TY - JOUR
AU - A. S. Serdyuk
AU - O. I. Stepanets
AU - A. L. Shydlich
PY - 2007/10/25
Y2 - 2024/11/08
TI - On some new criteria for infinite differentiability of periodic functions
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 59
IS - 10
SE - Research articles
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/3398
AB - 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.
ER -