- Research Article
- Open Access
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 .
- Positive Constant
- Recurrence Relation
- Positive Root
- Interpolation Polynomial
- Lagrange Interpolation
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 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 .
Let be even and satisfy the following properties.
(a) is continuous in , with .
(b) exists and is positive in .
then we write .
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.
Let and let be a positive integer. Assume that is -times continuously differentiable on and satisfies the followings.
(a) exists and , are positive for .
(c)There exist constants and such that on
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 ,
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 ).
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.
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 ]).
where is the polynomial of degree with .
When is odd, since , (2.6) is proved.
For the higher-order differential equation for orthonormal polynomials, we see that for and
Let for nonnegative integer . In the following theorem, we show the higher-order differential equation for orthonormal polynomials.
It comes from Theorem 2.1 and (2.13).
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 higher-order derivatives of orthonormal polynomials at the zeros of orthonormal polynomials with respect to exponential-type weights.
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 ]).
with as .
In the following, we have the estimation of the higher-order derivatives of orthonormal polynomials.
We note that for large enough,
because we know that from [3, Theorem ] and
To prove these results we need some lemmas.
(d) Let . Then for
For and , there exists satisfying as such that
(b) For and , there exists satisfying as such that
(c) For and , there exists satisfying as such that
(d) For and , there exists satisfying as such that
Since , we prove it by Theorem 3.1.
(b) For , we see
Consequently we have (b).
(c) Next we estimate . Suppose . Let us set . By (3.6) and (3.20) we have
(d) It is similar to (c). Consequently we have the following lemma.
Moreover, we can obtain (3.38) for from the above easily.
Therefore, we have (3.44) for .
Therefore, we have the results.
Proof of Theorem 3.3.
we proved the results.
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.
Therefore, from (3.65) we prove the result easily.
Proof of Theorem 3.6.
Let be nonnegative integers with . For we define the -order Hermite-Fejér 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
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 .
For define and for
Theorem 4.2 (cf. [10, Lemma ]).
Especially, for we define the -order Hermite-Fejér interpolation polynomials as the -order Hermite-Fejér interpolation polynomials . Then we know that
Then for the convergence theorem with respect to we have the following corollary.
Proof of Theorem 4.1.
Proof of Theorem 4.2.
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.
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.
(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 ]).
Lemma 4.8 (see[10, Lemma ]).
Then the following theorem is important to show a divergence theorem with respect to where is an odd integer.
we see that . Therefore, we proved the result.
- Levin E, Lubinsky DS: Orthogonal Polynomials for Exponential Weights, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 4. Springer, New York, NY, USA; 2001:xii+476.View ArticleGoogle Scholar
- Jung HS, Sakai R: Inequalities with exponential weights. Journal of Computational and Applied Mathematics 2008, 212(2):359–373. 10.1016/j.cam.2006.12.011MathSciNetView ArticleMATHGoogle Scholar
- Jung HS, Sakai R: Orthonormal polynomials with exponential-type weights. Journal of Approximation Theory 2008, 152(2):215–238. 10.1016/j.jat.2007.12.004MathSciNetView ArticleMATHGoogle Scholar
- Jung HS, Sakai R: Specific examples of exponential weights. Communications of the Korean Mathematical Society 2009, 24(2):303–319. 10.4134/CKMS.2009.24.2.303MathSciNetView ArticleMATHGoogle Scholar
- Kasuga T, Sakai R: Uniform or mean convergence of Hermite-Fejér interpolation of higher order for Freud weights. Journal of Approximation Theory 1999, 101(2):330–358. 10.1006/jath.1999.3371MathSciNetView ArticleMATHGoogle Scholar
- Kasuga T, Sakai R: Orthonormal polynomials with generalized Freud-type weights. Journal of Approximation Theory 2003, 121(1):13–53. 10.1016/S0021-9045(02)00041-2MathSciNetView ArticleMATHGoogle Scholar
- Kasuga T, Sakai R: Orthonormal polynomials for generalized Freud-type weights and higher-order Hermite-Fejér interpolation polynomials. Journal of Approximation Theory 2004, 127(1):1–38. 10.1016/j.jat.2004.01.006MathSciNetView ArticleMATHGoogle Scholar
- Kasuga T, Sakai R: Orthonormal polynomials for Laguerre-type weights. Far East Journal of Mathematical Sciences 2004, 15(1):95–105.MathSciNetMATHGoogle Scholar
- Kasuga T, Sakai R: Conditions for uniform or mean convergence of higher order Hermite-Fejér interpolation polynomials with generalized Freud-type weights. Far East Journal of Mathematical Sciences 2005, 19(2):145–199.MathSciNetMATHGoogle Scholar
- Kanjin Y, Sakai R: Pointwise convergence of Hermite-Fejér interpolation of higher order for Freud weights. The Tohoku Mathematical Journal 1994, 46(2):181–206. 10.2748/tmj/1178225757MathSciNetView ArticleMATHGoogle Scholar
- Kanjin Y, Sakai R: Convergence of the derivatives of Hermite-Fejér interpolation polynomials of higher order based at the zeros of Freud polynomials. Journal of Approximation Theory 1995, 80(3):378–389. 10.1006/jath.1995.1024MathSciNetView ArticleMATHGoogle Scholar
- Sakai R: Hermite-Fejér interpolation. In Approximation Theory (Kecskemét, 1990), Colloquia Mathematica Societatis Janos Bolyaiá. Volume 58. North-Holland, Amsterdam, The Netherlands; 1991:591–601.Google Scholar
- Sakai R: Hermite-Fejér interpolation prescribing higher order derivatives. In Progress in Approximation Theory. Edited by: Nevai P, Pinkus A. Academic Press, Boston, Mass, USA; 1991:731–759.Google Scholar
- Sakai R: Certain unbounded Hermite-Fejér interpolatory polynomial operators. Acta Mathematica Hungarica 1992, 59(1–2):111–114. 10.1007/BF00052097MathSciNetView ArticleMATHGoogle Scholar
- Sakai R, Vértesi P: Hermite-Fejér interpolations of higher order. III. Studia Scientiarum Mathematicarum Hungarica 1993, 28(1–2):87–97.MathSciNetMATHGoogle Scholar
- Sakai R, Vértesi P: Hermite-Fejér interpolations of higher order. IV. Studia Scientiarum Mathematicarum Hungarica 1993, 28(3–4):379–386.MathSciNetMATHGoogle Scholar
- Jung HS, Sakai R: Derivatives of integrating functions for orthonormal polynomials with exponential-type weights. Journal of Inequalities and Applications 2009, 2009:-22.Google Scholar
- Freud G: Orthogonal Polynomials. Pergamon Press, Oxford, UK; 1971.MATHGoogle Scholar
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