Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders

TL;DR

This paper develops a comprehensive multisummability framework for formal power series within Carleman ultraholomorphic classes, extending existing theories to broader classes characterized by nonzero proximate orders and distinct growth indices.

Contribution

It introduces a general multisummability theory for Carleman ultraholomorphic classes using weight sequences with nonzero proximate orders, expanding the scope of previous Gevrey-level theories.

Findings

Extended multisummability theory to Carleman classes

Provided analytical and cohomological reconstruction formulas

Generalized Laplace-like operators for multisum reconstruction

Abstract

We introduce a general multisummability theory of formal power series in Carleman ultraholomorphic classes. The finitely many levels of summation are determined by pairwise comparable, nonequivalent weight sequences admitting nonzero proximate orders and whose growth indices are distinct. Thus, we extend the powerful multisummability theory for finitely many Gevrey levels, developed by J.-P. Ramis, J. \'Ecalle and W. Balser, among others. We provide both the analytical and cohomological approaches, and obtain a reconstruction formula for the multisum of a multisummable series by means of iterated generalized Laplace-like operators.