A new class of Borel monogenic functions

Abstract

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.

arXiv commentTo 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