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Mathematical Notes

, Volume 103, Issue 3–4, pp 645–648 | Cite as

On the Convergence of Block Fourier Series of Functions of Bounded Variation in Two Variables

  • S. A. Telyakovskii
Article
  • 18 Downloads

Abstract

We present a necessary and sufficient condition for the series of absolute values of blocks of Fourier series elements and blocks of series of summands in Parseval’s identity to converge in the class of two-variable functions of bounded variation in the sense of Hardy.

Keywords

functions of bounded variation in two variables Fourier coefficients Parseval’s identity 

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References

  1. 1.
    W. H. Young, “On the integration of Fourier series,” Proc. London Math. Soc. (2) 9, 449–462 (1911).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. A. Telyakovskii, “On partial sums of Fourier series of functions of bounded variation,” in Trudy Mat. Inst. Steklov, Vol. 219: Theory of Approximations. Harmonic Analysis (Nauka, Moscow, 1997), pp. 378–386 [Proc. Steklov Inst.Math. 219, 372–381 (1997)].Google Scholar
  3. 3.
    A. S. Belov and S. A. Telyakovskii, “Refinement of the Dirichlet–Jordan and Young’s theorems on Fourier series of functions of bounded variation,” Mat. Sb. 198 (6), 25–40 (2007) [Sb. Math. 198 (5–6), 777–791 (2007)].CrossRefzbMATHGoogle Scholar
  4. 4.
    F. Móricz, “Pointwise behavior of double Fourier series of functions of bounded variation,” Monatsh. Math. 148 (1), 51–59 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    X. Z. Krasniqi, “On a class of double trigonometric series of functions of bounded variation,” East J. Approx. 17 (4), 377–384 (2011).MathSciNetzbMATHGoogle Scholar
  6. 6.
    F. Móricz, “Pointwise convergence of double Fourier integrals of functions of bounded variation over R 2,” J. Math. Anal. Appl. 424 (2), 1530–1543 (2015).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

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