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 $\psi_1$ and $\psi_2$. In particular, it is established that every function $f$ from the set $\mathcal{D}^{\infty}$ has at least one derivative whose parameters $\psi_1$ and $\psi_2$ decrease faster than any power function. At the same time, for an arbitrary function $f \in \mathcal{D}^{\infty}$ different from a trigonometric polynomial, there exists a pair $\psi$ whose parameters $\psi_1$ and $\psi_2$ have the same rate of decrease and for which the $\overline{\psi}$-derivative no longer exists.
We also obtain new criteria for $2 \pi$-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.