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Derivatives of Orthonormal Polynomials and Coefficients of HermiteFejér Interpolation Polynomials with ExponentialType Weights
Journal of Inequalities and Applications volume 2010, Article number: 816363 (2010)
Abstract
Let , and let be an even function. In this paper, we consider the exponentialtype weights , and the orthonormal polynomials of degree with respect to . So, we obtain a certain differential equation of higher order with respect to and we estimate the higherorder derivatives of and the coefficients of the higherorder HermiteFejér interpolation polynomial based at the zeros of .
1. Introduction
Let and . Let be an even function and let be such that for all For , we set
Then we can construct the orthonormal polynomials of degree with respect to . That is,
We denote the zeros of by
A function is said to be quasiincreasing if there exists such that for . For any two sequences and of nonzero real numbers (or functions), we write if there exists a constant independent of (or ) such that for being large enough. We write if and . We denote the class of polynomials of degree at most by .
Throughout denote positive constants independent of , and polynomials of degree at most . The same symbol does not necessarily denote the same constant in different occurrences.
We shall be interested in the following subclass of weights from [1].
Definition 1.1.
Let be even and satisfy the following properties.
(a) is continuous in , with .
(b) exists and is positive in .
(c)One has
(d)The function
is quasiincreasing in with
(e)There exists such that
Then we write . If there also exist a compact subinterval of and such that
then we write .
In the following we introduce useful notations.
(a)MhaskarRahmanovSaff (MRS) numbers is defined as the positive roots of the following equations:
(b)Let
(c)The function is defined as the following:
In [2, 3] we estimated the orthonormal polynomials associated with the weight and obtained some results with respect to the derivatives of orthonormal polynomials . In this paper, we will obtain the higher derivatives of . To estimate of the higher derivatives of the orthonormal polynomials sequence, we need further assumptions for as follows.
Definition 1.2.
Let and let be a positive integer. Assume that is times continuously differentiable on and satisfies the followings.
(a) exists and , are positive for .
(b)There exist positive constants such that for
(c)There exist constants and such that on
Then we write . Furthermore, and satisfies one of the following.
(a) is quasiincreasing on a certain positive interval .
(b) is nondecreasing on a certain positive interval .
(c)There exists a constant such that on .
Then we write .
Now, consider some typical examples of . Define for and ,
More precisely, define for , , and ,
where if , otherwise , and define
In the following, we consider the exponential weights with the exponents . Then we have the following examples (see [4]).
Example 1.3.
Let be a positive integer. Let . Then one has the following.
(a) belongs to .
(b)If and , then there exists a constant such that is quasiincreasing on .
(c)When , if , then there exists a constant such that is quasiincreasing on , and if , then is quasidecreasing on .
(d)When and , is nondecreasing on a certain positive interval .
In this paper, we will consider the orthonormal polynomials with respect to the weight class . Our main themes in this paper are to obtain a certain differential equation for of higherorder and to estimate the higherorder derivatives of at the zeros of and the coefficients of the higherorder HermiteFejér interpolation polynomials based at the zeros of . More precisely, we will estimate the higherorder derivatives of at the zeros of for two cases of an odd order and of an even order. These estimations will play an important role in investigating convergence or divergence of higherorder HermiteFejér interpolation polynomials (see [5–16]).
This paper is organized as follows. In Section 2, we will obtain the differential equations for of higherorder. In Section 3, we will give estimations of higherorder derivatives of at the zeros of in a certain finite interval for two cases of an odd order and of an even order. In addition, we estimate the higherorder derivatives of at all zeros of for two cases of an odd order and of an even order. Furthermore, we will estimate the coefficients of higherorder HermiteFejér interpolation polynomials based at the zeros of , in Section 4.
2. HigherOrder Differential Equation for Orthonormal Polynomials
In the rest of this paper we often denote and simply by and , respectively. Let if is odd, otherwise, and define the integrating functions and with respect to as follows:
where and . Then in [3, Theorem ] we have a relation of the orthonormal polynomial with respect to the weight :
Theorem 2.1 (cf. [6, Theorem ]).
Let and . Then for one has the secondorder differential relation as follows:
Here, one knows that for any integer ,
where
Especially, when is odd, one has
where is the polynomial of degree with .
Proof.
We may similarly repeat the calculation [6, Proof of Theorem ], and then we obtain the results. We stand for simply. Applying (2.2) to we also see
and so if we use the recurrence formula
and use (2.2) too, then we obtain the following:
We differentiate the left and right sides of (2.2) and substitute (2.2) and (2.9). Then consequently, we have, for ,
Using the recurrence formula (2.8) and , we have
because is an odd function. Therefore, we have
When is odd, since , (2.6) is proved.
For the higherorder differential equation for orthonormal polynomials, we see that for and
Let for nonnegative integer . In the following theorem, we show the higherorder differential equation for orthonormal polynomials.
Theorem 2.2.
Let and . Let and . Then one has the following equation for :
where
and for and
and for
Proof.
It comes from Theorem 2.1 and (2.13).
Corollary 2.3.
Under the same assumptions as Theorem 2.1, if is odd, then
where and for
Proof.
Let be odd. Then we will consider (2.6). Since , we have
and we have
Therefore, we have the result from (2.6).
In the rest of this paper, we let and for positive integer and assume that for and
where is defined in (1.13).
In Section 3, we will estimate the higherorder derivatives of orthonormal polynomials at the zeros of orthonormal polynomials with respect to exponentialtype weights.
3. Estimation of HigherOrder Derivatives of Orthonormal Polynomials
From [3, Theorem ] we know that there exist and such that for and ,
If is unbounded, then (2.22) is trivially satisfied. Additionally we have, from [17, Theorem ], that if we assume that is nondecreasing, then for with
where there exists a constant such that
Here, and for some .
For the higher derivatives of and , we have the following results in [17, Theorem ].
Theorem 3.1 (see[17, Theorem ]).
For and
Moreover, there exists such that for and ,
with as .
Corollary 3.2.
Let . Then there exists a positive constant such that one has for and ,
In the following, we have the estimation of the higherorder derivatives of orthonormal polynomials.
Theorem 3.3.
Let and . Then for the following equality holds for large enough:
where
and . Moreover, for
Here,
Corollary 3.4.
Suppose the same assumptions as Theorem 3.3. Given any , there exists a small fixed positive constant such that (3.8) holds satisfying and
for .
Corollary 3.5.
For and
Theorem 3.6.
Let and let , . Then
and especially if is even, then
We note that for large enough,
because we know that from [3, Theorem ] and
To prove these results we need some lemmas.
Lemma 3.7.

