Convergence of differentiable functions on closed sets and remarks on the proofs of the ``converse approximation lemmas''
type: Item
instance of: scholarly article
title: Convergence of differentiable functions on closed sets and remarks on the proofs of the ``converse approximation lemmas''
zbMATH Open document ID: 1223.26027
author: Rafael de la Llave
author: Xuemei Li
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2011-01-19T00:00:00Z
review text: In KAM theory and other areas of analysis, one is often led to consider a decreasing sequence of open sets \(U_n\subset{\mathbb R}^d\) on which a function \(f_n\) is defined. Assuming that \(U_\infty=\bigcap_{n\geq 0}U_n\) is closed, one asks whether \(f=\sum_{n\geq 0}f_n\) is Whitney differentiable on \(U_\infty\). A direct verification of the Whitney conditions is often difficult in applications, hence the need for convenient sufficient conditions. In the present paper, the authors give examples showing that even if the series and its derivatives converge fairly quickly, \(f\) can fail to be Lipschitz. Then they present suitable conditions involving not only the speed of convergence of derivatives, but also the geometry of the sets \(U_n\).
reviewed by: Vincent Thilliez
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5838574
zbMATH Keywords: Whitney differentiability
zbMATH Keywords: decreasing domains
zbMATH Keywords: KAM theory
MaRDI profile type: Publication
full work available at URL: dcdss.2010.3.623
OpenAlex ID: W2317612169
DOI: 10.3934/DCDSS.2010.3.623
description: scientific article; zbMATH DE number 5838574