\(C^{m,\omega}\) extension by bounded-depth linear operators
type: Item
instance of: scholarly article
title: \(C^{m,\omega}\) extension by bounded-depth linear operators
zbMATH Open document ID: 1195.47005
author: Garving K. Luli
published in: Advances in Mathematics
publication date: 2010-07-01T00:00:00Z
review text: Let \(S\subset\mathbb{R}^n\) be a closed set and \(C^{k,\omega}[S]\) be the trace of the Banach space \(C^{k,\omega}(\mathbb{R}^n)\) to \(S\). It was proved by \textit{H.\,Whitney} [Trans.\ Am.\ Math.\ Soc.\ 36, 63--89 (1934; JFM 60.0217.01 and Zbl 0008.24902)] for \(n= 1\), \(k\geq 1\), \textit{Yu.\,A.\thinspace Brudnyi} and \textit{P.\,Shvartsman} [J.~Geom.\ Anal.\ 7, No.\,4, 515--574 (1997; Zbl 0937.58007)] for \(n\geq 1\), \(k= 1\) and \textit{Ch.\,Fefferman} [Rev.\ Mat.\ Iberoam.\ 25, No.\,1, 1--48 (2009; Zbl 1173.46014)] for the general case that there exists a linear extension operator \(L: C^{k,\omega}[S]\to C^{k,\omega}(\mathbb{R}^n)\) whose norm is bounded by a constant depending only on \(k\) and \(n\). The operators constructed in the first two cited papers are well-localized in the sense that for all \(x\in \mathbb{R}^n\) and \(f\in C^{k,\omega}[S]\) there exist points \(x^i\in S\) and numbers \(\lambda_i\), \(1\leq i\leq d\), such that \[ (Lf)(x)= \sum^\infty_{i=1} \lambda_i f(x^i), \] where the \(x_i,\lambda_i\) depend only on \(x\), while \(d\) depends only on \(k\) and \(n\). The minimal number \(d\) is called depth of \(L\). As for the Fefferman construction, it was proved that the operator \(L\) has finite depth depending only on \(k\) and \(n\) if the set \(S\) is finite, see \textit{Ch.\,Fefferman} [loc.\,cit.]. Adding some new ingredients to the Fefferman construction and following the main line of his proof, the author constructs a linear extension operator \(L: C^{k,\omega}[S]\to C^{k,\omega}(\mathbb{R}^n)\) of finite depth \(d= d(k,n)\) for any closed subset \(S\). Moreover, he proves some additional properties of \(L\) that may be seen as the Whitney divided difference construction. For instance, for every non-limit point \(x\) of \(S\), there exists a subset \(\Sigma_x\subset S\) of cardinality at most \(d(k,n)\) depending only on \(x\) such that the Taylor polynomial \(T^k_x(Lf)\) is uniquely and linearly defined by \(f/\Sigma_x\).
reviewed by: Yuri A. Brudnyi
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 58A40
zbMATH DE Number: 5729703
zbMATH Keywords: bounded depth
zbMATH Keywords: linear extension operator
zbMATH Keywords: Whitney extension
zbMATH Keywords: \(C^{m,\omega}\) extension
MaRDI profile type: Publication
full work available at URL: j.aim.2010.01.027
OpenAlex ID: W2094943296
cites work: Q4317270
cites work: The Whitney problem of existence of a linear extension operator
cites work: Whitney's extension problem for multivariate \(C^{1,\omega}\)-functions
cites work: Q4274009
cites work: A generalized sharp Whitney theorem for jets
cites work: Interpolation and extrapolation of smooth functions by linear operators
cites work: A sharp form of Whitney's extension theorem
cites work: Whitney's extension problem for \(C^m\)
cites work: \(C^m\) extension by linear operators
cites work: The structure of linear extension operators for \(C^m\)
cites work: Extension of \(C^{m, \omega}\)-smooth functions by linear operators
cites work: Q5566770
cites work: Q5607484
DOI: 10.1016/J.AIM.2010.01.027
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description: scientific article; zbMATH DE number 5729703