Mathematical Notes

, Volume 99, Issue 5–6, pp 861–869 | Cite as

On the divergence of Fourier series in the spaces ϕ(L) containing L

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Abstract

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[−π, π], but narrower than L p = L p [−π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ϕ(L) of Kolmogorov’s theorem on the convergence of Fourier series in L p , p ∈ (0, 1), cannot be improved.

Keywords

Fourier series the space ϕ(Lthe spaces Lp p ∈ (0 1) convergence of Fourier series integrable function 

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References

  1. 1.
    H. K. Bari, Trigonometric Series (Fizmatgiz, Moscow, 1961) [in Russian].Google Scholar
  2. 2.
    A. Kolmogoroff, “Sur les fonctions harmoniques conjuguées et les séries de Fourier,” Fundamenta 7, 24–29 (1925).MATHGoogle Scholar
  3. 3.
    V. I. Filippov, “On the Kolmogorov theorems on Fourier series and conjugate functions,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 7, 21–34 (2012) [Russian Math. (Iz. VUZ) 56 (7), 18–29 (2012)].MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceUral Federal UniversityEkaterinburgRussia
  2. 2.Krasovskii Institute of Mathematics and Mechanics, Ural DivisionRussian Academy of SciencesEkaterinburgRussia

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