Invariant functions in Denjoy-Carleman classes (Q1016363)

In this paper, the author presents a version of \textit{G. W. Schwarz}'s theorem [Topology 14, 63--68 (1975; Zbl 0297.57015)] in the framework of Denjoy-Carleman classes. Section 2 is devoted to the review of some notions and results on Denjoy-Carleman classes: Denjoy-Carleman classes of differentiable functions, quasianalytic function classes, non-quasianalytic function classes, strong non-quasianalytic function classes, moderate growth, strong regularity, Whitney's extension theorem, Gevrey function, spaces of \(C^M\) - functions, polynomial density in \(C^M(U)\), closed ideals, \(C^M\)-functions and \(C^M\)-forms on manifolds. The main result is contained in Theorem 3.4 and it is related to other results: Hilbert's theorem concerning the algebra of G-invariant polynomials, symmetric functions in Denjoy-Carleman classes, invariant classes in Denjoy-Carleman classes, finite reflection groups. An interesting application of the main result to the representation of equivariant mappings in Denjoy-Carleman classes is given in Section 4. In Section 5 an extension of Theorem 3.4 to polar representations is given and in Section 6 an analog result in Denjoy-Carleman classes for real analytic proper Riemannian \(G\)-manifolds is presented.