On some new criteria for infinite differentiability of periodic functions

  • A. S. Serdyuk
  • O. I. Stepanets
  • A. L. Shydlich


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.

How to Cite
Serdyuk, A. S., O. I. Stepanets, and A. L. Shydlich. “On Some New Criteria for Infinite Differentiability of Periodic Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, no. 10, Oct. 2007, pp. 1399–1409, https://umj.imath.kiev.ua/index.php/umj/article/view/3398.
Research articles