Functions with Ultradifferentiable Powers

Nonlinear Conditions for Ultradifferentiability

“…The Roumieu parts of the results in the first and the fifth row are due to Thilliez [30]; see Sects. 2.4, 2.5.…”

“…We remark that non-quasianalytic Denjoy-Carleman classes E {M} , where the weight sequence M lacks moderate growth, do not have property (D) in general; see [30,Remark 2.2.3]. The moderate growth condition is rather restrictive (e.g., it implies that the class E {M} is contained in a Gevrey class).…”

“…Additional regularity properties for M endow E [M] (U ) with additional structure, e.g., log-convexity of m implies closedness under composition of functions. A crucial assumption in [30] is moderate growth of M, which reads as follows…”

“…There is a slight mismatch between our notation (also used in [9]) and that of [30] (and [22]). We write M j = m j j!…”

“…Only recently Thilliez [30] showed that Joris's result carries over to Denjoy-Carleman classes of Roumieu type E {M} . These are ultradifferentiable classes of smooth functions defined by certain growth properties imposed upon the sequence of iterated derivatives in terms of a weight sequence M (which in view of the Cauchy estimates measures the deviation from analyticity).…”