On E. Borel's theorem
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 \textit{R. Meise} and \textit{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.
non-quasi-analytic class of minimal resp. maximal type
existence of a continuous linear extension operator
solution of the Denjoy- Carleman problem
scientific article; zbMATH DE number 4030240
scientific article