Hermite and Hermite-Fejér interpolation for Stieltjes polynomials

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

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.