Intersection properties of Gevrey classes and of certain other classes (Q1273818)

Given an open subset \(\Omega\) of \(\mathbb R^n\) and \(s>0\), let us denote by \(C_\Omega\{p!^{1+s}\}\) the Gevrey class of functions associated with the sequence \(\{(p!)^s\}_{p\geq 0}\). Let \(\widehat{G}(\Omega)\) be the set of functions defined on \(\Omega\) that belong to \(C_\Omega\{p!^{1+s}\}\) for any \(s>0\). In this interesting paper the authors show that, contrary to an individual class \(C_\Omega\{p!^{1+s}\}\), the intersection \(\widehat{G}(\Omega)\) inherits the same good properties as those of the space \(C^\infty(\Omega)\) of infinitely differentiable functions on \(\Omega\). To be more specific, they prove versions for \(\widehat{G}(\Omega)\) of the classical Whitney extension theorem, Łojasiewicz regular separation of compact sets, division and preparation type theorems as well as a Whitney spectral theorem. Actually, the authors are working with more general Carleman classes defined by replacing the sequence \(\{(p!)^s\}_{p\geq 0}\) with a sequence \(\{M_p\}_{p\geq 0}\) satisfying appropriate growth conditions.