\({\mathcal C}^m\)-norms on finite sets and \({\mathcal C}^m\) extension
type: Item
instance of: scholarly article
title: \({\mathcal C}^m\)-norms on finite sets and \({\mathcal C}^m\) extension
zbMATH Open document ID: 1116.58005
DOI: 10.1215/S0012-7094-07-13711-6
author: Edward Bierstone
author: Pierre Milman
published in: Duke Mathematical Journal
publication date: 2007-04-23T00:00:00Z
review text: In recent papers [Ann. Math. (2) 161, No.~1, 509--577 (2005; Zbl 1102.58005); Ann. Math. (2) 164, No.~1, 313--359 (2006; Zbl 1109.58016)] \textit{C. L. Fefferman} has given criteria for a function defined on a compact subset \( E \) of \( {\mathbb R}^n \) to extend to a \( C^m \) or \( C^{m,\omega} \) function of \( {\mathbb R}^n \) (here, \( \omega \) denotes a regular modulus of continuity). These criteria rely on uniform bounds for the \( C^m \) or \( C^{m,\omega} \) norms of extensions from finite subsets \( S \) of \( E \) of cardinality at most a large integer \( k^\sharp \) depending only on \( m \) and \( n \). If \( d \) denotes the dimension of the vector space of real polynomials of degree at most \( m \) on \( {\mathbb R}^n \), Fefferman shows that one can take \( k^\sharp=(d+1)^{3\cdot 2^d} \) in the \( C^{m,\omega} \) case, and he gives a higher bound in the \( C^m \) case. In the present article, the authors show that one can take \( k^\sharp=2^d \) in both cases. They also show that the geometric \( C^m \) paratangent bundle of \( E \) that plays a key role in the dual formulation of \( C^m \) extension criteria [\textit{E. Bierstone, P. D. Milman}, and \textit{W. Pawłucki}, Ann. Math. (2) 164, No. 1, 361--370 (2006; Zbl 1109.58015)] can be defined using limits of distributions supported on \( 2^{d-1} \) points. The proofs involve the combinatorics of clusterings of finite subsets \( S \), based on partitions \( S=\bigcup S_i \) where the distances between points of each \( S_i \) are comparable to the diameter of \( S_i \), and the distances between the subsets \( S_i \) are relatively large in comparison with their diameters, both properties being uniform with respect to \( S \).
reviewed by: Vincent Thilliez
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 26B35
Mathematics Subject Classification ID: 58A20
zbMATH DE Number: 5145497
zbMATH Keywords: extension problems
zbMATH Keywords: differentiable functions
zbMATH Keywords: Whitney jets
zbMATH Keywords: paratangent spaces
MaRDI profile type: Publication
cites work: Examples concerning Whitney's \({\mathcal C}^m\) extension problem
cites work: Differentiable functions defined in closed sets. A problem of Whitney
cites work: Higher-order tangents and Fefferman's paper on Whitney's extension problem
cites work: Whitney's extension problem for multivariate \(C^{1,\omega}\)-functions
cites work: A sharp form of Whitney's extension theorem
cites work: Whitney's extension problem for \(C^m\)
cites work: Étude de quelques algèbres tayloriennes
cites work: \(C^1\) extensions of functions and stabilization of Glaeser refinements
cites work: Q5566771
cites work: The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces
cites work: Differentiable Functions Defined in Closed Sets. I
full work available at URL: s0012-7094-07-13711-6
OpenAlex ID: W1485553355
Recommended article: The \(C^m\) norm of a function with prescribed jets. I.
Recommended article: Whitney’s extension problems and interpolation of data
Recommended article: Q4008419
Recommended article: Plurisubharmonic functions characterized by one-variable extremal functions
Recommended article: Q3816493
Recommended article: Norms of polynomials and capacities on Banach spaces
Recommended article: The Whitney extension theorem in high dimensions
Recommended article: On the \(\varepsilon\)-entropy of nearly critical values
Recommended article: Analogues of the problem of analytic continuation from circles for some differential equations
Recommended article: Falconer's \((K, d)\) distance set conjecture can fail for strictly convex sets \(K\) in \(\mathbb{R}^d\)
description: scientific article; zbMATH DE number 5145497