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
,
-
(c)
(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)
(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
.
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
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.