Hermite and Hermite–Fejér interpolation for Stieltjes polynomials
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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{eqnarray*} \int _{-1}^1 w_{\lambda }(x) P_n^{(\lambda )}(x)E_{n+1}^{(\lambda )}(x) x^m dx \begin {cases} =0, & 0 \le m < n+1,\\ \neq 0, & m=n+1. \end{cases} \end{eqnarray*} 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 $\|H_{n+1}\|_{\infty }(0 \le \lambda \le 1)$ and $\|{\mathcal H}_{2n+1}\|_{\infty } (0 \le \lambda \le 1/2)$ 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.References
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Additional Information
- H. S. Jung
- Affiliation: Division of Applied Mathmatics, KAIST, 373-1 Gusongdong, Yusongku, Taejon 305-701, Korea
- Email: hsjung@amath.kaist.ac.kr
- 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.
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 75 (2006), 743-766
- MSC (2000): Primary 41A05, 65D05
- DOI: https://doi.org/10.1090/S0025-5718-05-01795-3
- MathSciNet review: 2196990