(a)
For
(3.18)
(b) For
(c) For
(d) Let . Then for
and for
Proof.

(a)
It is [1, Lemma (c)]. (b) It is [1, Lemma (c)]. (c) It comes from (3.1). (d) Since , is increasing. So, we obtain (d) by (1.12).
Lemma 3.8.
Let , and , , be defined in Theorem 2.1.

(a)
For and , there exists satisfying as such that
Moreover, for and ,
(b) For and , there exists satisfying as such that
Moreover, for and ,
(c) For and , there exists satisfying as such that
Moreover, for and ,
(d) For and , there exists satisfying as such that
Moreover, for and ,
Proof.

(a)
Since , we prove it by Theorem 3.1.
(b) For , we see
From (3.18), we know that . Therefore by (3.19), (3.21), and (3.6) we have for
and for we have by (3.21) and (3.22)
Consequently we have (b).
(c) Next we estimate . Suppose . Let us set . By (3.6) and (3.20) we have
For , we obtain the same estimate as
For , we have similarly to the case of
(d) It is similar to (c). Consequently we have the following lemma.
Lemma 3.9.
Let , , and . Let . Then
where is defined in Theorem 3.3 and for
Moreover, for ,
Proof.
Since
we have (3.39) for by (3.5). For we have from (3.6) and (3.19) that
Moreover, we can obtain (3.38) for from the above easily.
Lemma 3.10.
Let and . Let . Then for
with , where , , and are defined in Theorem 3.3. For one has
On the other hand, one has for ,
Proof.
First, we know that
Suppose . Since from (3.18) and (3.19)
we have from (3.6)
Since
we know from (3.6) that
Therefore we have for
Since from (3.3)
and similarly
we have
Then we have
Therefore, since
there exist constants with such that we have for
Especially, from the above estimates we can see (3.43) for . On the other hand, suppose . Then since from Theorem 2.1 and (3.5)
and , we have from Lemma 3.8
Therefore, we have (3.44) for .
Lemma 3.11.
Let and . Let . Then for , there exists satisfying as such that
Moreover, one has for ,
Proof.
For we have from Lemma 3.8 that there exists satisfying as such that
Similarly, for and ,
Therefore, we have the results.
Proof of Theorem 3.3.
First we know that the following differential equation is satisfied:
Suppose . Then since we see from (3.63) and (3.38) that
we have by (3.63) and mathematical induction
Next, suppose . More precisely, from Lemma 3.9 we have
Then by (3.63), (3.42), and (3.66) there exists a constant with
such that we have that
Suppose that there exist constants with such that
Then we have by (3.38) and (3.70)
and we have by (3.42) and (3.69)
where and . Also, we have by (3.59) that for
Therefore, there exists satisfying such that
Moreover, we have by (3.37) and (3.65)
and by (3.43) and (3.70)
Also we obtain by (3.59) and (3.65) that for
Therefore, since we have by (3.63) that
we proved the results.
Proof.
From (3.3), Theorem 3.1, and the definitions of in Theorem 3.3, if for any we choose a fixed constant small enough, then there exists an integer such that we can make , , and small enough for with .
Proof of Corollary 3.5.
Since we have from Lemma 3.8 that , for and for , we obtain using the mathematical induction that
Therefore, from (3.65) we prove the result easily.
Proof of Theorem 3.6.
We know that from (3.39)
and from (3.44)
Suppose
Then since
we have
Here, we used that . Similarly, since
we have
4. Estimation of the Coefficients of HigherOrder HermiteFejér Interpolation
Let be nonnegative integers with . For we define the order HermiteFejér interpolation polynomials as follows: for each ,
Especially for each we see . The fundamental polynomials , of are defined by
Here, is fundamental Lagrange interpolation polynomial of degree (cf. [18, page 23]) given by
and satisfies
Then
In this section, we often denote and if it does not confuse us. Then we will first estimate for . Since we have
by induction on , we can estimate .
Theorem 4.1.
Let . Then one has for
In addition, one has that for
and if is odd, then one has that for
For define and for
Theorem 4.2 (cf. [10, Lemma ]).
Let and let . Then for there exists uniquely a sequence of positive numbers
and . Moreover, one has for
Theorem 4.3.
Suppose the same assumptions as Theorem 4.2. Given any , there exists a small fixed positive constant such that (4.11) holds satisfying and
