On the Stability under Convolution of Resurgent Functions

MR3157149 David SAUZIN David SAUZIN On the Stability under Convolution of Resurgent Functions Funkcialaj Ekvacioj. Serio Internacia 56 2013 397--413 http://fe.math.kobe-u.ac.jp/FE/FullPapers/56-3/56_397.pdf http://www.ams.org/mathscinet-getitem?mr=MR3157149 This article introduces, for any closed discrete subset $\Omega$ of $\mathbb{C}$, the definition of $\Omega$-continuability, which is a particular case of Écalle's resurgence: $\Omega$-continuable functions are required to be holomorphic near $0$ and to admit analytic continuation along any path which avoids $\Omega$. 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 $\Omega$ under addition. Resurgent functions, Convolution algebras. 30D05, 37F99. 56-397 2013 On the Stability under Convolution of Resurgent Functions David SAUZIN David SAUZIN 1 Candelpergher, B.; Nosmas, J.-C.; Pham. F. Approche de la résurgence Actualités Mathématiques, Hermann, Paris 1993 MR1250603 2 Dudko, A.; Sauzin, D., Écalle-Voronin invariants via resurgence and alien calculus, in preparation 3 Écalle, J. Les fonctions résurgentes, Vol. 1 Publications Mathématiques d'Orsay, 81-5, Université de Paris-Sud, Département de Mathématiques, Orsay 1981 MR0670417 4 Écalle, J. Les fonctions résurgentes, Vol. 2 Publications Mathématiques d'Orsay, 81-6, Université de Paris-Sud, Département de Mathématiques, Orsay 1981 MR0670418 5 Écalle, J. Les fonctions résurgentes, Vol. 3 Publications Mathématiques d'Orsay, 85-5, Université de Paris-Sud, Département de Mathématiques, Orsay 1985 MR0852210 6 Écalle, J. Cinq applications des fonctions résurgentes Publications Mathématiques d'Orsay, 84-62, Université de Paris-Sud, Département de Mathématiques, Orsay 1984 7 Écalle, J. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac Actualités Math., Hermann, Paris 1992 MR1399559

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