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

Abstract

Let , and let be an even function. In this paper, we consider the exponential-type 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 higher-order derivatives of and the coefficients of the higher-order Hermite-Fejér 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

(1.1)

Then we can construct the orthonormal polynomials of degree with respect to . That is,

(1.2)

We denote the zeros of by

(1.3)

A function is said to be quasi-increasing 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

(1.4)

(d)The function

(1.5)

is quasi-increasing in with

(1.6)

(e)There exists such that

(1.7)

Then we write . If there also exist a compact subinterval of and such that

(1.8)

then we write .

In the following we introduce useful notations.

(a)Mhaskar-Rahmanov-Saff (MRS) numbers is defined as the positive roots of the following equations:

(1.9)

(b)Let

(1.10)

(c)The function is defined as the following:

(1.11)

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

(1.12)

(c)There exist constants and such that on

(1.13)

Then we write . Furthermore, and satisfies one of the following.

(a) is quasi-increasing 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 ,

(1.14)

More precisely, define for , , and ,

(1.15)

where if , otherwise , and define

(1.16)

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 quasi-increasing on .

(c)When , if , then there exists a constant such that is quasi-increasing 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 higher-order and to estimate the higher-order derivatives of at the zeros of and the coefficients of the higher-order Hermite-Fejér interpolation polynomials based at the zeros of . More precisely, we will estimate the higher-order 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 higher-order Hermite-Fejér interpolation polynomials (see [5–16]).

This paper is organized as follows. In Section 2, we will obtain the differential equations for of higher-order. In Section 3, we will give estimations of higher-order 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 higher-order 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 higher-order Hermite-Fejér interpolation polynomials based at the zeros of , in Section 4.

2. Higher-Order 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:

(2.1)

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

(2.2)

Theorem 2.1 (cf. [6, Theorem ]).

Let and . Then for one has the second-order differential relation as follows:

(2.3)

Here, one knows that for any integer ,

(2.4)

where

(2.5)

Especially, when is odd, one has

(2.6)

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

(2.7)

and so if we use the recurrence formula

(2.8)

and use (2.2) too, then we obtain the following:

(2.9)

We differentiate the left and right sides of (2.2) and substitute (2.2) and (2.9). Then consequently, we have, for ,

(2.10)

Using the recurrence formula (2.8) and , we have

(2.11)

because is an odd function. Therefore, we have

(2.12)

When is odd, since , (2.6) is proved.

For the higher-order differential equation for orthonormal polynomials, we see that for and

(2.13)

Let for nonnegative integer . In the following theorem, we show the higher-order differential equation for orthonormal polynomials.

Theorem 2.2.

Let and . Let and . Then one has the following equation for :

(2.14)

where

(2.15)

and for and

(2.16)

and for

(2.17)

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

(2.18)

where and for

(2.19)

Proof.

Let be odd. Then we will consider (2.6). Since , we have

(2.20)

and we have

(2.21)

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

(2.22)

where 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 ] we know that there exist and such that for and ,

(3.1)

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

(3.2)

where there exists a constant such that

(3.3)

(3.4)

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

(3.5)

Moreover, there exists such that for and ,

(3.6)

with as .

Corollary 3.2.

Let . Then there exists a positive constant such that one has for and ,

(3.7)

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

Theorem 3.3.

Let and . Then for the following equality holds for large enough:

(3.8)

where

(3.9)

and . Moreover, for

(3.10)

Here,

(3.11)

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

(3.12)

for .

Corollary 3.5.

For and

(3.13)

Theorem 3.6.

Let and let , . Then

(3.14)

and especially if is even, then

(3.15)

We note that for large enough,

(3.16)

because we know that from [3, Theorem ] and

(3.17)

To prove these results we need some lemmas.

Lemma 3.7.

• (a) For

  

  (3.18)

 (b) For

(3.19)

 (c) For

(3.20)

 (d) Let . Then for

(3.21)

and for

(3.22)

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

(3.23)

Moreover, for and ,

(3.24)

 (b) For and , there exists satisfying as such that

(3.25)

Moreover, for and ,

(3.26)

 (c) For and , there exists satisfying as such that

(3.27)

Moreover, for and ,

(3.28)

 (d) For and , there exists satisfying as such that

(3.29)

Moreover, for and ,

(3.30)

Proof.

• (a) Since , we prove it by Theorem 3.1.

  (b) For , we see

(3.31)

From (3.18), we know that . Therefore by (3.19), (3.21), and (3.6) we have for

(3.32)

and for we have by (3.21) and (3.22)

(3.33)

Consequently we have (b).

  (c) Next we estimate . Suppose . Let us set . By (3.6) and (3.20) we have

(3.34)

For , we obtain the same estimate as

(3.35)

For , we have similarly to the case of

(3.36)

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

Lemma 3.9.

Let , , and . Let . Then

(3.37)

where is defined in Theorem 3.3 and for

(3.38)

Moreover, for ,

(3.39)

Proof.

