Classification of infinitely differentiable periodic functions

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

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.
Published
25.12.2008
How to Cite
SerdyukA. S., StepanetsO. I., and ShydlichA. L. “Classification of Infinitely Differentiable Periodic Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, no. 12, Dec. 2008, pp. 1686–1708, https://umj.imath.kiev.ua/index.php/umj/article/view/3282.
Section
Research articles