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On E. Borel’s theorem. (English) Zbl 0633.46033
E. Borel’s theorem states that the map $$R: C^{\infty}[-1,1]\to {\mathbb{C}}^{{\mathbb{N}}_ 0}$$, $$R(f):=(f^{(p)}(0))_{p\geq 0}$$, is surjective. B. Mitiagin observed that R does not admit a continuous linear right inverse $$E: {\mathbb{C}}^{{\mathbb{N}}_ 0}\to C^{\infty}[-1,1]$$, i.e. there is no continuous linear operator which extends all $$C^{\infty}$$-jets defined at a single point into a neighbourhood. In this article R is considered as a map from $${\mathcal E}_{M_ p}[-1,1]$$ into $$\Lambda _{M_ p}$$ where $${\mathcal E}_{(M_ p)}$$ resp. $${\mathcal E}_{\{M_ p\}}$$ denotes a non-quasi-analytic class of minimal resp. maximal type and $$\Lambda _{(M_ p)}$$ resp. $$\Lambda _{\{M_ p\}}$$ are the sequence spaces the elements of which satisfy the same growth conditions in terms of the sequence $$(M_ p)$$ as the derivatives of the elements of $${\mathcal E}_{(M_ p)}$$ resp. $${\mathcal E}_{\{M_ p\}}.$$
In both cases we give conditions on $$(M_ p)$$ equivalent to the surjectivity of $$R: {\mathcal E}_{M_ p}[-1,1]\to \Lambda _{M_ p}$$ and to the existence of a continuous linear extension operator $$E: \Lambda$$ $${}_{M_ p}\to {\mathcal E}_{M_ p}[-1,1]$$. These results imply for Gevrey sequences $$M_ p=p^ s$$, $$s>1$$, that R is surjective for both types and admits a continuous linear right inverse for the minimal but not for the maximal type. The proofs rely on those convolution methods which provide the solution of the Denjoy-Carleman problem.
For Gevrey sequences the surjectivity of R was proved by many authors. Concerning continuous linear extension D. Vogt seems to have observed first the non-existence for the maximal Gevrey type while R. Meise and B. A. Taylor [C. R. Acad. Sci., Paris, Sér. I 302, 219-222 (1986; Zbl 0587.46038)] provide linear extension operators for many minimal type cases including Gevrey sequences.
Reviewer: H.-J.Petzsche

##### MSC:
 46F05 Topological linear spaces of test functions, distributions and ultradistributions 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 46F10 Operations with distributions and generalized functions
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