Open Access

Some Properties of Orthogonal Polynomials for Laguerre-Type Weights

Journal of Inequalities and Applications20112011:372874

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

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 .

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 .

Definition 1.1 (see [1, 2]).

We assume that has the following properties. Let ,

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

(b) exists in , while is positive in ,
  1. (c)
    (11)
     
(d)the function
(12)
is quasi-increasing in , with
(13)
(e)there exists such that
(14)
Then, we write . If there also exists a compact subinterval of , and such that
(15)

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 , ,

(b) exists in ,
  1. (c)
    (16)
     
(d)the function
(17)
is quasi-increasing in , with
(18)
(e)there exists such that
(19)
There exists a compact subinterval of , and such that
(110)

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]).

Let and . There exists such that uniformly for , ,
(111)

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

(112)
(c)there exist positive constants , and such that for
(113)

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

(d1) is quasi-increasing on ,

(d2) is nondecreasing on .

Then, we write .

Example 1.4 (see [1, 4]).

Let be a fixed integer. There are some typical examples satisfying all conditions of Definition 1.3 constructed as follows: let , , where is an integer. Then, we define
(114)

where is the th iterated exponential.

(1)When , we consider , then .

(2)When , is an integer, we define
(115)

Then, .

In the rest of this paper, we consider the classes and ; let or ( ). For , we set . Then, we can construct the orthonormal polynomials of degree with respect to . That is,
(116)
Let us denote the zeros of by
(117)
The Mhaskar-Rahmanov-Saff numbers are defined as follows:
(118)

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, 517]).

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

Theorem 1.5.

Let and . For each and one has
(119)

Theorem 1.6.

Let and . Assume that for , and if is bounded, then assume that
(120)
where is defined in (1.13). For each and , one has
(121)
and in particular if is even, then
(122)

Theorem 1.7.

Let and . Let , , and , . Then, under the same conditions as the assumptions of Theorem 1.6, there exist , for absolute constants , such that the following equality holds:
(123)

and as and .

Define
(124)
Let us define
(125)

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]).

Assume that . Let . Then uniformly for ,
(126)

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.

Let . Let and . Then one has for ,
(127)

Theorem 1.9.

Let . Let and . Then one has for ,
(128)

Theorem 1.10.

Let , and . Suppose that . For and , one has
(129)
and for
(130)

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 ,

(b) exists and is positive in ,
  1. (c)
    (21)
     
(d)the function
(22)
is quasi-increasing in , with
(23)
(e)there exists such that
(24)
Then, we write . If there also exists a compact subinterval of , and such that
(25)

then, we write .

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

Lemma 2.2.

Let . Then one has
(26)

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

(27)
(c)there exist positive constants , and such that for
(28)

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

(d1) is quasi-increasing on ,

(d2) is nondecreasing on .

Then, we write .

Let and . For , we set
(29)
Then, we can construct the orthonormal polynomials of degree with respect to . That is,
(210)
Let us denote the zeros of by
(211)

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 [46, 1921]. 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]).

Let with . Then, the orthonormal polynomials on can be entirely reduced to the orthonormal polynomials in as follows: for ,
(212)
In this paper, we will use the fact that is transformed into as meaning that
(213)

Theorem 2.5.

Let . Then one has
(214)
In particular, one has
(215)

where is defined in (1.13).

Proof.

Let . Then, from Lemma 2.2, one has . Let denote the maximum integer as (Gaussian symbol). For , one has
(216)
Therefore, we easily see that (a) of Definition 2.3 holds. Let . Since is increasing for and , there exists with such that for , 
(217)
Then, since for , 
(218)
one has by (b) of Definition 1.3 that
(219)
Similarly, one has by (2.16), (d) of Definition 1.2, and (b) of Definition 1.3 that
(220)
Consequently, one has (b) in Definition 2.3. We know that
(221)
and since on , one has from (1.13) that
(222)
Therefore, one has by (2.16)
(223)

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

For convenience, in the rest of this paper, we put as follows:
(31)
and . Then, we know that and
(32)

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:

(33)
(b)Let
(34)

Then, one has the following.

