A convergence theorem for the Pál method of interpolation on the roots of Hermite polynomials
type: Item
instance of: scholarly article
title: A convergence theorem for the Pál method of interpolation on the roots of Hermite polynomials
zbMATH Open document ID: 0605.41002
DOI: 10.1007/BF02633193
published in: Analysis Mathematica
publication date: 1985-01-01T00:00:00Z
review text: Let \(H_ n\) be the Hermite polynomial of degree n and let \(x_{i,n}\) \((i=1,2,...,n)\), \(x^*_{i,n}\) \((i=1,2,...,n-1)\) be the roots of the polynomials \(H_ n\) and \(H_ n'\), respectively. If \(f\in C^ 1(R)\) is such that \(f(0)=0\), \(\lim_{| x| \to +\infty}x^{2r}e^{-x^ 2/2} f(x)=0\) \((r=1,2,...)\) and \(\lim_{| x| \to +\infty}f'(x)e^{-x^ 2/2}=0,\) then for the sequence \((R_ n)\), \(n=2,4,6,..\). of interpolation polynomials of degree n, verifying \(R_ n(x_{i,n})=f(x_{i,n})\), \(R_ n'(x^*_{i,n})=f'(x^*_{i,n})\), the relation \(e^{-\nu /w^ 2}| f(x)-R_ n(x)| =O(\omega (f',n^{-1/2})\log n)\) is satisfied, where \(\nu >1\) and O does not depend on x.
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 3977589
zbMATH Keywords: Hermite polynomial
zbMATH Keywords: interpolation polynomials
author: László Szili
MaRDI profile type: Publication
cites work: A new modification of the Hermite-Fejer interpolation
cites work: On polynomial approximation with the weight 363-1363-1363-1
cites work: On two polynomial inequalities. I
cites work: Notes on interpolation. VIII
cites work: ON THE MODIFIED HERMITE INTERPOLATION POLYNOMIALS
cites work: On the convergence of interpolation polynomials
Recommended article: Q3836584
Recommended article: On convergence of Pal-type interpolation polynomials
Recommended article: Q3781523
Recommended article: Convergence of the derivatives of Lagrange interpolating polynomials based on the roots of Hermite polynomials
Recommended article: On convergence of Hermite-Fejér interpolation polynomials
Recommended article: A generalization of Pál interpolation. III: The Hermite case
Recommended article: Q3989314
Recommended article: Q4727602
Recommended article: Q4016098
Recommended article: Q3319169
description: scientific article; zbMATH DE number 3977589