Hermite and Hermite–Fejér interpolation for Stieltjes polynomials

Jung, H.

doi:10.1090/S0025-5718-05-01795-3

Hermite and Hermite–Fejér interpolation for Stieltjes polynomials
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Math. Comp. 75 (2006), 743-766 Request permission

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.

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