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Some Properties of Orthogonal Polynomials for Laguerre-Type Weights
Journal of Inequalities and Applications volume 2011, Article number: 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 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 
,
(b)
exists in 
, while 
is positive in 
,
-
(c)
(11)
(d)the function
is quasi-increasing 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 quasi-increasing 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, Theorem  1.3] have given the following theorem.
Theorem A (see [3, Theorem  1.3]).
Let 
and 
. There exists 
such that uniformly for 
, 
,
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 
(c)there exist positive constants 
, 
and 
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 
.
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 Mhaskar-Rahmanov-Saff 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 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.
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, Theorem  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 
 
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 quasi-increasing 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, Lemma  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 
 
(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
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, 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]).
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 
, 
Then, since for 
, 
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 Mhaskar-Rahmanov-Saff 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, Theorem  2.2, Lemma  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, Theorem  2.5]).
Let 
and 
. Then, uniformly for 
,
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
where 
is defined in (2.8). If 
, then one has for 
 
and in particular, if 
is even, then
Remark 3.5.
Let 
. Then, from [19, Theorem  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 
 
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, 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:
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, theorem  2.6] we showed that
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
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 
, 
(c)for 
, 
where
Proof.
-
(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 
, 
For 
, we know by (4.7) and (4.8) that
Then, for 
 
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, Theorem  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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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 Laguerre-Type 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