Strong asymptotic expansions in a multidirection

TL;DR

This paper proves that strong asymptotic expansions in a single multidirection imply global expansions in polysectors for bounded holomorphic functions, extending previous one-variable results with new functional-analytic techniques.

Contribution

It generalizes one-variable asymptotic expansion results to multiple variables and introduces novel functional-analytic methods for Gevrey asymptotics.

Findings

Strong asymptotic expansion in one direction implies global expansion in polysectors.

Results apply to asymptotically bounded holomorphic functions in polysectors.

Includes specialization to Gevrey strong asymptotic expansions.

Abstract

In this paper we prove that, for asymptotically bounded holomorphic functions defined in a polysector in ${\mathbb C}^n$, the existence of a strong asymptotic expansion in Majima's sense following a single multidirection towards the vertex entails (global) asymptotic expansion in the whole polysector. Moreover, we specialize this result for Gevrey strong asymptotic expansions. This is a generalization of a result proved by A. Fruchard and C. Zhang for asymptotic expansions in one variable, but the proof, mainly in the Gevrey case, involves different techniques of a functional-analytic nature.