Some Properties of Orthogonal Polynomials for Laguerre-Type Weights
© H. Jung and R. Sakai 2011
Received: 3 November 2010
Accepted: 21 February 2011
Published: 14 March 2011
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.
First, we introduce some classes of weights.
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]).
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]).
Theorem B (see [18, Theorem 13.6]).
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.
Definition 2.1 (see ).
Then we see that and from [1, Lemma 2.2]. In addition, we easily have the following.
Definition 2.3 (cf. ).
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  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]).
3. Proofs of Theorems 1.5, 1.6, and 1.7
In the following, we introduce some useful notations.
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)]).
Proof of Theorem 1.5.
Proof of Theorem 1.6.
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.
Proof of Theorem 1.7..
4. Proofs of Theorems 1.8, 1.9, and 1.10
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.
Proof of Theorem 1.9.
Lemma 4.4 (see [21, Theorem 2.7]).
Proof of Theorem 1.10.
The authors thank the referees for many kind suggestions and comments. H. S. Jung was supported by SEOK CHUN Research Fund, Sungkyunkwan University, 2010.
- 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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