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Hermite and Hermite-Fejér interpolation for Stieltjes polynomials

Author: H. S. Jung
Journal: Math. Comp. 75 (2006), 743-766
MSC (2000): Primary 41A05, 65D05
Published electronically: November 3, 2005
MathSciNet review: 2196990
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Abstract: Let $ w_{\lambda}(x):=(1-x^2)^{\lambda-1/2}$ and $ P_n^{(\lambda)}$ be the ultraspherical polynomials with respect to $ w_{\lambda}(x)$. Then we denote by $ E_{n+1}^{(\lambda)}$ the Stieltjes polynomials with respect to $ w_{\lambda}(x)$ satisfying

\begin{displaymath}\int_{-1}^1 w_{\lambda}(x) P_n^{(\lambda)}(x)E_{n+1}^{(\lambd... ...gin{cases} =0, & 0 \le m < n+1,\\ \neq 0, & m=n+1. \end{cases}\end{displaymath}      

In this paper, we show uniform convergence of the Hermite-Fejér interpolation polynomials $ H_{n+1}[\cdot]$ and $ {\mathcal H}_{2n+1}[\cdot]$ based on the zeros of the Stieltjes polynomials $ E_{n+1}^{(\lambda)}$ and the product $ E_{n+1}^{(\lambda)}P_n^{(\lambda)}$ for $ 0 \le \lambda \le 1$ and $ 0 \le \lambda \le 1/2$, respectively. To prove these results, we prove that the Lebesgue constants of Hermite-Fejér interpolation operators for the Stieltjes polynomials $ E_{n+1}^{(\lambda)}$ and the product $ E_{n+1}^{(\lambda)}P_n^{(\lambda)}$ are optimal, that is, the Lebesgue constants $ \Vert H_{n+1}\Vert _{\infty}(0 \le \lambda \le 1)$ and $ \Vert{\mathcal H}_{2n+1}\Vert _{\infty} (0 \le \lambda \le 1/2)$ have optimal order $ O(1)$. In the case of the Hermite-Fejér interpolation polynomials $ {\mathcal H}_{2n+1}[\cdot]$ for $ 1/2 < \lambda \le 1$, we prove weighted uniform convergence. Moreover, we give some convergence theorems of Hermite-Fejér and Hermite interpolation polynomials for $ 0 \le \lambda \le 1$ in weighted $ L_p$ norms.

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Additional Information

H. S. Jung
Affiliation: Division of Applied Mathmatics, KAIST, 373-1 Gusongdong, Yusongku, Taejon 305-701, Korea

Keywords: Generalized Stieltjes polynomial, Hermite--Fej\'er and Hermite interpolations, convergence, Lebesgue constant
Received by editor(s): March 9, 2004
Received by editor(s) in revised form: January 12, 2005
Published electronically: November 3, 2005
Additional Notes: This work was supported by Korea Research Foundation Grant (KRF-2002-050-C00003). The research on this project started when the author visited Professor Sven Ehrich in GSF-IBB. The author thanks Professor Sven Ehrich, Professor G. Mastroianni, and the referees for many kind suggestions and comments.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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