Stability under product and composition for uniform Carleman asymptotic expansions

TL;DR

This paper investigates the stability of Carleman classes of holomorphic functions under point-wise product and composition, establishing necessary and sufficient conditions for stability in various settings.

Contribution

It generalizes previous results by providing comprehensive conditions for stability in Carleman classes, including necessary and sufficient criteria in Roumieu and Beurling frameworks.

Findings

Conditions of algebrability and Faà di Bruno ensure stability.

Necessary conditions are identified for Roumieu stability.

Constructs characteristic functions based on classical results.

Abstract

We study the stability under point-wise product and under composition in Carleman classes of holomorphic functions, defined on sectors of the Riemann surface of the logarithm, and admitting a uniform asymptotic expansion with remainders controlled by a given sequence of positive real numbers $\mathbf{M}$. On the one hand, the well-known conditions of algebrability and Fa\`a di Bruno, imposed on the sequence $\mathbf{M}$, ensure the desired stability with respect to each operation in both the Roumieu and the Beurling settings. On the other hand, these conditions turn out to be necessary for the corresponding stability in the Roumieu case as long as the existence of suitable characteristic functions, in a precise sense, is guaranteed within the class. The construction of such functions rests on classical results of B. Rodr\'iguez-Salinas, and is given in detail. Our results are inspired by, and thoroughly generalize, several partial statements by G.~Auberson and G.~Mennessier for Gevrey classes of order 1.