To appear in Arnold Mathematical Journal. v3: revised version. We corrected the threshold of the quasi-analyticity theorem, which now holds under a geometric condition on the poles, and we prove the uniqueness of the polar decomposition (Borel property). New title, historical introduction and new examples illustrating the quasi-analytic threshold
Spaces of quasi-analytic classes are defined by the existence and uniqueness of Taylor expansions, which are not necessarily convergent. First examples were given by Borel in his theory of monogenic functions, a generalisation of holomorphic functions defined on locally closed sets. Denjoy and Carleman then gave simpler examples of quasi-analytic classes which are now widely known. Unfortunately, in most examples coming from mathematical physics and number theory, the power series are neither of Borel nor Denjoy-Carleman's classes. In this paper we introduce a quasi-analytic class which is relevant to perturbation theory and especially to KAM theory and dynamical systems. Our theorems also explain geometrically the divergence of most perturbative expansions by the presence of accumulation points of poles.
Monogenic functions over real alternative *-algebras: the several hypercomplex variables case
School of Mathematics, Hefei University of Technology（合肥工业大学数学学院）；Center for Pure Mathematics, School of Mathematical Sciences, Anhui University（安徽大学数学科学学院基础数学中心）；School of Science, Tianjin University of Technology and Education（天津职业技术师范大学理学院）
Continuous right inverses for the asymptotic Borel map in ultraholomorphic classes via a Laplace-type transform
在窄扇区上的一般Carleman超全纯类的框架下，给出了渐近Borel映射的线性连续右逆的一个新的构造。V.Thilliez利用J.Chaumat和A. M. Chollet，但我们的方法是完全不同的，依赖于引入一个合适的截断拉普拉斯型变换。这种技术更适合于将这些结果推广到多个变量设置。此外，它非常类似于Gevrey类的经典过程，因此指出了引入可和性概念的方法，该概念推广了J.P. Ramis开发的$k-$可和性理论。
