On some new criteria for infinite differentiability of 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 $\psi_2$ .
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.
English version (Springer): Ukrainian Mathematical Journal 59 (2007), no. 10, pp 1569-1580.
Citation Example: Serdyuk A. S., Shydlich A. L., Stepanets O. I. On some new criteria for infinite differentiability of periodic functions // Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1399–1409.
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