ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 904169, 31 pagesdoi:10.5402/2012/904169

Research Article

Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Hee Sun Jung,¹ Gou Nakamura,² Ryozi Sakai,³ and Noriaki Suzuki³

¹Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea²Science Division, Center for General Education, Aichi Institute of Technology, Yakusa-cho, Toyota 470-0392, Japan³Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 30 December 2011; Accepted 26 February 2012

Academic Editors: B. Ricceri and W. Yu

Copyright © 2012 Hee Sun Jung et al. 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 let , where and is an even function. Then we can construct the orthonormal polynomials of degree for . In this paper for an even integer we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros of . Moreover, for an odd integer , we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros of .

1. Introduction

Let , and let be an even function. Consider the weight , and define, for , Suppose that , for all . Then we can construct the orthonormal polynomials of degree for ; that is, We write by and denote the zeros of by

Let denote the class of polynomials with degree at most . For we define the higher-order Hermite-Fejér interpolation polynomial based at the zeros as follows: We note that is the Lagrange interpolation polynomial, is the ordinary Hermite-Fejér interpolation polynomial, and is the Krylov-Stayermann polynomial. For the general cases Kanjin and Sakai [1, 2] started to investigate the so-called Freud-type weights. The fundamental polynomials for the higher-order Hermite-Fejér interpolation polynomial are defined as follows: Using them, we can write Furthermore, we extend the operator . Let be a nonnegative integer, and let . For we define the -order Hermite-Fejér interpolation polynomials as follows: for each , Especially, is equal to , and for each we see ; that is, for , is the Hermite interpolation polynomial. The fundamental polynomials , , of are defined by Then we have For the ordinary Hermite and Hermite-Fejér interpolation polynomial , and the related approximation process, Lubinsky [3] gave some interesting convergence theorems.

Our purpose in this paper is to study and as certain analogies of the Lubinsky theorems in [3] and the related approximation process for the exponential-type weights. Kasuga and Sakai [4–8] investigated the convergence and divergence theorems for the Freud-type weights. Then for an even integer we give the convergence theorems for them; moreover, for an odd integer , we obtain a certain divergence theorem with respect to . In Section 1, we give the preliminaries for these studies, and in Section 2 we write some preliminary description. In Section 3 we report our theorems with some lemmas, and in Section 4 we prove the theorems. Finally, in Section 5, for an odd integer we obtain a certain divergence theorem with respect to .

In what follows we abbreviate several notations as , , , and if there is no confusion. For arbitrary nonzero real-valued functions and , we write if there exist constants independent of such that for all . For arbitrary positive sequences and , we define similarly.

Throughout this paper denote positive constants independent of , or polynomials , and the same symbol does not necessarily denote the same constant in different occurrences.

2. Preliminaries

A function is said to be quasi-increasing if there exists such that for .

Definition 2.1 (see [9]). Let be a continuous even function satisfying the following properties.(a) is continuous in and . (b) exists and is positive in . (c). (d) The function is quasi-increasing in , with for .(e) There exists such that Then we say that is in the class . Besides, if there exist a compact subinterval of and such that then we say that is in the class .

Example 2.2. We present some typical examples of satisfying . If a continuous exponential satisfies with some constants , , then is called a Freud weight. The class contains the Freud weights. (See [9]) For and a nonnegative integer , we put where, for , and . (See [10]) For with , we put where for we suppose if ; otherwise. For we suppose and . (See [10]) For , we put For , we define the Mhaskar-Rakhmanov-Saff number by the equation We have the following estimates for the coefficients in (1.6) or (1.9).

Lemma 2.3 (see [11, Theorem 2.6]). Let . For each and , we have ,

If we consider the higher-order Hermite-Fejér interpolation polynomial on a certain finite interval, then we can see a remarkable difference between the parity of , for example, the Lagrange interpolation polynomial and the ordinary Hermite-Fejér interpolation polynomial ([12–17]). Also, we can see a similar phenomenon in the case of the infinite intervals ([1, 2, 4–8]). To describe these aspects, however, we need a further strengthened definition for than Definition 2.1.

Definition 2.4. Let , and let be an integer. Assume that is a -times continuously differentiable function on and satisfies the following.(a) exists and , , are positive for . (b)There exist constants such that(c)There exist and such that Then we say that is in the class . (d)Suppose one of the following.(d-1) is quasi-increasing on a certain positive interval .(d-2) is nondecreasing on a certain positive interval .(d-3)There exist constants and such that on .Then one says that is in the class .

Example 2.5 (cf. [10, Theorem 3.1]). Let be a positive integer, and let be defined in (2.7).Let and be nonnegative even integers with . Then . (a)If , then we see that is quasi-increasing on a certain positive interval and is nondecreasing on . (b)If , then we see that , is nondecreasing on . Hence, . Let . Then , and one has the following. (c)If and , then there exists a constant such that is quasi-increasing on . (d)Let . If , then there exists a constant such that is quasi-increasing on , and, if , then is quasi-decreasing on . (e) Let and , then is nondecreasing on a certain positive interval on .Hence, .

