Classification of infinitely differentiable periodic functions

Abstract

The set $ \mathcal{D}^\infty $ of infinitely differentiable periodic functions is studied in terms of generalized $ \overline \psi $ -derivatives defined by a pair $ \overline \psi = (\psi_1, \psi_2)$ of sequences ψ 1 and ψ 2. In particular, we establish that every function f from the set $ \mathcal{D}^\infty $ 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 ∈ $ \mathcal{D}^\infty $ 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 $ \overline \psi $ -derivative no longer exists. We also obtain new criteria for 2π-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.