Sample numbers and optimal Lagrange interpolation in Sobolev spaces
type: Item
instance of: scholarly article
title: Sample numbers and optimal Lagrange interpolation in Sobolev spaces
zbMATH Open document ID: 1475.41003
author: Hui Wang
author: Guiqiao Xu
published in: Rocky Mountain Journal of Mathematics
publication date: 2021-09-10T00:00:00Z
review text: This paper studies approximations in weighted Sobolev spaces with respect to the Chebyshev norm. For this purpose we may choose a space embedded in the continuous functions over a compact domain which is a Banach space, and we wish to approximate functions given only at a finite number of nodes. The sampling numbers which are also computed in this article (this is the main point of the work in the paper) are the smallest errors for algorithms based on this number of knots for the worst errors with respect to the choice of approximants in the unit ball of the mentioned spaces. The sampling numbers are closely related to the well-known theory of the so-called \(n\)-widths. The authors are interested in optimal recovery from such spaces when the weight functions are integrable on \((-1,1)\). They then choose Chebyshev points of the first kind as interpolation nodes and show that those are optimal if Lagrange interpolation is used. A variation of this theme is the search for the optimal interpolation algorithm if the Chebyshev points are used \textbf{and} the end-points of the unit interval, and this is also investigated by the authors.
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 65D05
zbMATH DE Number: 7393769
zbMATH Keywords: sampling numbers
zbMATH Keywords: optimal Lagrange interpolation algorithm
zbMATH Keywords: Sobolev space
reviewed by: Martin D. Buhmann
MaRDI profile type: Publication
full work available at URL: rmj.2021.51.347
OpenAlex ID: W3168974851
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DOI: 10.1216/RMJ.2021.51.347
description: scientific article; zbMATH DE number 7393769