Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Open Access
Research Article

Abstract

Let Open image in new window , and let Open image in new window be an even function. In this paper, we consider the exponential-type weights Open image in new window , and the orthonormal polynomials Open image in new window of degree Open image in new window with respect to Open image in new window . So, we obtain a certain differential equation of higher order with respect to Open image in new window and we estimate the higher-order derivatives of Open image in new window and the coefficients of the higher-order Hermite-Fejér interpolation polynomial based at the zeros of Open image in new window .

Keywords

Positive Constant Recurrence Relation Positive Root Interpolation Polynomial Lagrange Interpolation 

1. Introduction

Let Open image in new window and Open image in new window . Let Open image in new window be an even function and let Open image in new window be such that Open image in new window for all Open image in new window For Open image in new window , we set

Then we can construct the orthonormal polynomials Open image in new window of degree Open image in new window with respect to Open image in new window . That is,

We denote the zeros of Open image in new window by

A function Open image in new window is said to be quasi-increasing if there exists Open image in new window such that Open image in new window for Open image in new window . For any two sequences Open image in new window and Open image in new window of nonzero real numbers (or functions), we write Open image in new window if there exists a constant Open image in new window independent of Open image in new window (or Open image in new window ) such that Open image in new window for Open image in new window being large enough. We write Open image in new window if Open image in new window and Open image in new window . We denote the class of polynomials of degree at most Open image in new window by Open image in new window .

Throughout Open image in new window denote positive constants independent of Open image in new window , and polynomials of degree at most Open image in new window . 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 Open image in new window be even and satisfy the following properties.

(a) Open image in new window is continuous in Open image in new window , with Open image in new window .

(b) Open image in new window exists and is positive in Open image in new window .

(c)One has
(d)The function
is quasi-increasing in Open image in new window with
(e)There exists Open image in new window such that
Then we write Open image in new window . If there also exist a compact subinterval Open image in new window of Open image in new window and Open image in new window such that

then we write Open image in new window .

In the following we introduce useful notations.

(a)Mhaskar-Rahmanov-Saff (MRS) numbers Open image in new window is defined as the positive roots of the following equations:
(c)The function Open image in new window is defined as the following:

In [2, 3] we estimated the orthonormal polynomials Open image in new window associated with the weight Open image in new window and obtained some results with respect to the derivatives of orthonormal polynomials Open image in new window . In this paper, we will obtain the higher derivatives of Open image in new window . To estimate of the higher derivatives of the orthonormal polynomials sequence, we need further assumptions for Open image in new window as follows.

Definition 1.2.

Let Open image in new window and let Open image in new window be a positive integer. Assume that Open image in new window is Open image in new window -times continuously differentiable on Open image in new window and satisfies the followings.

(a) Open image in new window exists and Open image in new window , Open image in new window are positive for Open image in new window .

(b)There exist positive constants Open image in new window such that for Open image in new window

(c)There exist constants Open image in new window and Open image in new window such that on Open image in new window

Then we write Open image in new window . Furthermore, Open image in new window and Open image in new window satisfies one of the following.

(a) Open image in new window is quasi-increasing on a certain positive interval Open image in new window .

(b) Open image in new window is nondecreasing on a certain positive interval Open image in new window .

(c)There exists a constant Open image in new window such that Open image in new window on Open image in new window .

Then we write Open image in new window .

Now, consider some typical examples of Open image in new window . Define for Open image in new window and Open image in new window ,

More precisely, define for Open image in new window , Open image in new window , Open image in new window and Open image in new window ,

where Open image in new window if Open image in new window , otherwise Open image in new window , and define

In the following, we consider the exponential weights with the exponents Open image in new window . Then we have the following examples (see [4]).

Example 1.3.

Let Open image in new window be a positive integer. Let Open image in new window . Then one has the following.

(a) Open image in new window belongs to Open image in new window .

(b)If Open image in new window and Open image in new window , then there exists a constant Open image in new window such that Open image in new window is quasi-increasing on Open image in new window .

(c)When Open image in new window , if Open image in new window , then there exists a constant Open image in new window such that Open image in new window is quasi-increasing on Open image in new window , and if Open image in new window , then Open image in new window is quasidecreasing on Open image in new window .

(d)When Open image in new window and Open image in new window , Open image in new window is nondecreasing on a certain positive interval Open image in new window .

In this paper, we will consider the orthonormal polynomials Open image in new window with respect to the weight class Open image in new window . Our main themes in this paper are to obtain a certain differential equation for Open image in new window of higher-order and to estimate the higher-order derivatives of Open image in new window at the zeros of Open image in new window and the coefficients of the higher-order Hermite-Fejér interpolation polynomials based at the zeros of Open image in new window . More precisely, we will estimate the higher-order derivatives of Open image in new window at the zeros of Open image in new window 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 higher-order Hermite-Fejér interpolation polynomials (see [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]).

