On Whitney's extension theorem for ultradifferentiable functions
type: Item
instance of: scholarly article
title: On Whitney's extension theorem for ultradifferentiable functions
zbMATH Open document ID: 1264.46028
author: Manuel Valdivia
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2012-12-05T00:00:00Z
review text: The author considers ultradistributions and ultradifferentiable functions of Beurling and of Roumieu type in the sense of Komatsu, assuming on the sequence of positive numbers \((M_n)_n\) the conditions of \(M_0=1\), logarithmic convexity, stability for ultradifferentiable operators and non strong quasi-analyticity. For a nonempty open subset \(\Omega\) of the \(k\)-dimensional Euclidean space \(\mathbb{R}^k\), let \(\mathcal{E}^{(M_n)}(\Omega)\) denote the set of ultradifferentiable functions of Beurling type, \(\mathcal{E}^{\{M_n\}}(\Omega)\) the corresponding set for the Roumieu case and write \(\mathcal{E}^{*}(\Omega)\) when \(*\) is either \((M_n)\) or \(\{M_n\}\). In this setting, the author gives some results related to Whitney's extension theorem and he also gives some structure theorems for ultradistributions with compact support. The following theorems are given in this paper. \noindent Theorem. Let \(M\) be a bounded subset of \(\mathbb{R}^k\) whose closure is \(K\). Let \(F := ( f_\alpha)_{\alpha \in \mathbb{N}_0^k}\) be a jet in \(K\) such that, for each \(h > 0\), there is a constant \(C\) (respectively, there are constants \(h\) and \(C\)) so that \[ | f_\alpha (x)| \leq C \, h^{|\alpha |} M_{|\alpha |}, \;\alpha \in \mathbb{N}_0^k, \;x \in K. \] If \(F\) is a Whitney jet in every compact subset of \(M\), and \(M\) is Whitney-regular, then there is a function f in \(\mathcal{E}^{(M_n)}(\mathbb{R}^k)\) (respectively, \(\mathcal{E}^{\{M_n\}}(\mathbb{R}^k))\) such that \(D^{\alpha} f_{|K} = f_{\alpha}\), \(\alpha \in \mathbb{N}_0^k\). \noindent Theorem. Let \(K\) be a nonempty compact subset of \(\mathbb{R}^k\). If \(K\) is Whitney-regular, then \[ \Phi: \mathcal{E}^{*}(\mathbb{R}^k) \longrightarrow \mathcal{E}^{*}(K) , \Phi(f)=(D^{\alpha} f_{|K})_{\alpha \in \mathbb{N}_0^k} \] is a topological homomorphism. \noindent Theorem. Let \(K\) be a compact subset of \(\Omega\) and let \(T\) be an element of \( \mathcal{E}^{(M_n)'}(\Omega)\) (respectively, \(\ma…
reviewed by: María Carmen Gómez-Collado
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46F20
zbMATH DE Number: 6112713
zbMATH Keywords: Whitney jet
zbMATH Keywords: ultradistributions
MaRDI profile type: Publication
full work available at URL: s13398-011-0045-3
OpenAlex ID: W2012301472
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DOI: 10.1007/S13398-011-0045-3
description: scientific article; zbMATH DE number 6112713