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

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Ukrainian Mathematical Journal Aims and scope

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:

$$A_{k} \to 0,\quad {\left| {\Delta a_{k} } \right|} \leq A_{k} \quad \forall k \geq 0,\quad {\sum\limits_{k = 0}^\infty {{\left( {k + 1} \right)}{\left| {\Delta A_{k} } \right|}} } < \infty \,.$$

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References

  1. S. Sidon, “Hinreichende Bedingungen für den Fourier-Charakter einer Trigonometrischen Reihe,” J. London Math. Soc., 14, No. 5, 158–160 (1939).

    Article  MATH  Google Scholar 

  2. S. A. Telyakovskii, “On one sufficient Sidon condition for the integrability of trigonometric series,” Mat. Zametki, 14, No. 3, 317–328 (1973).

    MathSciNet  Google Scholar 

  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. P. V. Zaderei, “On conditions for the integrability of multiple trigonometric series,” Ukr. Mat. Zh., 44, No. 3, 340–365 (1992).

    Article  MathSciNet  Google Scholar 

  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).

  6. S. A. Telyakovskii, “On conditions for the integrability of multiple trigonometric series,” Tr. Mat. Inst. Akad. Nauk SSSR, 164, 180–188 (1983).

    MathSciNet  Google Scholar 

  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).

  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).

    MathSciNet  Google Scholar 

  9. O. I. Kuznetsova, “On one class of N-dimensional trigonometric series,” Mat. Zametki, 63, No. 3, 402–406 (1998).

    MathSciNet  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 579–585, May, 2008.

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Zaderei, P.V., Pelagenko, E.N. & Ivashchuk, O.V. On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series. Ukr Math J 60, 663–670 (2008). https://doi.org/10.1007/s11253-008-0094-8

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  • DOI: https://doi.org/10.1007/s11253-008-0094-8

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