This paper is organized as follows. In Section 2, we will obtain the differential equations for Open image in new window of higher-order. In Section 3, we will give estimations of higher-order derivatives of Open image in new window at the zeros of Open image in new window in a certain finite interval for two cases of an odd order and of an even order. In addition, we estimate the higher-order derivatives of Open image in new window at all zeros of Open image in new window for two cases of an odd order and of an even order. Furthermore, we will estimate the coefficients of higher-order Hermite-Fejér interpolation polynomials based at the zeros of Open image in new window , in Section 4.

2. Higher-Order Differential Equation for Orthonormal Polynomials

In the rest of this paper we often denote Open image in new window and Open image in new window simply by Open image in new window and Open image in new window , respectively. Let Open image in new window if Open image in new window is odd, Open image in new window otherwise, and define the integrating functions Open image in new window and Open image in new window with respect to Open image in new window as follows:

where Open image in new window and Open image in new window . Then in [3, Theorem Open image in new window ] we have a relation of the orthonormal polynomial Open image in new window with respect to the weight Open image in new window :

Theorem 2.1 (cf. [6, Theorem Open image in new window ]).

Let Open image in new window and Open image in new window . Then for Open image in new window one has the second-order differential relation as follows:
Here, one knows that for any integer Open image in new window ,
Especially, when Open image in new window is odd, one has

where Open image in new window is the polynomial of degree Open image in new window with Open image in new window .

Proof.

We may similarly repeat the calculation [6, Proof of Theorem Open image in new window ], and then we obtain the results. We stand for Open image in new window simply. Applying (2.2) to Open image in new window 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 Open image in new window ,
Using the recurrence formula (2.8) and Open image in new window , we have
because Open image in new window is an odd function. Therefore, we have

When Open image in new window is odd, since Open image in new window , (2.6) is proved.

For the higher-order differential equation for orthonormal polynomials, we see that for Open image in new window and Open image in new window

Let Open image in new window for nonnegative integer Open image in new window . In the following theorem, we show the higher-order differential equation for orthonormal polynomials.

Theorem 2.2.

Proof.

It comes from Theorem 2.1 and (2.13).

Corollary 2.3.

Under the same assumptions as Theorem 2.1, if Open image in new window is odd, then

Proof.

Let Open image in new window be odd. Then we will consider (2.6). Since Open image in new window , we have
and we have

Therefore, we have the result from (2.6).

In the rest of this paper, we let Open image in new window and Open image in new window for positive integer Open image in new window and assume that Open image in new window for Open image in new window and

where Open image in new window is defined in (1.13).

In Section 3, we will estimate the higher-order derivatives of orthonormal polynomials at the zeros of orthonormal polynomials with respect to exponential-type weights.

3. Estimation of Higher-Order Derivatives of Orthonormal Polynomials

From [3, Theorem Open image in new window ] we know that there exist Open image in new window and Open image in new window such that for Open image in new window and Open image in new window ,

If Open image in new window is unbounded, then (2.22) is trivially satisfied. Additionally we have, from [17, Theorem Open image in new window ], that if we assume that Open image in new window is nondecreasing, then for Open image in new window with Open image in new window

where there exists a constant Open image in new window such that

Here, Open image in new window and Open image in new window for some Open image in new window .

For the higher derivatives of Open image in new window and Open image in new window , we have the following results in [17, Theorem Open image in new window ].

Theorem 3.1 (see[17, Theorem Open image in new window ]).

with Open image in new window as Open image in new window .

Corollary 3.2.

Let Open image in new window . Then there exists a positive constant Open image in new window such that one has for Open image in new window and Open image in new window ,

In the following, we have the estimation of the higher-order derivatives of orthonormal polynomials.

Theorem 3.3.

Corollary 3.4.

Suppose the same assumptions as Theorem 3.3. Given any Open image in new window , there exists a small fixed positive constant Open image in new window such that (3.8) holds satisfying Open image in new window and

for Open image in new window .

Corollary 3.5.

Theorem 3.6.

and especially if Open image in new window is even, then

We note that for Open image in new window large enough,

because we know that Open image in new window from [3, Theorem Open image in new window ] and

To prove these results we need some lemmas.

 (b) For Open image in new window

 (c) For Open image in new window

 (d) Let Open image in new window . Then for Open image in new window

Proof.
  1. (a)

    It is [1, Lemma Open image in new window (c)]. (b) It is [1, Lemma Open image in new window (c)]. (c) It comes from (3.1). (d) Since Open image in new window , Open image in new window is increasing. So, we obtain (d) by (1.12).

     

Lemma 3.8.

