Extension of ultradifferentiable functions of Roumieu type
Let \(K\subset {\mathbb{R}}^ n\) be a compact convex set and let \(E_{\{M_ j\}}(K)\) denote the ultradifferentiable functions of Roumieu type on K. It is shown, that there is no continuous linear extension operator \(T: E_{\{M_ j\}}(K)\to E_{\{M_ j\}}(J)\), if the sequence \((M_ j)\) is regular (\(K\subset\overset\circ J\subset {\mathbb{R}}^ n)\). This especially holds for the Gevrey sequence \(M_ j=(j!)^ s\), \(s>1\). The proof uses the category of tame (F)-spaces and an appropriate variant of the property (DN), which was introduced by D. Vogt to characterize the subspaces of (nuclear) power series spaces of finite type.
ultradifferentiable functions of Roumieu type
scientific article; zbMATH DE number 4030233