Definition 2.6. One uses the following notation:

Lemma 2.7 (see [18, Corollary 4.5]). Let , . If and , then and, for , where For one has

Remark 2.8. In [19, Theorem 2.2] we see that if is large enough. Therefore Lemma 2.7 holds for all , .

Levin and Lubinsky (see [9, Lemma 3.7]) showed that there exists such that for some and for large enough ,

In [20] we have the following estimations.

Lemma 2.9 (see [20, Theorem 1.6]). Let . Let be unbounded. Then, for any , there exists such that, for , Let be the constant in Definition 2.1(e), that is, If , then there exists such that and, if , then for any there exists such that

Remark 2.10. If is bounded, then is called a Freud-type weight, and, if is unbounded, then is called an Erdős-type weight. In (2.20) and (2.21), we set and Then (2.17) holds. If then we have (2.21). Note that all the examples in Example 2.5 satisfy this inequality. For the Freud-type exponent , we have The inequality (2.18) implies

3. Theorems

In the rest of this paper we assume the following for the weight .

Assumption 3.1. Consider the weight , , . (a)(cf. [20, Theorem 1.4]) If is bounded, then we suppose, for in (2.12), (b)There exist and such that here, if is bounded, that is, a Freud-type weight, then we set , and if is unbounded, that is, an Erdős-type weight, then we set . Define

Remark 3.2. If is unbounded, then (3.1) holds because of Lemma 2.9 (2.21). (3.2) holds for In (3.3) we note that as .

We shall state our theorems. Put Furthermore, we consider the class , and construct the following interpolation polynomial:

Then we have Define Here we note that, for some ,

Moreover, we define

Lemma 3.3. Let . Let be fixed. Then one has the following: (a)(See [9, Lemma 3.5(a)]) Uniformly for , (b)(See [9, Lemma 3.5(b)]) Uniformly for , Moreover, (c)(See [9, Lemma 3.11 (3.52)]) Uniformly for , (d)(See [9, Lemma 3.4 (3.17), (3.18)]) Uniformly for , one has

Lemma 3.4. For , we have

Proof. Let . By Lemma 3.3(d) we have So, we have Now, if , then So, we have Let . Then we have

Let denote a positive constant depending only on .

Assumption 3.5. Let and , . Suppose the following.(A-1)Let satisfy that, for a given ,where we suppose . (A-2)Let for a certain . Then we suppose that satisfies (A-3) Let . Then we suppose that there exists a constant such that

Remark 3.6. In (3.22), we have the following. and For some positive constant , we have . Hence, from (3.22), it follows that We have a chain of results under Assumption 3.1.

Proposition 3.7. Let . For satisfying (3.26), one has

Proposition 3.8. Let . For satisfying (3.25), one has where is defined by (3.3).

Proposition 3.9. Let . For satisfying (3.23), one has and for satisfying (3.24) one has

Proposition 3.10. Let . Let be fixed. Then one has

Proposition 3.11. Let . Let with be fixed. Then one has

Theorem 3.12. Let . For satisfying (3.22), one has

Theorem 3.13. Let . For satisfying (3.22) and (3.23), one has

Corollary 3.14. Let . For satisfying (3.22), one has

Theorem 3.15. Let . For satisfying (3.22), (3.23) and (3.24), one has

Define where equals to one of the following: One also defines

Theorem 3.16. Let . Let satisfy (3.22), (3.23) and (3.24). If then we have

For example, we can take in the following way. If is unbounded, we have, for , (see [10, Lemma 2.1(b)]). Then we set If is bounded and , , then we take as so that (3.43) can hold. Then defined by (3.43) satisfies (3.40).

4. Proof of Theorems

For constants , the same symbol does not necessarily denote the same constant in different occurrences.

Lemma 4.1. One has the following. (See [19, Theorem 2.3]) Let and . Then, uniformly for one has and for , (See [19, Theorem 2.5(d)]) Let and . Let and . Then (See [19, Theorem 2.5(c)]) Let and . Then one has (See [19, Theorem 2.5(a)]) Let and . For we have

Lemma 4.2 (see [19, Theorem 2.2]). Let and . For the zeros , one has the following: For and , For the minimum positive zero ( is the largest integer ⩽ ), one has and for large enough , (See [19, Lemma 4.7]) , where is defined by (1.3).

Lemma 4.3. Let . Then there exist such that

Proof. The first inequality follows from Lemma 3.4. We show the second inequality. Noting (4.8), from Lemma 4.1 (4.1) we have From (4.2) we see that Therefore we have the result.

