On a problem concerning quasianalytic local rings

Hassan Sfouli¹

¹ Département de Mathématiques Faculté des Sciences Université Ibn Tofail BP 133 Kénitra, Maroc

Annales Polonici Mathematici, Tome 111 (2014) no. 1, pp. 13-20.

Résumé

Let $(\mathcal{C}_{n})_{n}$ be a quasianalytic differentiable system. Let $m\in\mathbb{N}$. We consider the following problem: let $f\in\mathcal{C}_{m}$ and $\widehat{f}$ be its Taylor series at $0\in\mathbb{ R}^{m}$. Split the set $\mathbb{N}^{m}$ of exponents into two disjoint subsets $A$ and $B$, $\mathbb{N}^{m}= A\cup B$, and decompose the formal series $\widehat{f}$ into the sum of two formal series $G$ and $H$, supported by $A$ and $B$, respectively. Do there exist $g, h\in \mathcal{C}_{m}$ with Taylor series at zero $G$ and $H$, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some $m\geq2$, then the system $(\mathcal{C}_{n})_{n}$ is contained in the system of analytic germs. As an application of this result, we give a simple proof of Carleman's theorem (on the non-surjectivity of the Borel map in the quasianalytic case), under the condition that the quasianalytic classes considered are closed under differentiation, for $n\geq2$.

Détail

DOI : 10.4064/ap111-1-2

Keywords: mathcal quasianalytic differentiable system mathbb consider following problem mathcal widehat its taylor series mathbb split set mathbb exponents disjoint subsets mathbb cup decompose formal series widehat sum formal series supported respectively there exist mathcal taylor series zero respectively main result paper following have positive answer above problem geq system mathcal contained system analytic germs application result simple proof carlemans theorem non surjectivity borel map quasianalytic under condition quasianalytic classes considered closed under differentiation geq