$L_p$-Convergence of higher order Hermite or Hermite-Fej\'er interpolation polynomials with exponential-type weights

TL;DR

This paper establishes $L_p$-convergence results for higher order Hermite-Fejér interpolation polynomials associated with exponential-type weights, focusing on their behavior at zeros of orthonormal polynomials.

Contribution

It provides new $L_p$-convergence theorems for higher order Hermite-Fejér interpolation polynomials with exponential weights, extending previous results to more general weights and interpolation orders.

Findings

Proved $L_p$-convergence of Hermite-Fejér interpolation polynomials.

Derived convergence theorems for polynomials at zeros of orthonormal polynomials.

Extended convergence results to exponential-type weights with specific conditions.

Abstract

Let $\mathbb{R}=(-\infty,\infty)$, and let $Q\in C^1(\mathbb{R}): \mathbb{R}\rightarrow \mathbb{R^+}=[0,\infty)$ be an even function, which is an exponent. We consider the weight $w_\rho(x)=|x|^{\rho} e^{-Q(x)}$, $\rho\geqslant 0$, $x\in \mathbb{R}$, 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-Fej\'er interpolation polynomials at the zeros $\left\{x_{k,n,\rho}\right\}_{k=1}^n$ of $p_{n}(w_\rho ^2;x)$.