 (b) For Open image in new window and Open image in new window , there exists Open image in new window satisfying Open image in new window as Open image in new window such that

 (c) For Open image in new window and Open image in new window , there exists Open image in new window satisfying Open image in new window as Open image in new window such that

 (d) For Open image in new window and Open image in new window , there exists Open image in new window satisfying Open image in new window as Open image in new window such that

Proof.
  1. (a)

    Since Open image in new window , we prove it by Theorem 3.1.

     

  (b) For Open image in new window , we see

From (3.18), we know that Open image in new window . Therefore by (3.19), (3.21), and (3.6) we have for Open image in new window
and for Open image in new window we have by (3.21) and (3.22)

Consequently we have (b).

  (c) Next we estimate Open image in new window . Suppose Open image in new window . Let us set Open image in new window . By (3.6) and (3.20) we have

For Open image in new window , we have similarly to the case of Open image in new window

  (d) It is similar to (c). Consequently we have the following lemma.

Lemma 3.9.

where Open image in new window is defined in Theorem 3.3 and for Open image in new window

Proof.

we have (3.39) for Open image in new window by (3.5). For Open image in new window we have from (3.6) and (3.19) that

Moreover, we can obtain (3.38) for Open image in new window from the above easily.

Lemma 3.10.

On the other hand, one has for Open image in new window ,

Proof.

First, we know that
Suppose Open image in new window . Since from (3.18) and (3.19)
we have from (3.6)
we know from (3.6) that
Since from (3.3)
and similarly
Then we have
Therefore, since
Especially, from the above estimates we can see (3.43) for Open image in new window . On the other hand, suppose Open image in new window . Then since from Theorem 2.1 and (3.5)
and Open image in new window , we have from Lemma 3.8

Therefore, we have (3.44) for Open image in new window .

Lemma 3.11.

Moreover, one has for Open image in new window ,

Proof.

For Open image in new window we have from Lemma 3.8 that there exists Open image in new window satisfying Open image in new window as Open image in new window such that

Therefore, we have the results.

Proof of Theorem 3.3.

First we know that the following differential equation is satisfied:
Suppose Open image in new window . Then since we see from (3.63) and (3.38) that
we have by (3.63) and mathematical induction
Next, suppose Open image in new window . More precisely, from Lemma 3.9 we have
Then by (3.63), (3.42), and (3.66) there exists a constant Open image in new window with
such that we have that
Suppose that there exist constants Open image in new window with Open image in new window such that
Then we have by (3.38) and (3.70)
and we have by (3.42) and (3.69)
Therefore, there exists Open image in new window satisfying Open image in new window 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 Open image in new window
Therefore, since we have by (3.63) that

we proved the results.

Proof.

From (3.3), Theorem 3.1, and the definitions of Open image in new window in Theorem 3.3, if for any Open image in new window we choose a fixed constant Open image in new window small enough, then there exists an integer Open image in new window such that we can make Open image in new window , Open image in new window , and Open image in new window small enough for Open image in new window with Open image in new window .

Proof of Corollary 3.5.

Since we have from Lemma 3.8 that Open image in new window , Open image in new window for Open image in new window and Open image in new window for Open image in new window , 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)
Then since
Here, we used that Open image in new window . Similarly, since

4. Estimation of the Coefficients of Higher-Order Hermite-Fejér Interpolation

Let Open image in new window be nonnegative integers with Open image in new window . For Open image in new window we define the Open image in new window -order Hermite-Fejér interpolation polynomials Open image in new window as follows: for each Open image in new window ,

Especially for each Open image in new window we see Open image in new window . The fundamental polynomials Open image in new window , Open image in new window of Open image in new window are defined by

Here, Open image in new window is fundamental Lagrange interpolation polynomial of degree Open image in new window (cf. [18, page 23]) given by

and Open image in new window satisfies

Then

In this section, we often denote Open image in new window and Open image in new window if it does not confuse us. Then we will first estimate Open image in new window for Open image in new window . Since we have

by induction on Open image in new window , we can estimate Open image in new window .

Theorem 4.1.

In addition, one has that for Open image in new window

For Open image in new window define Open image in new window and for Open image in new window

Theorem 4.2 (cf. [10, Lemma Open image in new window ]).

Let Open image in new window and let Open image in new window . Then for Open image in new window there exists uniquely a sequence Open image in new window of positive numbers

Theorem 4.3.

Suppose the same assumptions as Theorem 4.2. Given any Open image in new window , there exists a small fixed positive constant Open image in new window such that (4.11) holds satisfying Open image in new window and

for Open image in new window .

Theorem 4.4.

On the other hand, one has for Open image in new window
Especially, if Open image in new window is odd, then one has

Especially, for Open image in new window we define the Open image in new window -order Hermite-Fejér interpolation polynomials Open image in new window as the Open image in new window -order Hermite-Fejér interpolation polynomials Open image in new window . Then we know that

where Open image in new window and

Then for the convergence theorem with respect to Open image in new window we have the following corollary.

