Classification of infinitely differentiable periodic functions

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.