Proof of Proposition 3.7. We recall the definition of : Using Lemma 2.3, we may estimate, for and , Let or . For simplicity, we assume , and let for a fixed small enough. Then we can assume that there exists a constant such that Assume that . Then we first estimate . By Lemma 4.1(3) and the definition of , we have and by Remark 3.6 Therefore, we have Next, we estimate . For , , we have, by Lemma 4.3 and Lemma 4.1(4), Then, since we know from Remark 3.6 that we have, by Lemma 3.4, By the definition of and by Lemma 3.3(a) , we have By Lemma 4.2, we have Here, we note from (4.14) and (4.8) that Thus, we have, for , Therefore, we have because of .Remark 4.4. If we consider the estimate of with (4.13), then we obtain We continue the proof Proposition 3.7. We need to estimate for . Now, suppose . Then similarly to (4.23), we have Similarly to (4.21), we have using (4.22) because of and . Consequently we achieve the result.

Remark 4.5. The above proof implies the following: there exists a constant such that

Proof of Proposition 3.8. We use the same method to prove of Proposition 3.7. So, we let or . For simplicity, we assume , and let for a fixed small enough and there exists a constant such that . Assume that . Since , we may leave out the term with . Hence we consider only the term of . Noting Assumption 3.1, (3.2), and Lemma 2.7, we estimate , where First we estimate . By Lemma 4.2, (4.16), and the definition of , we have Since here for some positive constant , we know that and by (3.25) Therefore, using (3.3) and (4.15), we have Noting (4.24), we have Next, we estimate . Noting (4.19) and (4.22), we have Then by (4.33) and (3.25) (noting (4.34)), using the notation (3.3) and (4.24), we have Therefore, since we know from (4.23) that noting (4.24), we have Finally, we estimate for . Suppose . Then similar to the above computations, we have Then, since we know by (4.28) that we have Then, since for , we have Therefore, the result is proved.

Proof of Proposition 3.9. By Lemma 2.3 we see that, for a constant , where, We set and hereafter we wrote (*) as . Now, we repeat the proof of Proposition 3.7 by exchanging with , , and we note (3.23) (and (3.26)). Then, for we obtain For , we use the estimate for in the proof of Proposition 3.7; furthermore we use Remark 4.4. Then we have Consequently, we have Similarly, we have

Proof of Proposition 3.10. Let be fixed. From Proposition 3.9 we see

Proof of Proposition 3.11. Let with . Then satisfies the condition (A-1). By Proposition 3.8, we see So, from Propositions 3.10 and 3.8 (noting (3.3)) we have

Proof of Theorem 3.12. Since satisfies (3.22), we see For a given , there exists a polynomial with such that In fact, by [21, Theorem 1.4], there exists a polynomial such that Let . Noting that and , we have that is, we have (4.55). Here, we know that Therefore, by Propositions 3.7, 3.11, and 3.8, we have for large enough Here we see that, by Proposition 3.7 with (constant depending only on ) and (4.55), and for large enough, we have Consequently, noting (3.3), we have Then we have Theorem 3.12.

Proof of Theorem 3.13. From Theorem 3.12 and Proposition 3.9, we have

Proof of Corollary 3.14. From Theorem 3.12 and Proposition 3.8, we have

Proof of Theorem 3.15. From Theorem 3.13 and Proposition 3.10 we have

Proof of Theorem 3.16. We use Theorems 3.12, 3.13, and Corollary 3.14, and Theorem 3.15. Then we have

5. Divergence Theorem

If is a positive odd integer, then we obtain the unboundedness of . We define

Theorem 5.1 (cf. [1, Theorem 2]). Let be an odd integer. Then there exists a constant and such that for

Let , and let us define Then we will show that for

Remark 5.2. For the interpolation polynomial , we can see a remarkable difference between the cases of an odd number and of an even number . Let us consider any continuous function . Then we can extend to a continuous function which satisfies (3.22) and , . Then from Theorem 3.12 for an even positive integer , we see On the other hand, the standard argument (cf, [22, Theorem 4.3]) leads us to the following. Theorem 5.1 means that there exists a certain function such that that is, for , we see that the interpolation polynomials do not converge to . We also remark that the polynomial interpolates at only .To prove the theorem we use the following lemma.

Lemma 5.3 (see [18, Theorem 11]). For , there is a polynomial of degree such that for , and the following relation holds. Let . Then one has an expression for and : where for , , , , there exist the constants such that and satisfies for with , and for .

Proof of Theorem 5.1. To get a lower bound of , it suffices to consider a lower bound (5.4) of with . Let and we consider the intervals . If we consider large enough, then we have . See the expression Then we have It follows from Remark 4.5 that . Hence, it is enough to show that . Let ; then by Lemma 4.2 and the definition of there exists such that For we consider only such that, for some positive integer , Then for each we define Here, we will see that Let Then we have where and are integers. On the other hand, Therefore, we have Now, we take a positive integer such that Consequently, we have the following. By Lemmas 4.2, Lemma 4.1, , , and Lemma 2.3, we have Here, using Lemma 5.3 and (5.15), we have Here, there exists such that . Therefore we see So we have (5.4), and consequently the proof of Theorem 5.1 is completed.

Acknowledgment

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

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