- Research Article
- Open Access

# Some Properties of Orthogonal Polynomials for Laguerre-Type Weights

- HeeSun Jung
^{1}Email author and - Ryozi Sakai
^{2}

**2011**:372874

https://doi.org/10.1155/2011/372874

© H. Jung and R. Sakai 2011

**Received:**3 November 2010**Accepted:**21 February 2011**Published:**14 March 2011

## Abstract

Let , let be a continuous, nonnegative, and increasing function, and let be the orthonormal polynomials with the weight . For the zeros of we estimate , where is a positive integer. Moreover, we investigate the various weighted -norms ( ) of .

## Keywords

- Positive Constant
- Orthogonal Polynomial
- Order Derivative
- Large Integer
- Interpolation Polynomial

## 1. Introduction and Main Results

Let and . Let be a continuous, nonnegative, and increasing function. Consider the exponential weights , , and then we construct the orthonormal polynomials with the weight . In this paper, for the zeros of we estimate , where is a positive integer. Moreover, we investigate the various weighted -norms ( ) of .

We say that is quasi-increasing if there exists such that for . The notation means that there are positive constants such that for the relevant range of , . The similar notation is used for sequences and sequences of functions.

Throughout, denote positive constants independent of . The same symbol does not necessarily denote the same constant in different occurrences. We denote the class of polynomials with degree by .

First, we introduce some classes of weights.

Levin and Lubinsky [1, 2] introduced the class of weights on as follows. Let , where .

We assume that
has the following properties. Let
*,*

(a) is continuous in , with limit 0 at 0 and ,

then we write .

We consider the case , that is, the space , and we strengthen Definition 1.1 slightly.

Definition 1.2.

We assume that has the following properties:

(a) are continuous, positive in , with , ,

then we write .

Let us consider the weight in Definition 1.2. Levin and Lubinsky [2, Theorem 1.3] have given the following theorem.

Theorem A (see [3, Theorem 1.3]).

Now, we will estimate the higher-order derivatives of the orthonormal polynomials . However, we need to focus on a smaller class of weights.

Definition 1.3.

Let
and
be an integer. For the exponent
*,* we assume the following:

(a) , for and , and , .

(b)there exist positive constants such that for

(d)there exists such that we have one among the following:

(d1) is quasi-increasing on ,

(d2) is nondecreasing on .

Then, we write .

where is the th iterated exponential.

(1)When , we consider , then .

Then, .

In this paper, we will consider the orthonormal polynomials with respect to the weight class . Our main themes in this paper are to estimate the higher-order derivatives of at the zeros of and to investigate the various weighted -norms ( ) of . More precisely, we will estimate the higher-order derivatives of at all zeros of for two cases of an odd order and of an even order. In addition, we will give asymptotic relation of the odd order derivatives of at the zeros of in a certain finite interval. These estimations will play an important role in investigating convergence or divergence of higher-order Hermite-Fejér interpolation polynomials (see [3, 5–17]).

Then, our main purpose is to obtain estimations with respect to , , as follows.

Theorem 1.5.

Theorem 1.6.

Theorem 1.7.

and as and .

We consider the class of weights, , which is defined in Definition 2.1 below. Levin and Lubinsky have obtained the following theorem.

Theorem B (see [18, Theorem 13.6]).

We remark that Levin and Lubinsky have shown Theorem B for more wider class . In the following, we investigate the various weighted -norms ( ) of .

Theorem 1.8.

Theorem 1.9.

Theorem 1.10.

This paper is organized as follows. In Section 2, we will introduce the weight class as an analogy of the class and the known results of orthonormal polynomials with respect to in order to prove the main results. In Section 3, we will prove Theorems 1.5, 1.6, and 1.7. Finally, we will prove the results for the various weighted -norms ( ) of , that is, Theorems 1.8, 1.9, and 1.10, in Section 4.

## 2. Preliminaries

Levin and Lubinsky introduced the classes and as an analogy of the classes and which they already defined on . They defined the following.

Definition 2.1 (see [18]).

We assume that has the following properties:

(a) is continuous in , with ,

then, we write .

Then we see that and from [1, Lemma 2.2]. In addition, we easily have the following.

Lemma 2.2.

where .

On , we can consider the corresponding class to as follows.

Definition 2.3 (cf. [19]).

Let
and
be an integer. Let
be a continuous and even function on
*.* For the exponent
, we assume the following:

(a) , for and ,

(b)there exist positive constants such that for

(d)there exists such that one has one among the following:

(d1) is quasi-increasing on ,

(d2) is nondecreasing on .

Then, we write .

Jung and Sakai [5, Theorems 3.3 and 3.6] estimate , , , and we will obtain analogous estimations with respect to , , in Theorems 1.6 and 1.7.

There are many properties of with respect to , , of Definition 2.3 in [4–6, 19–21]. They were obtained by transformations from the results in [1, 2]. In this paper, we consider and . In [5] we got the estimations of , , with the weight . By a transformation of the results with respect to , we estimate , , . In order to it we will give the transformation theorems in this section. In the following, we will give some applications of them.

Theorem 2.4 (see [21, Theorem 2.1]).

Theorem 2.5.

where is defined in (1.13).

Proof.

where is defined in (1.13). The inequalities (d1) and (d2) of Definition 2.3 follow easily from (d1) and (d2) of Definition 1.3. Therefore, one has (2.14).

## 3. Proofs of Theorems 1.5, 1.6, and 1.7

In the following, we introduce some useful notations.

(a)The Mhaskar-Rahmanov-Saff numbers and are defined as the positive roots of the following equations:

Then, one has the following.

Lemma 3.1 (see [1, (2.5), (2.7), (2.9)]).

To prove Theorem 1.6, we need some lemmas as follows.

Lemma 3.2 (see [21, Theorem 2.2, Lemma 3.7]).

Lemma 3.3 (see [6, Theorem 2.5]).

Lemma 3.4 (see [5, Theorem 3.6 and Lemma 3.7 (3.20)]).

Remark 3.5.

In addition, since , we know that

(i) is bounded is bounded,

(ii) is unbounded for any ,

(iii) for some constant .

Lemma 3.6.

Proof.

It is easily proved, using the mathematical induction on .

Proof of Theorem 1.5.

Proof of Theorem 1.6.

Since , we know that and we know that by Theorem 2.5 and from (3.1), (3.2), and Lemma 3.1 that

(i) ,

(ii) for for ,

(iii) .

Next, we will prove Theorem 1.7. To prove it, we need two lemmas as follows.

Lemma 3.7 ([5, Theorem 3.3]).

and as and .

From Lemma 3.3, we easily have the following.

Lemma 3.8.

Proof of Theorem 1.7..

Finally, if we let , then the result is proved.

## 4. Proofs of Theorems 1.8, 1.9, and 1.10

Lemma 4.1.

Proof.

But, seeing our proof of [21, Theorem 2.6] carefully, we can easily prove the first equivalence.

Lemma 4.2 (see [21, Theorem 2.4]).

Proof of Theorem 1.8.

Consequently, using Lemma 3.1, one has the result.

Lemma 4.3.

- (a)

Proof of Theorem 1.9.

From Remark 3.5(iii), we see that . So, consequently, one has the result.

Lemma 4.4 (see [21, Theorem 2.7]).

Proof of Theorem 1.10.

## Declarations

### Acknowledgments

The authors thank the referees for many kind suggestions and comments. H. S. Jung was supported by SEOK CHUN Research Fund, Sungkyunkwan University, 2010.

## Authors’ Affiliations

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This article is published under license to BioMed Central Ltd. 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.