Classification of infinitely differentiable periodic functions
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 t...
Збережено в:
| Дата: | 2008 |
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| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3282 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | 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. |
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