$L_p$-Convergence of higher order Hermite or Hermite-Fejér interpolation polynomials with exponential-type weights

Abstract

Let $\mathbb{R}=(-\infty ,\infty )$, and let $Q\in C^1(\mathbb{R}):\mathbb{R}\to {\mathbb{R}}^{+}=[0,\infty )$ be an even function, which is an exponent. We consider the weight $w_{\rho }(x)=|x{|}^{\rho }{e}^{-Q(x)}$, $\rho ⩾0$, $x\in \mathbb{R}$, and then we can construct the orthonormal polynomials $p_n(w_{\rho }^2;x)$ of degree n for $w_{\rho }^2(x)$. In this paper we obtain $L_p$-convergence theorems of even order Hermite-Fejér interpolation polynomials at the zeros ${\left\{x_{k,n,\rho }\right\}}_{k=1}^n$ of $p_n(w_{\rho }^2;x)$.