Binary bases of spaces of continuous functions.
type: Item
instance of: scholarly article
title: Binary bases of spaces of continuous functions.
zbMATH Open document ID: 1058.41004
DOI: 10.1016/S0022-247X(02)00514-0
author: Clyde Martin
author: Philip W. Smith
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-01-06T00:00:00Z
review text: The authors consider the following problem. Let \(\alpha=(\alpha_0, \alpha_1,\dots)\) and set \(S_\alpha=\text{ span} \{t^n-\alpha_nt^{n+1}:n=0,1,\dots\}\), and let \(C[a,b]\) be the Banach space of continuous functions on the finite interval \([a,b]\) with norm defined by \(\| f\| =\max_{t\in[a,b]}| f(t)| \). This paper addresses the question of density of \(S_\alpha\) in \(C[a,b]\). The authors show that this problem is quite complex and, in fact, variants of this problem are equivalent to the general moment problem. Let \(C[a,b]^*\) be the topological dual of \(C[a,b]\). Fix \([a,b]\). The moment problem \(\beta_k=F(t^k)\) for \(k=0,1,\dots\), has a solution \(F\in C[a,b]^*\) if and only if \([\beta_kt^j-\beta_jt^k:0\leq k,j]\neq C[a,b]\). The authors also show that this problem is related to a series of operator questions and they explore the relation of these operator equations to the moment problem.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 44A60
zbMATH DE Number: 2022062
zbMATH Keywords: continuous functions
zbMATH Keywords: density
zbMATH Keywords: binomials
zbMATH Keywords: finite interval
zbMATH Keywords: moment problem
MaRDI profile type: Publication
cites work: Q5508534
cites work: Q5618011
cites work: Approximation with the output of linear control systems
cites work: Q5566063
cites work: Q4340161
cites work: Q3777859
cites work: Control theoretic smoothing splines
cites work: Convergence and Gibbs' phenomenon in the cubic spline interpolation of discontinuous functions
cites work: Splines and linear control theory
full work available at URL: s0022-247x(02)00514-0
OpenAlex ID: W2078413685
description: scientific article; zbMATH DE number 2022062