Moment Summation Methods and Non-Homogeneous Carleman Classes

We extend the classical theorems of F. Nevanlinna and Beurling by characterizing the image of various spaces of smooth functions under the generalized Laplace transform. To achieve this, we introduce and analyze novel non-homogeneous Carleman classes, which generalize the traditional homogeneous definitions. This characterization allows us to derive necessary and sufficient conditions for the applicability of moment summation methods within a given class of functions. Furthermore, we establish an extension of Écalle's concept of quasianalytic continuation and apply these results to the theory of multi-summability and Euler-type differential equations.