Optimal spline spaces of higher degree for \(L^2\) \(n\)-widths
type: Item
instance of: scholarly article
title: Optimal spline spaces of higher degree for \(L^2\) \(n\)-widths
zbMATH Open document ID: 1358.41007
author: Michael S. Floater
author: Espen Sande
published in: Journal of Approximation Theory
publication date: 2017-02-28T00:00:00Z
review text: In this article, Kolmogorov \(n\)-width are studied, that is, the \(n\)-width of a subset \(A\), say, of a normed linear space \((X,\|\cdot\|)\) with respect to that space \(X\). The \(n\) comes from the dimension \(n\) of subspaces \(X_n\) of \(X\) which are used to define the \(n\)-width by the expression \[ \inf_{X_n}\sup_{u\in A}\inf_{v\in X_n}\|u-v\|. \] Now, subspaces \(X_n\) are called ``optimal'' if the \(n\)-width is attained by using them in place of the \(\inf_{X_n}\). The authors show in this paper that spline subspaces of high degrees provide optimal subspaces for \(A\)s given by Sobolev spaces (with or without zero boundary conditions) and \(X\) being spaces of square-integrable functions on the unit interval.
reviewed by: Martin D. Buhmann
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 6690408
zbMATH Keywords: \(n\)-widths
zbMATH Keywords: splines
zbMATH Keywords: isogeometric analysis
zbMATH Keywords: total positivity
zbMATH Keywords: Green's functions
MaRDI profile type: Publication
full work available at URL: j.jat.2016.12.002
OpenAlex ID: W2560948221
cites work: On principles for the selection of shape functions for the generalized finite element method
cites work: \(n\)-widths, sup-infs, and optimality ratios for the \(k\)-version of the isogeometric finite element method
cites work: DUALITY OF CONVEX FUNCTIONS AND EXTREMUM PROBLEMS
cites work: Q5624248
cites work: Q5559978
cites work: Spline spaces are optimal for \(L^2\) n-width
cites work: On n-Widths in L ∞
cites work: Q3344608
cites work: Approximation error estimates and inverse inequalities for B-splines of maximum smoothness
DOI: 10.1016/J.JAT.2016.12.002
Recommended article: Optimal spline spaces for \(L^2\ n\)-width problems with boundary conditions
Recommended article: \({\mathcal L}\)-splines and widths
Recommended article: Q3314299
Recommended article: Q3315943
Recommended article: Upper and Lower Bounds on the Dimension of Multivariate Spline Spaces
Recommended article: Q4215110
Recommended article: Q3682942
Recommended article: Q3742050
Recommended article: Q3802010
Recommended article: Q4490277
description: scientific article; zbMATH DE number 6690408