Whitney's extension theorem for ultradifferentiable functions of Beurling type
type: Item
instance of: scholarly article
title: Whitney's extension theorem for ultradifferentiable functions of Beurling type
zbMATH Open document ID: 0683.46020
DOI: 10.1007/BF02386123
published in: Arkiv för Matematik
publication date: 1988-01-01T00:00:00Z
review text: The authors introduce classes of non-quasianalytic functions \({\mathcal E}_{\omega}({\mathbb{R}}^ n)\) similar to those treated by Beurling and Björck: Given a weight function \(\omega\) : \({\mathbb{R}}\to [0,\infty [\) (i.e. \(\omega\) is continuous, even, increasing on [0,\(\infty [\), satisfies \(\omega (0)=0\), lim \(\omega\) (t)\(=\infty\), \(\limsup_{t\to \infty}\omega (2t)/\omega (t)<\infty,\) \(\lim_{t\to \infty}\log t/\omega (t)=0,\int^{+\infty}_{-\infty}(\omega (t)/1+t^ 2)dt<\infty,\) and \(\phi\) : \(t\mapsto \omega (e^ t)\) is convex on \({\mathbb{R}})\), let \[ {\mathcal E}_{\omega}({\mathbb{R}}^ n):=\{f\in C^{\infty}({\mathbb{R}}^ n):\quad \forall_{m\in {\mathbb{N}}}\sup_{\alpha \in {\mathbb{N}}^ n_ 0}\sup_{| x| \leq m}| f^{(\alpha)}(x)| \exp (- m\phi^*(| \alpha | /m))<\infty \}, \] where \(\phi^*: x\mapsto \sup \{xy-\phi (y):\) \(y\geq 0\}\) denotes the Young conjugate of \(\phi\) \(| [0,\infty [.\) The main result of the present paper is the following version of Whitney's extension theorem - which includes also previous results of Ehrenpreis and Komatsu. Theorem (shortened form). For a weight function \(\omega\) the following are equivalent: (1) For all \(n\in {\mathbb{N}}\) and all \(K\subset {\mathbb{R}}^ n\) compact convex with \(\overset\circ K \neq \emptyset\) the restriction \(\rho_ K: {\mathcal E}_{\omega}({\mathbb{R}}^ n)\to {\mathcal E}_{\omega}(K)\), \(f\mapsto (f^{\alpha}| K)_{\alpha \in {\mathbb{N}}^ n_ 0}\) is surjective. (2) For all \(n\in {\mathbb{N}}\) and all bounded open sets \(\Omega \subset {\mathbb{R}}^ n\) with real-analytic boundary the map \(\rho_{{\bar \Omega}}: {\mathcal E}_{\omega}({\mathbb{R}}^ n)\to {\mathcal E}_{\omega}({\bar \Omega})\) is surjective. (3) For all \(n\in {\mathbb{N}}\) the Borel map \(B_ n: {\mathcal E}_{\omega}({\mathbb{R}}^ n)\to {\mathcal E}_{\omega}\{0\}\), \(B_ n(f):=(f^{(\alpha)}(0))_{\alpha \in {\mathbb{N}}^ n_ 0}\), is surjective. (4) There exists \(C>0\) with \(\int^{\infty}_{0}(\omega (yt/t^ 2…
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46F05
zbMATH DE Number: 4118980
zbMATH Keywords: ultradifferentiable functions
zbMATH Keywords: classes of non-quasianalytic functions
zbMATH Keywords: Young conjugate
zbMATH Keywords: Whitney's extension theorem
zbMATH Keywords: Hörmander's solution of the \({\bar \partial }\)-problem
zbMATH Keywords: Phragmén-Lindelöf principle
zbMATH Keywords: dual characterization of surjective linear maps between Fréchet-Schwartz- spaces
author: Reinhold Meise
author: B. A. Taylor
reviewed by: Susanne Dierolf
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4118980