The Hermite interpolation
type: Item
instance of: scholarly article
title: The Hermite interpolation
zbMATH Open document ID: 0822.41005
DOI: 10.1007/BF02575884
author: Biancamaria Della Vecchia
author: Giuseppe Mastroianni
published in: Calcolo
publication date: 1995-10-08T00:00:00Z
review text: Let \(f \in C^ 1[- 1,1]\) with the usual norm \(\max (\| f \|_ \infty, \| f' \|_ \infty)\) and let \(H_{2n} (f)\) be the Hermite interpolation polynomial of degree at most \(2n - 1\) interpolating \(f\) and \(f'\) at the zeros \(x_ k\), \(k = 1, \dots, n\) of the Jacobi polynomial with weight \((1 - x)^ \alpha (1 + x)^ \beta\), \(\alpha, \beta > - 1\), \(x \in (-1,1)\). While for Lagrange interpolation at the nodes \(x_ k\) the norm of the Lagrange operator (Lebesgue constant) is asymptotically \(\log n\) for \(\alpha\), \(\beta \geq 1/2\), the norm of the operator \(H_{2n}\) is known not to be asymptotically \(\log n\) for every choice of the parameter \(\alpha\) and \(\beta\). In the present paper the authors use as nodes for Hermite interpolation the zeros of generalized Jacobi polynomials together when further points in \([-1, +1]\) (which increases the degree of the interpolating polynomial) and a weighted norm for \(C[-1,1]\) with generalized Jacobi weights. In this way they obtain completely analogous results to Lagrange interpolation. The paper is very clear written and contains besides this many interesting details also concerning simultaneous approximation.
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 739151
zbMATH Keywords: Hermite interpolation
zbMATH Keywords: Jacobi polynomial
zbMATH Keywords: generalized Jacobi polynomials
reviewed by: Manfred Wolfgang Müller
MaRDI profile type: Publication
cites work: CONVERGENCE IN THE MEAN AND ALMOST EVERYWHERE OF FOURIER SERIES IN POLYNOMIALS ORTHOGONAL ON AN INTERVAL
cites work: Hermite interpolation and mean convergence of its derivatives
cites work: Q4026668
cites work: Q4010391
cites work: Q4027805
cites work: Q4326452
cites work: Lagrange interpolation on generalized Jacobi zeros with additional nodes
cites work: Mean Convergence of Derivatives of Extended Lagrange Interpolation with Additional Nodes
cites work: Q5551609
cites work: Bernstein's inequality in \(L^p\) for \(O<p<1\)
cites work: Mean Convergence of Lagrange Interpolation. III
cites work: Mean convergence of Hermite interpolation
cites work: On the order of magnitude of fundamental polynomials of Hermite interpolation
cites work: Weighted \(L^ p\) convergence of Hermite interpolation of higher order
cites work: Q4026710
full work available at URL: https://doi.org/10.1007/bf02575884
OpenAlex ID: W2008760046
description: scientific article; zbMATH DE number 739151