Ukrainian Mathematical Journal

, Volume 60, Issue 5, pp 663–670

# On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

• E. N. Pelagenko
• O. V. Ivashchuk
Article
For a trigonometric series
$${\sum\limits_{k = 0}^\infty {a_{k} } }{\sum\limits_{l \in kV\backslash {\left( {k - 1} \right)}V} {e^{{i{\left( {l,x} \right)}}} } },\quad \quad a_{k} \to 0,\quad \quad k \to \infty ,$$
defined on [−π, π) m , where V is a certain polyhedron in R m , we prove that
$${\int\limits_{T^{m} } {{\left| {{\sum\limits_{k = 0}^\infty {a_{k} } }{\sum\limits_{l \in kV\backslash {\left( {k - 1} \right)}V} {e^{{i{\left( {l,x} \right)}}} } }} \right|}} }\,dx \leq C{\sum\limits_{k = 0}^\infty {{\left( {k + 1} \right)}} }{\left| {\Delta A_{k} } \right|}$$
if the coefficients a k satisfy the following Sidon-Telyakovskii-type conditions:
$$A_{k} \to 0,\quad {\left| {\Delta a_{k} } \right|} \leq A_{k} \quad \forall k \geq 0,\quad {\sum\limits_{k = 0}^\infty {{\left( {k + 1} \right)}{\left| {\Delta A_{k} } \right|}} } < \infty \,.$$

## Keywords

Fourier Series Naukova Dumka Approximation Property Mathematical Journal Trigonometric Series
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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