Hermite and Hermite–Fejér interpolation for Stieltjes polynomials

Abstract

Let w λ ( x ) := ( 1 − x 2 ) λ − 1 / 2 w_{\lambda }(x):=(1-x^2)^{\lambda -1/2} and P n ( λ ) P_n^{(\lambda )} be the ultraspherical polynomials with respect to w λ ( x ) w_{\lambda }(x) . Then we denote by E n + 1 ( λ ) E_{n+1}^{(\lambda )} the Stieltjes polynomials with respect to w λ ( x ) w_{\lambda }(x) satisfying ∫ − 1 1 w λ ( x ) P n ( λ ) ( x ) E n + 1 ( λ ) ( x ) x m d x { = 0 , a m p ; 0 ≤ m > n + 1 , ≠ 0 , a m p ; m = n + 1. \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 [ ⋅ ] H_{n+1}[\cdot ] and H 2 n + 1 [ ⋅ ] {\mathcal H}_{2n+1}[\cdot ] based on the zeros of the Stieltjes polynomials E n + 1 ( λ ) E_{n+1}^{(\lambda )} and the product E n + 1 ( λ ) P n ( λ ) E_{n+1}^{(\lambda )}P_n^{(\lambda )} for 0 ≤ λ ≤ 1 0 \le \lambda \le 1 and 0 ≤ λ ≤ 1 / 2 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 ( λ ) E_{n+1}^{(\lambda )} and the product E n + 1 ( λ ) P n ( λ ) E_{n+1}^{(\lambda )}P_n^{(\lambda )} are optimal, that is, the Lebesgue constants ‖ H n + 1 ‖ ∞ ( 0 ≤ λ ≤ 1 ) \|H_{n+1}\|_{\infty }(0 \le \lambda \le 1) and ‖ H 2 n + 1 ‖ ∞ ( 0 ≤ λ ≤ 1 / 2 ) \|{\mathcal H}_{2n+1}\|_{\infty } (0 \le \lambda \le 1/2) have optimal order O ( 1 ) O(1) . In the case of the Hermite–Fejér interpolation polynomials H 2 n + 1 [ ⋅ ] {\mathcal H}_{2n+1}[\cdot ] for 1 / 2 > λ ≤ 1 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 ≤ λ ≤ 1 0 \le \lambda \le 1 in weighted L p L_p norms.