- 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

## 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 ,

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 , ,

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:

(b)there exist positive constants such that for

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

where is the th iterated exponential.

(1)When , we consider , 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.

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:

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

Lemma 2.2.

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:

(b)there exist positive constants such that for

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

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.

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

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

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

Lemma 3.7 ([5, Theorem 3.3]).

From Lemma 3.3, we easily have the following.

Lemma 3.8.

Proof of Theorem 1.7..

## 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

## References

- Levin E, Lubinsky D:
**Orthogonal polynomials for exponential weights****on**.*Journal of Approximation Theory*2005,**134**(2):199–256. 10.1016/j.jat.2005.02.006MathSciNetView ArticleMATHGoogle Scholar - Levin E, Lubinsky D:
**Orthogonal polynomials for exponential weights****on****—II.***Journal of Approximation Theory*2006,**139**(1–2):107–143. 10.1016/j.jat.2005.05.010MathSciNetView ArticleMATHGoogle Scholar - Kasuga T, Sakai R:
**Orthonormal polynomials for generalized Freud-type weights and higher-order Hermite-Fejér interpolation polynomials.***Journal of Approximation Theory*2004,**127**(1):1–38. 10.1016/j.jat.2004.01.006MathSciNetView ArticleMATHGoogle Scholar - Jung H, Sakai R:
**Specific examples of exponential weights.***Korean Mathematical Society. Communications*2009,**24**(2):303–319. 10.4134/CKMS.2009.24.2.303MathSciNetView ArticleMATHGoogle Scholar - Jung HS, Sakai R:
**Derivatives of orthonormal polynomials and coefficients of Hermite-Fejér interpolation polynomials with exponential-type weights.***Journal of Inequalities and Applications*2010,**2010:**-29.Google Scholar - Jung HS, Sakai R:
**The Markov-Bernstein inequality and Hermite-Fejér interpolation for exponential-type weights.***Journal of Approximation Theory*2010,**162**(7):1381–1397. 10.1016/j.jat.2010.02.006MathSciNetView ArticleMATHGoogle Scholar - Kasuga T, Sakai R:
**Uniform or mean convergence of Hermite-Fejér interpolation of higher order for Freud weights.***Journal of Approximation Theory*1999,**101**(2):330–358. 10.1006/jath.1999.3371MathSciNetView ArticleMATHGoogle Scholar - Kasuga T, Sakai R:
**Orthonormal polynomials with generalized Freud-type weights.***Journal of Approximation Theory*2003,**121**(1):13–53. 10.1016/S0021-9045(02)00041-2MathSciNetView ArticleMATHGoogle Scholar - Kasuga T, Sakai R:
**Orthonormal polynomials for Laguerre-type weights.***Far East Journal of Mathematical Sciences*2004,**15**(1):95–105.MathSciNetMATHGoogle Scholar - Kasuga T, Sakai R:
**Conditions for uniform or mean convergence of higher order Hermite-Fejér interpolation polynomials with generalized Freud-type weights.***Far East Journal of Mathematical Sciences*2005,**19**(2):145–199.MathSciNetMATHGoogle Scholar - Kanjin Y, Sakai R:
**Pointwise convergence of Hermite-Fejér interpolation of higher order for Freud weights.***The Tohoku Mathematical Journal*1994,**46**(2):181–206. 10.2748/tmj/1178225757MathSciNetView ArticleMATHGoogle Scholar - Kanjin Y, Sakai R:
**Convergence of the derivatives of Hermite-Fejér interpolation polynomials of higher order based at the zeros of Freud polynomials.***Journal of Approximation Theory*1995,**80**(3):378–389. 10.1006/jath.1995.1024MathSciNetView ArticleMATHGoogle Scholar - Sakai R:
**Hermite-Fejér interpolation.**In*Approximation Theory*.*Volume 58*. North-Holland, Amsterdam, The Netherlands; 1991:591–601.Google Scholar - Sakai R:
**The degree of approximation of differentiable by Hermite interpolation polynomials.**In*Progress in Approximation Theory*. Edited by: Nevai P, Pinkus A. Academic Press, Boston, Mass, USA; 1991:731–759.Google Scholar - Sakai R:
**Certain unbounded Hermite-Fejér interpolatory polynomial operators.***Acta Mathematica Hungarica*1992,**59**(1–2):111–114. 10.1007/BF00052097MathSciNetView ArticleMATHGoogle Scholar - Sakai R, Vértesi P:
**Hermite-Fejér interpolations of higher order. III.***Studia Scientiarum Mathematicarum Hungarica*1993,**28**(1–2):87–97.MathSciNetMATHGoogle Scholar - Sakai R, Vértesi P:
**Hermite-Fejér interpolations of higher order. IV.***Studia Scientiarum Mathematicarum Hungarica*1993,**28**(3–4):379–386.MathSciNetMATHGoogle Scholar - Levin E, Lubinsky DS:
*Orthogonal Polynomials for Exponential Weights*. Springer, New York, NY, USA; 2001:xii+476.View ArticleMATHGoogle Scholar - Jung HS, Sakai R:
**Derivatives of integrating functions for orthonormal polynomials with exponential-type weights.***Journal of Inequalities and Applications*2009,**2009:**-22.Google Scholar - Jung HS, Sakai R:
**Inequalities with exponential weights.***Journal of Computational and Applied Mathematics*2008,**212**(2):359–373. 10.1016/j.cam.2006.12.011MathSciNetView ArticleMATHGoogle Scholar - Jung HS, Sakai R:
**Orthonormal polynomials with exponential-type weights.***Journal of Approximation Theory*2008,**152**(2):215–238. 10.1016/j.jat.2007.12.004MathSciNetView ArticleMATHGoogle Scholar

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