Ukrainian Mathematical Journal

, Volume 60, Issue 5, pp 663–670 | Cite as

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

  • P. V. Zaderei
  • E. N. Pelagenko
  • O. V. Ivashchuk
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:
$$A_{k} \to 0,\quad {\left| {\Delta a_{k} } \right|} \leq A_{k} \quad \forall k \geq 0,\quad {\sum\limits_{k = 0}^\infty {{\left( {k + 1} \right)}{\left| {\Delta A_{k} } \right|}} } < \infty \,.$$


Fourier Series Naukova Dumka Approximation Property Mathematical Journal Trigonometric Series 
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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • P. V. Zaderei
    • 1
  • E. N. Pelagenko
    • 1
  • O. V. Ivashchuk
    • 1
  1. 1.Kiev National University of Technology and DesignKievUkraine

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