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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
Article
  • 21 Downloads
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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References

  1. 1.
    S. Sidon, “Hinreichende Bedingungen für den Fourier-Charakter einer Trigonometrischen Reihe,” J. London Math. Soc., 14, No. 5, 158–160 (1939).zbMATHCrossRefGoogle Scholar
  2. 2.
    S. A. Telyakovskii, “On one sufficient Sidon condition for the integrability of trigonometric series,” Mat. Zametki, 14, No. 3, 317–328 (1973).MathSciNetGoogle Scholar
  3. 3.
    Yu. L. Nosenko, “On Sidon-type conditions for the integrability of double trigonometric series,” in: Theory of Functions and Mappings [in Russian], Naukova Dumka, Kiev (1979), pp. 132–149.Google Scholar
  4. 4.
    P. V. Zaderei, “On conditions for the integrability of multiple trigonometric series,” Ukr. Mat. Zh., 44, No. 3, 340–365 (1992).CrossRefMathSciNetGoogle Scholar
  5. 5.
    O. I. Kuznetsova, Lebesgue Constants and Approximation Properties of Linear Means of Multiple Fourier Series [in Russian], Candidate-Degree Series (Physics and Mathematics), Donetsk (1985).Google Scholar
  6. 6.
    S. A. Telyakovskii, “On conditions for the integrability of multiple trigonometric series,” Tr. Mat. Inst. Akad. Nauk SSSR, 164, 180–188 (1983).MathSciNetGoogle Scholar
  7. 7.
    A. N. Kolmogorov, “Sur l’ordre be grandeur des coefficients de la serie be Fourier-Lebesgue,” Bull. Acad. Pol. Sci. (A), 83–86 (1923).Google Scholar
  8. 8.
    A. N. Podkorytov, “Order of Lebesgue constants of Fourier sums with respect to polyhedra,” Vestn. Leningrad. Univ., Ser. Mat. Mekh. Astron., 7, 110–111 (1982).MathSciNetGoogle Scholar
  9. 9.
    O. I. Kuznetsova, “On one class of N-dimensional trigonometric series,” Mat. Zametki, 63, No. 3, 402–406 (1998).MathSciNetGoogle Scholar

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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