Skip to main content
SpringerLink
Log in

Unconditional Convergence of General Fourier Series

  • Research Articles
  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We consider problems of unconditional convergence of Fourier series of \(\operatorname{Lip}1\) functions with respect to general orthonormal systems (ONSs). Sufficient conditions on the functions of an ONS are found under which the Fourier series of every \(\operatorname{Lip}1\) function with respect to this system converges unconditionally. We show that some of the obtained results are sharp. We also prove that from any ONS \((\varphi_n)\) one can extract a subsequence \((\varphi_{n_k})\) with respect to which the Fourier series of every \(\operatorname{Lip}1\) function converges unconditionally.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. S. Banach, “Sur la divergence des séries orthogonales,” Stud. Math. 9, 139–155 (1940).

    Article  MATH  Google Scholar 

  2. S. V. Bochkarev, “Absolute convergence of Fourier series with respect to complete orthonormal systems,” Russ. Math. Surv. 27 (2), 55–81 (1972) [transl. from Usp. Mat. Nauk 27 (2), 53–76 (1972)].

    Article  Google Scholar 

  3. G. Cagareishvili, “General Fourier coefficients and problems of summability almost everywhere,” Ann. Polon. Math. 126 (2), 113–128 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  4. L. D. Gogoladze and G. Cagareishvili, “General Fourier coefficients and convergence almost everywhere,” Izv. Math. 85 (2), 228–240 (2021) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 85 (2), 60–72 (2021)].

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Gogoladze and V. Tsagareishvili, “Some classes of functions and Fourier coefficients with respect to general orthonormal systems,” Proc. Steklov Inst. Math. 280, 156–168 (2013) [transl. from Tr. Mat. Inst. Steklova 280, 162–174 (2013)].

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Gogoladze and V. Tsagareishvili, “Summability of general orthonormal Fourier series,” Stud. Sci. Math. Hung. 52 (4), 511–536 (2015).

    MathSciNet  MATH  Google Scholar 

  7. L. D. Gogoladze and V. Sh. Tsagareishvili, “Unconditional convergence of Fourier series for functions of bounded variation,” Sib. Math. J. 59 (1), 65–72 (2018) [transl. from Sib. Mat. Zh. 59 (1), 86–94 (2018)].

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Gogoladze and V. Tsagareishvili, “Differentiable functions and general orthonormal systems,” Moscow Math. J. 19 (4), 695–707 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Kaczmarz, “Über die Konvergenz der Reihen von Orthogonalfunktionen,” Math. Z. 23, 263–270 (1925).

    Article  MathSciNet  MATH  Google Scholar 

  10. B. S. Kashin, “On Weyl’s multipliers for almost everywhere convergence of orthogonal series,” Anal. Math. 2 (4), 249–266 (1976).

    Article  MathSciNet  Google Scholar 

  11. B. S. Kashin and A. A. Saakyan, Orthogonal Series, 2nd ed. (AFTs, Moscow, 1999). Engl. transl. of the 1st ed.: Orthogonal Series (Am. Math. Soc., Providence, RI, 1989), Transl. Math. Monogr. 75.

    MATH  Google Scholar 

  12. J. R. McLaughlin, “Integrated orthonormal series,” Pac. J. Math. 42, 469–475 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Menchoff, “Sur les séries de fonctions orthogonales. I: La convergence,” Fundam. Math. 4, 82–105 (1923).

    Article  MATH  Google Scholar 

  14. A. M. Olevskii, “Orthogonal series in a complete system,” Am. Math. Soc., Transl., Ser. 2, 43, 91–138 (1964) [transl. from Mat. Sb. 58 (2), 707–748 (1962)].

    Google Scholar 

  15. W. Orlicz, “Zur theorie der Orthogonalreihen,” Bull. Int. Acad. Pol. Sci., Cracovie, 81–115 (1927).

    MATH  Google Scholar 

  16. H. Rademacher, “Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen,” Math. Ann. 87, 112–138 (1922).

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Tandori, “Über die orthogonalen Funktionen. I,” Acta Sci. Math. 18, 57–130 (1957).

    MathSciNet  MATH  Google Scholar 

  18. V. Sh. Tsagaraeishvili and G. Tutberidze, “Multipliers of absolute convergence,” Math. Notes 105 (3–4), 439–448 (2019) [transl. from Mat. Zametki 105 (3), 433–443 (2019)].

    Article  MathSciNet  MATH  Google Scholar 

  19. V. Tsagaraeishvili and G. Tutberidze, “Absolute convergence factors of Lipschitz class functions for general Fourier series,” Georgian Math. J. 29 (2), 309–315 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  20. V. A. Zorich, Mathematical Analysis II (Springer, Berlin, 2016).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ivane Javakhishvili Tbilisi State University, 1 Ilia Tchavtchavadze Ave., 0179, Tbilisi, Georgia

    L. Gogoladze & V. Tsagareishvili

Authors
  1. L. Gogoladze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Tsagareishvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Gogoladze.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 319, pp. 83–93 https://doi.org/10.4213/tm4286.

Translated by E. Shubik

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gogoladze, L., Tsagareishvili, V. Unconditional Convergence of General Fourier Series. Proc. Steklov Inst. Math. 319, 74–84 (2022). https://doi.org/10.1134/S0081543822050078

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543822050078

Keywords

Search

Navigation