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.
English version (Springer): Ukrainian Mathematical Journal 60 (2008), no. 12, pp 1982-2005.
Citation Example: Serdyuk A. S., Shydlich A. L., Stepanets O. I. Classification of infinitely differentiable periodic functions // Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1686–1708.
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