Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

Abstract

Let $w_{\lambda }(x):=(1-x^2{)}^{\lambda -1/2}$ and $P_{\lambda ,n}$ be the ultraspherical polynomials with respect to $w_{\lambda }(x)$. Then, we denote the Stieltjes polynomials $E_{\lambda ,n+1}$ with respect to $w_{\lambda }(x)$ satisfying ${\int }_{-1}^1\mathrm{‍}{w}_{\lambda }\left(x\right)P_{\lambda ,n}\left(x\right)E_{\lambda ,n+1}\left(x\right)x^{m}d\mathrm{x\hspace{0.17em}}$. In this paper, we consider the higher-order Hermite-Fejér interpolation operator $H_{n+1,m}$ based on the zeros of $E_{\lambda ,n+1}$ and the higher order extended Hermite-Fejér interpolation operator ${\mathscr{H}}_{2n+1,m}$ based on the zeros of $E_{\lambda ,n+1}{P}_{\lambda ,n}$. When $m$ is even, we show that Lebesgue constants of these interpolation operators are and , respectively; that is, and . In the case of the Hermite-Fejér interpolation polynomials ${\mathscr{H}}_{2n+1,m}[·]$ for , we can prove the weighted uniform convergence. In addition, when $m$ is odd, we will show that these interpolations diverge for a certain continuous function on $[-\mathrm{1,1}]$, proving that Lebesgue constants of these interpolation operators are similar or greater than log $n$.