Norms of projectors onto spaces with Riesz bases
type: Item
instance of: scholarly article
title: Norms of projectors onto spaces with Riesz bases
zbMATH Open document ID: 0961.41008
DOI: 10.1016/S0377-0427(00)00379-4
published in: Journal of Computational and Applied Mathematics
publication date: 2001-05-17T00:00:00Z
review text: Let \(\Omega\subseteq \mathbb R^d, d=1,2,\dots{}\) be a closed connected set, let \(X\) be a linear space of bounded, real-valued functions on \(\Omega\) and \(\langle\cdot,\cdot\rangle :X\times X\rightarrow\mathbb R\) be a semi-definite inner product on \(X.\) Let \(\mathcal S\subset C(\Omega)\cap X\) be a Hilbert subspace of \(X\) with respect to \(\langle\cdot,\cdot\rangle.\) Then for each \(f\in X\) there exists a unique best approximant \(u_f\in \mathcal S.\) Let \(P:X\rightarrow\mathcal S\) be the metric projection defined by \(Pf=u_f\) and let \(\|P\|_\infty:= \sup_{f\neq 0}\|Pf\|_{L_\infty(\Omega)}/\|f\|_{L_\infty(\Omega)}.\) Under certain conditions the author obtains upper bounds for \(\|P\|_\infty,\) in terms of the properties of a Riesz bases in \(\mathcal S.\) Better upper bounds are obtained in the particular case when \(X=C[a,b], \mathcal S= \mathcal S_r(\Delta)\subset C^{r-2}[a,b]\) is the space of polynomial splines of order \(r\) with knots \(\Delta: a=x_0<\dots{}<x_{n+1}=b\) and \(\langle f,g\rangle=\int_a^b f(x)g(x) dx.\) The case when \(X=L_\infty[a,b],\mathcal S=\mathcal S_r(\Delta)\) and \(\langle f,g\rangle=\sum_{i=1}^nf(t_i)g(t_i),\) is also treated.
reviewed by: Ioan Şerb
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1491257
zbMATH Keywords: orthogonal projection
zbMATH Keywords: operator norm
zbMATH Keywords: spline space
author: Manfred von Golitschek
MaRDI profile type: Publication
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full work available at URL: s0377-0427(00)00379-4
OpenAlex ID: W1971406023
description: scientific article; zbMATH DE number 1491257