Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights
© H. S. Jung and R. Sakai. 2010
Received: 10 November 2009
Accepted: 14 January 2010
Published: 14 April 2010
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 .
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 .
We shall be interested in the following subclass of weights from .
In the following we introduce useful notations.
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.
In the following, we consider the exponential weights with the exponents . Then we have the following examples (see ).
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
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 ]).
It comes from Theorem 2.1 and (2.13).
Therefore, we have the result from (2.6).
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 ,
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
For the higher derivatives of and , we have the following results in [17, Theorem ].
Theorem 3.1 (see[17, Theorem ]).
In the following, we have the estimation of the higher-order derivatives of orthonormal polynomials.
because we know that from [3, Theorem ] and
To prove these results we need some lemmas.
Consequently we have (b).
(d) It is similar to (c). Consequently we have the following lemma.
Therefore, we have the results.
Proof of Theorem 3.3.
we proved the results.
Proof of Corollary 3.5.
Therefore, from (3.65) we prove the result easily.
Proof of Theorem 3.6.
4. Estimation of the Coefficients of Higher-Order Hermite-Fejér Interpolation
Here, is fundamental Lagrange interpolation polynomial of degree (cf. [18, page 23]) given by
Theorem 4.2 (cf. [10, Lemma ]).
Proof of Theorem 4.1.
Proof of Theorem 4.2.
Proof of Theorem 4.3.
It is proved by the same reason as the proof of Corollary 3.4.
Proof of Theorem 4.4.
Proof of Corollary 4.5.
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
By Theorem 4.2 we have the following.
Lemma 4.7 (see[10, Lemma ]).
Lemma 4.8 (see[10, Lemma ]).
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