Classification of infinitely differentiable periodic functions - Serdyuk, A. S.; Stepanets, O. I.; Shydlich, A. L. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
The set $\mathcal{D}^{∞}$ of infinitely differentiable periodic functions is studied in terms of generalized$\overline{ψ}$-derivatives defined by a pair $\overline{ψ} = (ψ_1, ψ_2)$ of sequences $ψ_1$ and $ψ_2$. In particular, it is established that every function $f$ from the set $\mathcal{D}^{∞}$ 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 $f \in \mathcal{D}^{∞}$ 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 $\overline{ψ}$-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.