On the best conditioned bases of quadratic polynomials
type: Item
instance of: scholarly article
title: On the best conditioned bases of quadratic polynomials
zbMATH Open document ID: 1060.41022
author: Simon Foucart
published in: Journal of Approximation Theory
publication date: 2004-10-14T00:00:00Z
review text: Let \(u=(u_1,\dots,u_n)\) be a basis of a subspace \(U\) in a Banach space \(X\). The \(\ell_\infty\)-condition number of \(u\) is the number \[ K_\infty(u)= \sup_{a\in \ell^n_\infty}\frac {\|\sum^n_1a_iu_i \|} {\| a\|_\infty} \sup_{a\in\ell^n_\infty} \frac{\| a \|_\infty} {\|\sum^n_1a_iu_i \|}. \] The author finds a basis \(u^0\) of the space of polynomials of degree two \({\mathcal P}_2\Subset[-1,1]\) such that \(K_\infty(u^0)\approx 1.24839\). For arbitrary \(u\) one has in this case \(K_\infty({\mathcal P}_2)\leq 1.25\) and the upper is achieved for some Lagrange basis. The result in particular, means that there is no Lagrange basis for \({\mathcal P}_2\) of the optimal condition number.
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 2107875
zbMATH Keywords: polynomial basis
zbMATH Keywords: Banach-Mazur distance
zbMATH Keywords: basis condition number
reviewed by: Yuri A. Brudnyi
MaRDI profile type: Publication
full work available at URL: j.jat.2004.07.002
OpenAlex ID: W2030009906
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DOI: 10.1016/J.JAT.2004.07.002
description: scientific article; zbMATH DE number 2107875