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

(35)

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

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

For the minimum positive zero ( is the largest integer ), one has
(36)
and for the maximum zero , one has for large enough ,
(37)
Moreover, for some constant , one has
(38)

Lemma 3.3 (see [6, Theorem  2.5]).

Let and . Then, uniformly for ,
(39)

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

Let and , . Assume that for and if is bounded, then assume
(310)
where is defined in (2.8). If , then one has for
(311)
and in particular, if is even, then
(312)

Remark 3.5.

Let . Then, from [19, Theorem  1.6] we know that when is unbounded, for any , there exists such that for ,
(313)

In addition, since , we know that

(i) is bounded is bounded,

(ii) is unbounded for any ,

(iii) for some constant .

Lemma 3.6.

For , one has
(314)
where satisfy that for ,
(315)
and for
(316)

Proof.

It is easily proved, using the mathematical induction on .

Proof of Theorem 1.5.

By Lemmas 3.3, 3.6 and (3.2), one has
(317)
Since by Lemma 3.2
(318)
one has from Lemma 3.1 that
(319)

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) .

Then, using Remark 3.5, we can apply Lemma 3.4 to , . In a similar way to the proof of Theorem 1.5, one has from Lemma 3.4 and Lemma 3.1
(320)
Let be even. Then, one has from Lemma 3.4 that
(321)
Since by Lemma 3.2 and
(322)
one has
(323)
Therefore, when is even, one has by Lemma 3.1 that
(324)

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

Lemma 3.7 ([5, Theorem  3.3]).

Let , . Let , , and . Then, under the same conditions as the assumptions of Lemma 3.4, there exist , for absolute constants , such that the following equality holds:
(325)

and as and .

From Lemma 3.3, we easily have the following.

Lemma 3.8.

Let and . Then, uniformly for ,
(326)

Proof of Theorem 1.7..

By Lemmas 3.4, 3.6 and Theorem 2.4, one has
(327)
Since we know that
(328)
by the same reason as the proof of Theorem 1.6, we can apply Lemma 3.7 to . Then, using Lemmas 3.7 and 3.6, one has
(329)
Here, , . Since from (3.28) we see that for ,
(330)
one has that
(331)
as and . If we let
(332)
then one has
(333)
and as and . On the other hand, we obtain
(334)
Here, one has from Lemma 3.8 and (3.28) that
(335)

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

4. Proofs of Theorems 1.8, 1.9, and 1.10

Lemma 4.1.

Let , and let and . Then, one has for that
(41)

Proof.

In [21, theorem  2.6] we showed that
(42)

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]).

Let , and . Let . Then, given , there exists a positive constant such that one has for any polynomial  that
(43)

Proof of Theorem 1.8.

From Theorem 2.4 and Lemmas 4.2 and 4.1, one has
(44)
On the other hand, one has by Theorem 2.4 and Lemma 4.1 that
(45)

Consequently, using Lemma 3.1, one has the result.

Lemma 4.3.

Let , and let . Then, uniformly for and , one has the following:
  1. (a)
    (46)
     
(b)for and , 
(47)
(c)for , 
(48)
where
(49)
Proof.
  1. (a)

    It is from [1, Theorem  1.2]. (b) It is from [2, Theorem  1.3]. (c) It is from [2, Theorem  1.4].

     

Proof of Theorem 1.9.

By Theorem 1.8, one has for , 
(410)
For , we know by (4.7) and (4.8) that
(411)
Then, for
(412)
Therefore, one has
(413)
On the other hand, one has from Theorem 1.8 that
(414)
and by (4.6) that
(415)
Therefore, one has
(416)

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

Let
(417)
Then, we obtain by Lemma 3.1 that
(418)

Lemma 4.4 (see [21, Theorem  2.7]).

Let and . For and , one has
(419)

Proof of Theorem 1.10.

By Theorem 2.4, we can transform on to on .
(420)
Using Lemma 4.4 and noting (3.1), one has
(421)
On the other hand, by Lemma 4.2, we see
(422)
where is a constant. Therefore, using Lemma 4.4 and noting (3.1) and the definition of , one has
(423)
Therefore, one has
(424)

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

(1)
Department of Mathematics Education, Sungkyunkwan University
(2)
Department of Mathematics, Meijo University

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Copyright

© H. Jung and R. Sakai 2011

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.