Mathematical Notes

, Volume 52, Issue 3, pp 918–930 | Cite as

Convergence of subsequences of partial cubic sums of Fourier series in mean and almost everywhere

  • S. V. Konyagin


Under certain assumptions on the regularity of a function Φ necessary and sufficient conditions are found for Φ under which the integrability of Φ(¦f¦) implies, for every function f measurable onTd, the existence of a subsequence of cubic sums of the Fourier series of f that converges to f in mean or almost everywhere.


Fourier Fourier 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    S. V. Konyagin, “On the convergence of a subsequence of partial sums of multiple trigonometric Fourier series,” Trudy MIAN, 190, 102–116, Nauka, Moscow (1989).Google Scholar
  2. 2.
    R. Edwards, Fourier Series in Modern Applications, Vol. 2 [Russian translation], Mir, Moscow (1985).Google Scholar
  3. 3.
    R. D. Getsadze, “On convergence in measure of multiple Fourier series,” Soobshch. Akad. Nauk Gruz. SSR,122, No. 2, 269–271 (1986).Google Scholar
  4. 4.
    P. Stein, “On a theorem of M. Riesz,” J. Lond. Math. Soc.,8, 242–247 (1933).Google Scholar
  5. 5.
    J. Bourgain, “On the behavior of the constant in the Littlewood-Paley inequality,” Lect. Notes in Math.,1376, 202–208 (1989).Google Scholar
  6. 6.
    B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1981).Google Scholar
  7. 7.
    S. V. Konyagin, “On the divergence of subsequences of rectangular partial sums of dual trigonometric Fourier series almost everywhere,” Dokl. Rassh. Zased. IPM im. I. N. Vekua,5, No. 2, 71–74 (1990).Google Scholar
  8. 8.
    N. I. Akhiezer, Lectures on Theory of Approximation [in Russian], Nauka, Moscow (1965).Google Scholar
  9. 9.
    S. M. Nikol'skii, “Approximation in mean of functions by means of trigonometric polynomials,” Izv. Akad. Nauk SSSR. Ser. Mat.,10, No. 1, 207–256 (1946).Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • S. V. Konyagin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityUSSR

Personalised recommendations