Algebrability and Faà di Bruno conditions on M suffice for stability under product and composition in Roumieu and Beurling Carleman classes with uniform asymptotics, and are necessary in Roumieu when suitable characteristic functions exist in the class.
Rodríguez-Salinas, Clases de funciones analíticas, clases semianalíticas y cuasianalíticas, Rev
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Stability under product and composition for uniform Carleman asymptotic expansions
Algebrability and Faà di Bruno conditions on M suffice for stability under product and composition in Roumieu and Beurling Carleman classes with uniform asymptotics, and are necessary in Roumieu when suitable characteristic functions exist in the class.