@article{Serdyuk_Stepanets_Shydlich_2007, title={On some new criteria for infinite differentiability of periodic functions}, volume={59}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/3398}, abstractNote={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 $\psi_2$ . It is shown 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, and, 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.
<br><br>}, number={10}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Serdyuk, A. S. and Stepanets, O. I. and Shydlich, A. L.}, year={2007}, month={Oct.}, pages={1399–1409} }