Corollary 4.5.

On the other hand, one has for Open image in new window
Especially, if Open image in new window is odd, then one has

Proof of Theorem 4.1.

Theorem 4.1 is shown by induction with respect to Open image in new window . The case of Open image in new window follows from (4.6), Corollary 3.5, and Theorem 3.6. Suppose that for the case of Open image in new window the results hold. Then from the following relation:
we have (4.7) and (4.8). Moreover, we obtain (4.9) from the following: for Open image in new window

Proof of Theorem 4.2.

Similarly to Theorem 4.1, we use mathematical induction with respect to Open image in new window . From Theorem 3.3 we know that for Open image in new window
Then from the following relations:

we have the results by induction with respect to Open image in new window .

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 Open image in new window . From (4.2) and (4.4) we know that Open image in new window and the following recurrence relation; for Open image in new window
When Open image in new window , Open image in new window so that (4.14) and (4.15) are satisfied for Open image in new window . From (4.7), (4.8), (4.28), and assumption of induction on Open image in new window , for Open image in new window , we have the results easily. When Open image in new window is odd, we know that

Therefore, similarly we have (4.16) from (4.8), (4.9), (4.28), and assumption of induction on Open image in new window .

Proof of Corollary 4.5.

Since Open image in new window , it is trivial from Theorem 4.4.

We rewrite the relation (4.10) in the form for Open image in new window ,

and for Open image in new window ,

Now, for every Open image in new window we will introduce an auxiliary polynomial determined by Open image in new window as the following lemma.

Lemma 4.6 (see[10, Lemma Open image in new window ]).
  1. (i)
    For Open image in new window , there exists a unique polynomial Open image in new window of degree Open image in new window such that
     

 (ii) Open image in new window and Open image in new window , Open image in new window .

Since Open image in new window is a polynomial of degree Open image in new window , we can replace Open image in new window in (4.10) with Open image in new window , that is,

for an arbitrary Open image in new window and Open image in new window . We use the notation Open image in new window which coincides with Open image in new window if Open image in new window is an integer. Since Open image in new window , we have Open image in new window for Open image in new window in a neighborhood of Open image in new window and an arbitrary real number Open image in new window .

We can show that Open image in new window is a polynomial of degree at most Open image in new window with respect to Open image in new window for Open image in new window , where Open image in new window is the Open image in new window th partial derivative of Open image in new window with respect to Open image in new window at Open image in new window (see [6, page 199]). We prove these facts by induction on Open image in new window . For Open image in new window it is trivial. Suppose that it holds for Open image in new window . To simplify the notation, let Open image in new window and Open image in new window for a fixed Open image in new window . Then Open image in new window . By Leibniz's rule, we easily see that

which shows that Open image in new window is a polynomial of degree at most Open image in new window with respect to Open image in new window . Let Open image in new window , Open image in new window be defined by

Then Open image in new window is a polynomial of degree at most Open image in new window .

By Theorem 4.2 we have the following.

Lemma 4.7 (see[10, Lemma Open image in new window ]).

Lemma 4.8 (see[10, Lemma Open image in new window ]).

Lemma 4.9.

Proof.

If we let Open image in new window , then it suffices to show that Open image in new window . For every Open image in new window
By (4.24), (4.35), and (4.36), we see that the first sum Open image in new window has the form of
Then since
we know that Open image in new window . By (4.37) and (4.7), the second sum Open image in new window is bounded by Open image in new window . Here, we can make Open image in new window for arbitrary positive Open image in new window . Therefore, we obtain the following result: for every Open image in new window

Then the following theorem is important to show a divergence theorem with respect to Open image in new window where Open image in new window is an odd integer.

Theorem 4.10 (cf. [10, ( Open image in new window )] and [15]).

For Open image in new window , there is a polynomial Open image in new window of degree Open image in new window such that Open image in new window for Open image in new window and the following relation holds. Let Open image in new window . Then one has an expression for Open image in new window , and Open image in new window :

Proof.

We prove (4.44) by induction on Open image in new window . Since Open image in new window and Open image in new window , (4.44) holds for Open image in new window . From (4.28) we write Open image in new window in the form of
Then by (4.12) and (4.14), Open image in new window is bounded by Open image in new window . For Open image in new window we suppose (4.44) and (4.45). Then we have for Open image in new window
where Open image in new window and Open image in new window which are defined in (4.11) and (4.44). Then using Lemma 4.9 and Open image in new window we have the following form:
Here, since

we see that Open image in new window . Therefore, we proved the result.

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Copyright information

© H. S. Jung and R. Sakai. 2010

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.

Authors and Affiliations

  1. 1.Department of Mathematics EducationSungkyunkwan UniversitySeoulSouth Korea
  2. 2.Department of MathematicsMeijo UniversityNagoyaJapan

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