we have

Then, we obtain from (4.3), (2.2), and (4.56) that

Therefore, we have the result inductively.

(b) From (3.7), we know that

Therefore, by (3.4) and (4.61), we have

Suppose that, for an integer ,

Then from (3.5), (3.7), and (4.61)

Here, we see that for

Hence, we obtain, from (3.4),

Lemma 4.13. Let and . Then, for a nonnegative integer ,

Proof. Let . From (4.37) and Lemma 4.5, we see that

Here, we let when . To estimate the first term, we split it into two terms as follows:

Then, using , from (2.13) and (2.15), we obtain

and, from (4.56) and from (4.61),

Thus, we have, for ,

Similarly, noting , from (2.10) and (2.12)

and, from (2.8) and (4.55),

Then, we have, for ,

Therefore, we have, for ,

Thus, we have, for ,

Proof of Theorem 2.7. From (4.35), (3.25), (3.27), and Lemma 4.13, we have

Suppose that

Then, we obtain from (4.35)

Therefore, (2.16) is proved.

Acknowledgment

The authors thank the referees for many kind suggestions and comments.

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