On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series
For the trigonometric series $$\sum_{k=0}^{\infty}a_k\sum_{l\in kV \setminus (k-1)V}e^{i(l, x)}, \quad a_k\rightarrow 0,\quad k\rightarrow \infty,$$ given on $[-\pi, \pi)^m$, where $V$ is some polyhedron in $R^m$, we prove that the inequality $$\int\limits_{T^m}\left|\sum^{\infty}_{k=0} a_k \sum_{l\...
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| Datum: | 2008 |
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| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2008
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3177 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | For the trigonometric series
$$\sum_{k=0}^{\infty}a_k\sum_{l\in kV \setminus (k-1)V}e^{i(l, x)}, \quad a_k\rightarrow 0,\quad k\rightarrow \infty,$$
given on $[-\pi, \pi)^m$, where $V$ is some polyhedron in $R^m$, we prove that the inequality
$$\int\limits_{T^m}\left|\sum^{\infty}_{k=0} a_k \sum_{l\in kV\setminus(k-1)V}e^{i(l, x)} \right| dx \leq C \sum^{\infty}_{k=0} (k+1) |\Delta A_k|,$$
holds if the coefficients $a_k$ satisfy the following conditions of the Sidon - Telyakovskii type:
$$A_k\rightarrow\infty,\quad |\Delta a_k| \leq A_k, \quad \forall k \geq 0, \quad \sum^{\infty}_{k=0}(k+1) |\Delta A_k| |
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