How to reconstruct an analytic function from its asymptotic expansion?

Abstract

Asymptotic expansions of analytic functions at a boundary point of their domain of analyticity arise in a variety of problems in Applied Mathematics and Mathematical Physics. Relying on a concept of an asymptotic expansion which is slightly more restrictive than that of Poincaré, but still considerably weaker than that of a ‘strong asymptotic expansion’, it is shown when and how the analytic function can be reconstructed from the asymptotic expansion. As in the famous Watson-Nevanlinna reconstruction by Borel summability moment summation methods are used. We find an interesting new link between the growth rate of the expansion coefficients with n and the allowed geometrical form of the domain of analyticity near the expansion point. The general reconstruction result is illustrated by a class of special reconstructions, the Mittag-Leffler reconstructions.