On the stability by convolution product of some resurgent algebras

Abstract : Various functional spaces take place in Resurgence theory : multiplicative spaces of formal series expansions that one would like to sum; convolutive spaces of analytic functions, the elements of which coming from the former ones by formal Borel transformations; multiplicative spaces of analytic functions deduced from the previous ones by Laplace transformations, thus giving the Borel-Laplace transforms whose asymptotics give back the formal objects one started with. This thesis is devoted to the construction of convolution algebras. Our aim is to provide an original and self-contained proof of the stability under convolution products of the space of endlessly continuable functions, in a way understandable by any youngsearcher in the field. The second part of the thesis concentrates on the convolution space of endlessly continuable functions with simple singularities. We show how the use of the alien derivations bring deep knowlege on the singular structure. We end our work with some problems, still open according to us but of great importance in practice and for which we think that our methods could be applied as well.
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Yafei Ou. On the stability by convolution product of some resurgent algebras. Mathematics [math]. Université d'Angers, 2012. English. ⟨NNT : 2012ANGE0055⟩. ⟨tel-01277192⟩

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