Advertisement

Ukrainian Mathematical Journal

, Volume 71, Issue 2, pp 237–247 | Cite as

Quasiunconditional Basis Property of the Faber–Schauder System

  • G. M. Grigoryan
  • V. G. KrotovEmail author
Article
  • 6 Downloads
We prove that, for any 0 < 𝛿 < 1, there exists a measurable set E𝛿 ⊂ [0, 1], mes (E𝛿) > 1 𝛿, such that for any function fC[0, 1], one can find a function \( \tilde{f} \)C[0, 1] that coincides with f on E𝛿 such that the Fourier–Faber–Schauder series for this function \( \tilde{f} \) unconditionally converges in C[0, 1]. Moreover, the moduli of the nonzero Fourier–Faber–Schauder coefficients of the function \( \tilde{f} \) coincide with the elements of a given sequence {bn} satisfying the condition
$$ {b}_n\downarrow 0,\kern1em \sum \limits_{n=1}^{\infty}\frac{b_n}{n}=+\infty . $$

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], AFTs, Moscow (1999).zbMATHGoogle Scholar
  2. 2.
    G. Faber, “Über die Orthogonalenfunctionen des Herrn Haar,” Jahresber. Deutsch. Math. Verien., 19, 104–112 (1910).zbMATHGoogle Scholar
  3. 3.
    J. Schauder, “Zur Theorie stetiger Abbildungen in Functionalraumen,” Math. Z., 26, 47–65 (1927).MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Karlin, “Bases in Banach spaces,” Duke Math. J., 15, No. 4, 971–985 (1948).MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. G. Grigoryan and V. G. Krotov, “Luzin theorem on correction and the coefficients of Fourier expansion in the Faber–Schauder system,” Mat. Zametki, 93, No. 3, 172–178 (2013).CrossRefGoogle Scholar
  6. 6.
    V. G. Krotov, “Representation of measurable functions in the form of series in the Faber–Schauder system and universal series,” Izv. Akad. Nauk SSSR, Ser. Mat., 41, No. 1, 215–229 (1977).MathSciNetzbMATHGoogle Scholar
  7. 7.
    L. N. Galoyan, M. G. Grigoryan, and A. Kh. Kobelyan, “On the convergence of Fourier series in classical systems,” Mat. Sb., 206, No. 7, 55–94 (2015).MathSciNetCrossRefGoogle Scholar
  8. 8.
    N. N. Luzin, “On the main theorem of integral calculus,” Mat. Sb., 28, No. 2, 266–294 (1912).Google Scholar
  9. 9.
    D. Menchoff, “Sur la convergence uniforme des series de Fourier,” Mat. Sb., 53, No. 1, 67–96 (1942).MathSciNetGoogle Scholar
  10. 10.
    P. L. Ul’yanov, “On the N. N. Luzin works in the metric theory of functions,” Usp. Mat. Nauk, 40, No. 3, 15–70 (1985).zbMATHGoogle Scholar
  11. 11.
    V. G. Krotov, “On the universal Fourier series in the Faber–Schauder system,” Vestn. MGU, Ser. Mat. Mekh., No. 4, 53–58 (1975).Google Scholar
  12. 12.
    M. G. Grigoryan and A. A. Sargsyan, “Nonlinear approximation of continuous functions in the Faber–Schauder system,” Mat. Sb., 199, No. 5, 3–26 (2008).MathSciNetCrossRefGoogle Scholar
  13. 13.
    M. G. Grigoryan and A. A. Sargsyan, “Unconditional C-strong property of the Faber–Schauder system,” J. Math. Anal. Appl., 352, No. 2, 718–723 (2009).MathSciNetCrossRefGoogle Scholar
  14. 14.
    T. M. Grigoryan, “On the unconditional convergence of series with respect to the Faber–Schauder system,” Anal. Math., 39, No. 1, 56–67 (2013).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Yerevan State UniversityYerevanArmenia
  2. 2.Belarusian State UniversityMinskBelarus

Personalised recommendations