Lagrange interpolation for continuous piecewise smooth functions
type: Item
instance of: scholarly article
title: Lagrange interpolation for continuous piecewise smooth functions
zbMATH Open document ID: 1156.41002
author: Sergio Amat
author: S. Busquier
author: Antonio Escudero
author: J. C. Trillo
published in: Journal of Computational and Applied Mathematics
publication date: 2008-10-22T00:00:00Z
review text: Given \(y\in(a,b)\), let \(\gamma:[a,b]\to \mathbb{R}\) be an increasing continuous function with finite derivative for all \(x\in(a,b)\backslash \{y\}\), and \(\gamma'(y)=+\infty\). The following class of continuous piecewise smooth functions is constructed in the paper. \(\mathcal{F}_\gamma([a,b])\) denotes the class of functions \(f\) which are continuous in a finite interval \([a,b]\), and such that the following limit exists for all \(x\in[a,b]\). \[ \mathcal{D}(f)(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{\gamma(x+h)-\gamma(x)}. \] Taking into account the design of the derivative \(\mathcal{D}(f)\), the authors have extended the notion of divided differences, to obtain an explicit formulation for the error when the Lagrange interpolation is applied to \(f\in \mathcal{F}_\gamma([a,b])\). Let \(\gamma=\sum_{i=0}^m \gamma_i\), where \(\gamma_i\) is associated with \(y_i\in(a,b)\), \(i=0,...,m\). If \(f\) is smooth at all points \(x\in (y_i,y_{i+1})\), and has finite lateral derivatives at \(y_i\), \(i=0,...,m\), then the Lagrange interpolation formula holds true for \(f\).
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 5355630
zbMATH Keywords: Lagrange interpolation
zbMATH Keywords: smoothness
zbMATH Keywords: error estimates
reviewed by: J. Illán
MaRDI profile type: Publication
full work available at URL: j.cam.2007.10.011
OpenAlex ID: W2019923810
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DOI: 10.1016/J.CAM.2007.10.011
description: scientific article; zbMATH DE number 5355630