Classification of infinitely differentiable periodic functions - Serdyuk, A. S.; Stepanets, O. I.; Shydlich, A. L. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
The set D^∞ of infinitely differentiable periodic functions is studied in terms of generalized ψ-derivatives defined by a pair ψ = (ψ_1, ψ_2) of sequences ψ_1 and ψ_2. In particular, it is established that every function f from the set D^∞ has at least one derivative whose parameters ψ_1 and ψ_2 decrease faster than any power function. At the same time, for an arbitrary function f D^∞ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ_1 and ψ_2 have the same rate of decrease and for which the ψ-derivative no longer exists. We also obtain new criteria for 2 π-periodic functions real-valued on the real axis to belong to the set of functions analytic on the axis and to the set of entire functions.