A decomposition lemma for entire functions and its applications to spaces of ultradifferentiable functions. (English) Zbl 0708.46041
Let \(K\subset {\mathbb{R}}^ n\) be a compact set with \(\overset \circ K\neq \emptyset\) which is of the form \(K=\prod^{m}_{j=1}\bar G_ j\) where \(G_ j\subset {\mathbb{R}}^{n_ j}\), \(1\leq n_ j\leq n\), is a bounded open set with real-analytic boundary. For a weight function \(\omega\) : \({\mathbb{R}}_+\to {\mathbb{R}}_+\), let \({\mathcal E}_{\omega}(K)\), respectively \({\mathcal E}_{\{\omega \}}(K)\) be the \(\omega\)-Whitney jets of Beurling, respectively Roumieu type on K. It is proved that \({\mathcal E}_{\omega}(K)\), respectively \({\mathcal E}_{\{\omega \}}(K)\), is isomorphic to a power series space of infinite type, respectively finite type. The results ae generalizations of those of H. Komatsu [J. Fac. Sci. Tokyo Ser. I A 24, 607-628 (1977; Zbl 0385.46027)], M. Langenbruch [Math. Nachr. 140, 109-126 (1989; Zbl 0692.46030)], D. Vogt [Arch. Math. 38, 540-548 (1982; Zbl 0477.46014)] and those of the authors [Ark. Math. 26, 265-287 (1988; Zbl 0683.46020)].
Reviewer: I.Ciorănescu
MSC:
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
Keywords:
ultradifferentiable function; \(\omega \) -Whitney jets of Beurling, respectively Roumieu type; power series space of infinite typeReferences:
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