Since

(3.40)

we have (3.39) for by (3.5). For we have from (3.6) and (3.19) that

(3.41)

Moreover, we can obtain (3.38) for from the above easily.

Lemma 3.10.

Let and . Let . Then for

(3.42)

with , where , , and are defined in Theorem 3.3. For one has

(3.43)

On the other hand, one has for ,

(3.44)

Proof.

First, we know that

(3.45)

Suppose . Since from (3.18) and (3.19)

(3.46)

we have from (3.6)

(3.47)

Since

(3.48)

we know from (3.6) that

(3.49)

Therefore we have for

(3.50)

Since from (3.3)

(3.51)

and similarly

(3.52)

we have

(3.53)

Then we have

(3.54)

Therefore, since

(3.55)

there exist constants with such that we have for

(3.56)

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)

(3.57)

and , we have from Lemma 3.8

(3.58)

Therefore, we have (3.44) for .

Lemma 3.11.

Let and . Let . Then for , there exists satisfying as such that

(3.59)

Moreover, one has for ,

(3.60)

Proof.

For we have from Lemma 3.8 that there exists satisfying as such that

(3.61)

Similarly, for and ,

(3.62)

Therefore, we have the results.

Proof of Theorem 3.3.

First we know that the following differential equation is satisfied:

(3.63)

Suppose . Then since we see from (3.63) and (3.38) that

(3.64)

we have by (3.63) and mathematical induction

(3.65)

Next, suppose . More precisely, from Lemma 3.9 we have

(3.66)

Then by (3.63), (3.42), and (3.66) there exists a constant with

(3.67)

such that we have that

(3.68)

Suppose that there exist constants with such that

(3.69)

(3.70)

Then we have by (3.38) and (3.70)

(3.71)

and we have by (3.42) and (3.69)

(3.72)

where and . Also, we have by (3.59) that for

(3.73)

Therefore, there exists satisfying such that

(3.74)

Moreover, we have by (3.37) and (3.65)

(3.75)

and by (3.43) and (3.70)

(3.76)

Also we obtain by (3.59) and (3.65) that for

(3.77)

Therefore, since we have by (3.63) that

(3.78)

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

(3.79)

Therefore, from (3.65) we prove the result easily.

Proof of Theorem 3.6.

We know that from (3.39)

(3.80)

and from (3.44)

(3.81)

Suppose

(3.82)

Then since

(3.83)

we have

(3.84)

Here, we used that . Similarly, since

(3.85)

we have

(3.86)

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

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

(4.1)

Especially for each we see . The fundamental polynomials , of are defined by

(4.2)

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

(4.3)

and satisfies

(4.4)

Then

(4.5)

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

(4.6)

by induction on , we can estimate .

Theorem 4.1.

Let . Then one has for

(4.7)

In addition, one has that for

(4.8)

and if is odd, then one has that for

(4.9)

For define and for

(4.10)

Theorem 4.2 (cf. [10, Lemma ]).

Let and let . Then for there exists uniquely a sequence of positive numbers

(4.11)

and . Moreover, one has for

(4.12)

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

(4.13)

for .

Theorem 4.4.

Let . Then one has for

(4.14)

On the other hand, one has for

(4.15)

Especially, if is odd, then one has

(4.16)

Especially, for we define the -order Hermite-Fejér interpolation polynomials as the -order Hermite-Fejér interpolation polynomials . Then we know that

(4.17)

where and

(4.18)

Then for the convergence theorem with respect to we have the following corollary.

Corollary 4.5.

Let . Then one has for

(4.19)

On the other hand, one has for

(4.20)

Especially, if is odd, then one has

(4.21)

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:

(4.22)

we have (4.7) and (4.8). Moreover, we obtain (4.9) from the following: for

(4.23)

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

(4.24)

and for

(4.25)

where and

(4.26)

Then from the following relations:

(4.27)

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

(4.28)

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

(4.29)

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 ,

(4.30)

and for ,

(4.31)

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,

(4.33)

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

(4.34)

which shows that is a polynomial of degree at most with respect to . Let , be defined by

(4.35)

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

(4.36)

(4.37)

Lemma 4.8 (see[10, Lemma ]).

If , then for ,

(4.38)

Lemma 4.9.

For positive integers and with

(4.39)

Proof.

If we let , then it suffices to show that . For every

(4.40)

By (4.24), (4.35), and (4.36), we see that the first sum has the form of

(4.41)

Then since

(4.42)

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

(4.43)

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 :

(4.44)

where satisfies that for and for

(4.45)

Proof.

We prove (4.44) by induction on . Since and , (4.44) holds for . From (4.28) we write in the form of

(4.46)

Then by (4.12) and (4.14), is bounded by . For we suppose (4.44) and (4.45). Then we have for

(4.47)

where and which are defined in (4.11) and (4.44). Then using Lemma 4.9 and we have the following form:

(4.48)

Here, since

(4.49)

we see that . Therefore, we proved the result.