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Some Properties of Orthogonal Polynomials for LaguerreType Weights
Journal of Inequalities and Applications volumeÂ 2011, ArticleÂ number:Â 372874 (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 quasiincreasing 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 ,
(b) exists in , while is positive in ,

(c)
(11)
(d)the function
is quasiincreasing in , with
(e)there exists such that
Then, we write . If there also exists a compact subinterval of , and such that
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 ,

(c)
(16)
(d)the function
is quasiincreasing in , with
(e)there exists such that
There exists a compact subinterval of , and such that
then we write .
Let us consider the weight in Definition 1.2. Levin and Lubinsky [2, Theoremâ€‰â€‰1.3] have given the following theorem.
Theorem A (see [3, Theoremâ€‰â€‰1.3]).
Let and . There exists such that uniformly for , ,
Now, we will estimate the higherorder 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
(c)there exist positive constants , and such that for
(d)there exists such that we have one among the following:
(d1) is quasiincreasing on ,
(d2) is nondecreasing on .
Then, we write .
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
where is the th iterated exponential.
(1)When , we consider , then .
(2)When , is an integer, we define
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,
Let us denote the zeros of by
The MhaskarRahmanovSaff numbers are defined as follows:
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 higherorder derivatives of at the zeros of and to investigate the various weighted norms () of . More precisely, we will estimate the higherorder 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 higherorder HermiteFejÃ©r interpolation polynomials (see [3, 5â€“17]).
Then, our main purpose is to obtain estimations with respect to , , as follows.
Theorem 1.5.
Let and . For each and one has
Theorem 1.6.
Let and . Assume that for , and if is bounded, then assume that
where is defined in (1.13). For each and , one has
and in particular if is even, then
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:
and as and .
Define
Let us define
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, Theoremâ€‰â€‰13.6]).
Assume that . Let . Then uniformly for ,
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 ,
Theorem 1.9.
Let . Let and . Then one has for ,
Theorem 1.10.
Let , and . Suppose that . For and , one has
and for â€‰
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 ,

(c)
(21)
(d)the function
is quasiincreasing in , with
(e)there exists such that
Then, we write . If there also exists a compact subinterval of , and such that
then, we write .
Then we see that and from [1, Lemmaâ€‰â€‰2.2]. In addition, we easily have the following.
Lemma 2.2.
Let . Then one has
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
(c)there exist positive constants , and such that for â€‰
(d)there exists such that one has one among the following:
(d1) is quasiincreasing on ,
(d2) is nondecreasing on .
Then, we write .
Let and . For , we set
Then, we can construct the orthonormal polynomials of degree with respect to . That is,
Let us denote the zeros of by
Jung and Sakai [5, Theoremsâ€‰â€‰3.3 andâ€‰â€‰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 Definitionâ€‰â€‰2.3 in [4â€“6, 19â€“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, Theoremâ€‰â€‰2.1]).
Let with . Then, the orthonormal polynomials on can be entirely reduced to the orthonormal polynomials in as follows: for ,
In this paper, we will use the fact that is transformed into as meaning that
Theorem 2.5.
Let . Then one has
In particular, one has
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
Therefore, we easily see that (a) of Definition 2.3 holds. Let . Since is increasing for and , there exists with such that for ,â€‰
Then, since for ,â€‰
one has by (b) of Definition 1.3 that
Similarly, one has by (2.16), (d) of Definition 1.2, and (b) of Definition 1.3 that
Consequently, one has (b) in Definition 2.3. We know that
and since on , one has from (1.13) that
Therefore, one has by (2.16)
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:
and . Then, we know that and
In the following, we introduce some useful notations.
(a)The MhaskarRahmanovSaff numbers and are defined as the positive roots of the following equations:
(b)Let
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, Theoremâ€‰â€‰2.2, Lemmaâ€‰â€‰3.7]).
For the minimum positive zero ( is the largest integer ), one has
and for the maximum zero , one has for large enough ,
Moreover, for some constant , one has
Lemma 3.3 (see [6, Theoremâ€‰â€‰2.5]).
Let and . Then, uniformly for ,
Lemma 3.4 (see [5, Theoremâ€‰â€‰3.6 and Lemmaâ€‰â€‰3.7 (3.20)]).
Let and , . Assume that for and if is bounded, then assume
where is defined in (2.8). If , then one has for â€‰
and in particular, if is even, then
Remark 3.5.
Let . Then, from [19, Theoremâ€‰â€‰1.6] we know that when is unbounded, for any , there exists such that for ,
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
where satisfy that for ,
and for â€‰
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
Since by Lemma 3.2
one has from Lemma 3.1 that
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
Let be even. Then, one has from Lemma 3.4 that
Since by Lemma 3.2 and
one has
Therefore, when is even, one has by Lemma 3.1 that
Next, we will prove Theorem 1.7. To prove it, we need two lemmas as follows.
Lemma 3.7 ([5, Theoremâ€‰â€‰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:
and as and .
From Lemma 3.3, we easily have the following.
Lemma 3.8.
Let and . Then, uniformly for ,
Proof of Theorem 1.7..
By Lemmas 3.4, 3.6 and Theorem 2.4, one has
Since we know that
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
Here, , . Since from (3.28) we see that for ,
one has that
as and . If we let
then one has
and as and . On the other hand, we obtain
Here, one has from Lemma 3.8 and (3.28) that
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
Proof.
In [21, theoremâ€‰â€‰2.6] we showed that
But, seeing our proof of [21, Theoremâ€‰â€‰2.6] carefully, we can easily prove the first equivalence.
Lemma 4.2 (see [21, Theoremâ€‰â€‰2.4]).
Let , and . Let . Then, given , there exists a positive constant such that one has for any polynomial â€‰that
Proof of Theorem 1.8.
From Theorem 2.4 and Lemmas 4.2 and 4.1, one has
On the other hand, one has by Theorem 2.4 and Lemma 4.1 that
Consequently, using Lemma 3.1, one has the result.
Lemma 4.3.
Let , and let . Then, uniformly for and , one has the following:

(a)
(46)
(b)for and ,â€‰
(c)for ,â€‰
where
Proof.

(a)
It is from [1, Theoremâ€‰â€‰1.2]. (b) It is from [2, Theoremâ€‰â€‰1.3]. (c) It is from [2, Theoremâ€‰â€‰1.4].
Proof of Theorem 1.9.
By Theorem 1.8, one has for ,â€‰
For , we know by (4.7) and (4.8) that
Then, for â€‰
Therefore, one has
On the other hand, one has from Theorem 1.8 that
and by (4.6) that
Therefore, one has
From Remark 3.5(iii), we see that . So, consequently, one has the result.
Let
Then, we obtain by Lemma 3.1 that
Lemma 4.4 (see [21, Theoremâ€‰â€‰2.7]).
Let and . For and , one has
Proof of Theorem 1.10.
By Theorem 2.4, we can transform on to on .
Using Lemma 4.4 and noting (3.1), one has
On the other hand, by Lemma 4.2, we see
where is a constant. Therefore, using Lemma 4.4 and noting (3.1) and the definition of , one has
Therefore, one has
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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.
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Jung, H., Sakai, R. Some Properties of Orthogonal Polynomials for LaguerreType Weights. J Inequal Appl 2011, 372874 (2011). https://doi.org/10.1155/2011/372874
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DOI: https://doi.org/10.1155/2011/372874