On the Stability under Convolution of Resurgent Functions

Abstract

This article introduces, for any closed discrete subset Ω of C, the definition of Ω-continuability, a particular case of Écalle's resurgence: Ω-continuable functions are required to be holomorphic near 0 and to admit analytic continuation along any path which avoids Ω. We give a rigorous and self-contained treatment of the stability under convolution of this space of functions, showing that a necessary and sufficient condition is the stability of Ω under addition.