Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case.
type: Item
instance of: scholarly article
title: Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case.
zbMATH Open document ID: 1058.47016
DOI: 10.1016/S0022-1236(03)00109-5
published in: Journal of Functional Analysis
publication date: 2004-03-15T00:00:00Z
review text: Let \(W_{0}^{1,p}(I)\) be the Sobolev space of functions with zero trace, where \(1<p<\infty\), \(I \equiv (a,b)\), \(-\infty<a<b<\infty\). Suppose that \(L^{1,p}(I)\) is the space of all functions \(u\in L^p_{loc}(I)\) with finite pseudonorm \(\| u'\| _{p,I}\) which vanishes on the subspace of all constant functions. The symbol \(L^{1,p}(I)\setminus\{1\}\) denotes a factorization of the space \(L^{1,p}(I)\) with respect to constant functions, equipped with the norm \(\| u' \| _{p,I}\). In the paper under review, it is shown that the approximation numbers \(a_{n}(E_i)\) \((i=1,2)\) of the Sobolev embeddings \[ E_1: W^{1,p}_{0}(I) \to L^p(I), \quad E_2: L^{1,p}(I)/ \{1\} \to L^{1,p}(I) / \{1\} \] have the following properties: \[ a_n(E_1)= a_n(E_2) = \frac{c_p | I| }{n} , \] where \(c_p= \frac{(p' p^{p-1})^{1/p}}{2\pi p}, \,\, \pi_p =\frac{\pi}{\sin\pi/p}. \) Besides, it is proved that \[ \inf \{ \| E_3 - p_{n+1}\| : p_{n+1}:L^{1,p}(I) \to L^p(I); \text{ rank\,} p_{n+1} < n+1\} = \frac{c_p | I| }{n} \] for the unbounded Sobolev embedding \[ E_3: L^{1,p}(I) \to L^p(I). \]
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 47B07
zbMATH DE Number: 2057778
zbMATH Keywords: approximation numbers
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Hardy-type operators
zbMATH Keywords: Sobolev embedding
zbMATH Keywords: integral operators
author: D. E. Edmunds
author: Jan Lang
reviewed by: Alexander Meskhi
MaRDI profile type: Publication
cites work: Q3661894
cites work: A homotopic deformation along \(p\) of a Leray-Schauder degree result and existence for \((| u'| ^{p-2}u')'+f(t,u)=0\), \(u(0)=u(T)=0\), \(p>1\)
cites work: Q4343193
cites work: Q3764747
cites work: Q4347439
cites work: Remainder estimates for the approximation numbers of weighted Hardy operators acting on \(L^2\)
cites work: Entropy Numbers and Approximation Numbers in Function Spacess
cites work: Two-sided estimates for the approximation numbers of Hardy-type operators in $L^{∞}$ and L¹
cites work: On the approximation numbers of Sobolev embeddings on singular domains and trees
cites work: Improved estimates for the approximation numbers of Hardy-type operators.
cites work: Note on a nonlinear eigenvalue problem
cites work: Some remarkable sine and cosine functions
cites work: Approximation and entropy numbers of Volterra operators with application to Brownian motion
cites work: Two-sided estimates on singular values for a class of integral operators on the semi-axis
cites work: Q4169288
full work available at URL: s0022-1236(03)00109-5
OpenAlex ID: W2039621717
description: scientific article; zbMATH DE number 2057778