for .
Theorem 4.4.
Let . Then one has for
On the other hand, one has for
Especially, if is odd, then one has
Especially, for we define the order HermiteFejér interpolation polynomials as the order HermiteFejér interpolation polynomials . Then we know that
where and
Then for the convergence theorem with respect to we have the following corollary.
Corollary 4.5.
Let . Then one has for
On the other hand, one has for
Especially, if is odd, then one has
Proof of Theorem 4.1.
Theorem 4.1 is shown by induction with respect to . The case of follows from (4.6), Corollary 3.5, and Theorem 3.6. Suppose that for the case of the results hold. Then from the following relation:
we have (4.7) and (4.8). Moreover, we obtain (4.9) from the following: for
Proof of Theorem 4.2.
Similarly to Theorem 4.1, we use mathematical induction with respect to . From Theorem 3.3 we know that for
and for
where and
Then from the following relations:
we have the results by induction with respect to .
Proof of Theorem 4.3.
It is proved by the same reason as the proof of Corollary 3.4.
Proof of Theorem 4.4.
To prove the result, we proceed by induction on . From (4.2) and (4.4) we know that and the following recurrence relation; for
When , so that (4.14) and (4.15) are satisfied for . From (4.7), (4.8), (4.28), and assumption of induction on , for , we have the results easily. When is odd, we know that
Therefore, similarly we have (4.16) from (4.8), (4.9), (4.28), and assumption of induction on .
Proof of Corollary 4.5.
Since , it is trivial from Theorem 4.4.
We rewrite the relation (4.10) in the form for ,
and for ,
Now, for every we will introduce an auxiliary polynomial determined by as the following lemma.
Lemma 4.6 (see[10, Lemma ]).

(i)
For , there exists a unique polynomial of degree such that
(4.32)
(ii) and , .
Since is a polynomial of degree , we can replace in (4.10) with , that is,
for an arbitrary and . We use the notation which coincides with if is an integer. Since , we have for in a neighborhood of and an arbitrary real number .
We can show that is a polynomial of degree at most with respect to for , where is the th partial derivative of with respect to at (see [6, page 199]). We prove these facts by induction on . For it is trivial. Suppose that it holds for . To simplify the notation, let and for a fixed . Then . By Leibniz's rule, we easily see that
which shows that is a polynomial of degree at most with respect to . Let , be defined by
Then is a polynomial of degree at most .
By Theorem 4.2 we have the following.
Lemma 4.7 (see[10, Lemma ]).
Let , and let be a positive constant. If and , then
Lemma 4.8 (see[10, Lemma ]).
If , then for ,
Lemma 4.9.
For positive integers and with
Proof.
If we let , then it suffices to show that . For every
By (4.24), (4.35), and (4.36), we see that the first sum has the form of
Then since
we know that . By (4.37) and (4.7), the second sum is bounded by . Here, we can make for arbitrary positive . Therefore, we obtain the following result: for every
Then the following theorem is important to show a divergence theorem with respect to where is an odd integer.
Theorem 4.10 (cf. [10, ()] and [15]).
For , there is a polynomial of degree such that for and the following relation holds. Let . Then one has an expression for , and :
where satisfies that for and for
Proof.
We prove (4.44) by induction on . Since and , (4.44) holds for . From (4.28) we write in the form of
Then by (4.12) and (4.14), is bounded by . For we suppose (4.44) and (4.45). Then we have for
where and which are defined in (4.11) and (4.44). Then using Lemma 4.9 and we have the following form:
Here, since
we see that . Therefore, we proved the result.
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Jung, H., Sakai, R. Derivatives of Orthonormal Polynomials and Coefficients of HermiteFejér Interpolation Polynomials with ExponentialType Weights. J Inequal Appl 2010, 816363 (2010). https://doi.org/10.1155/2010/816363
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DOI: https://doi.org/10.1155/2010/816363
Keywords
 Positive Constant
 Recurrence Relation
 Positive Root
 Interpolation Polynomial
 Lagrange Interpolation