General Fourier coefficients for Lip α class functions


We investigate the behavior of Fourier coefficients of Lip α class functions with respect to general orthonormal systems (ONS). The generalizations in a certain sense of some theorems by S. Szász, B. I. Golubov and S. V. Bochkarev are obtained.

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


  1. 1.

    S. V. Bochkarev, Absolute convergence of Fourier series in complete orthogonal systems,Uspehi Mat. Nauk, 27 (1972), 53–76 (in Russian); translation in Russian Math. Surveys, 27 (1972), 55-81.

  2. 2.

    Ciesielski, Z.: Properties of the orthonormal Franklin system. Studia Math. 23, 141–157 (1963)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Gogoladze, L., Meskhia, R.: On the absolute convergence of trigonometric Fourier series. Proc. A. Razmadze Math. Inst. 141, 29–40 (2006)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Gogoladze, L., Tsagareishvili, V.: On the divergence of Fourier series of functions in several variables. Anal. Math. 39, 163–178 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Golubov, B.I.: On Fourier series of continuous functions with respect to a Haar system. Izv. Akad. Nauk SSSR Ser. Mat. 28, 1271–1296 (1964). (in Russian)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Hardy, G.H.: Weierstrass's non-differentiable function. Trans. Amer. Math. Soc. 17, 301–325 (1916)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    B. S. Kashin and A. A. Saakyan, Orthogonal Series, 2nd ed., Izd-vo Nauchno-Issledovatel'skogo Aktuarno-Finansovogo Tsentra (AFTs) (Moscow, 1999) (in Russian)

  8. 8.

    Konyuškov, A.A.: Convergence of certain series of Fourier coefficients. Uspehi Mat. Nauk 14, 189–196 (1959). (in Russian)

    MathSciNet  Google Scholar 

  9. 9.

    McLaughlin, J.R.: Integrated orthonormal series. Pacific J. Math. 42, 469–475 (1972)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Mityagin, B.S.: On the absolute convergence of the series of Fourier coefficients. Dokl. Akad. Nauk SSSR 157, 1047–1050 (1964). (in Russian)

    MathSciNet  Google Scholar 

  11. 11.

    Szász, O.: Fourier series and mean moduli of continuity. Trans. Amer. Math. Soc. 42, 366–395 (1937)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Tsagareishvili, V.: On the absolute convergence of Fourier series with respect to general orthonormal systems. Georgian Math. J. 24, 471–478 (2017)

    MathSciNet  Article  Google Scholar 

  13. 13.

    S. S. Volosivets and B. I. Golubov, Generalized absolute convergence of series of Fourier coefficients with respect to Haar type systems, Izv. Vyssh. Uchebn. Zaved. Mat., 62 (2018), 10–20 (in Russian); translation in Russian Math. (Iz. VUZ), 62 (2018), 7–16

  14. 14.

    A. Zygmund, Trigonometric Series. I, 2nd ed., Cambridge University Press (New York, 1959)

Download references

Author information



Corresponding author

Correspondence to G. Cagareishvili.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cagareishvili, G., Gogoladze, L. General Fourier coefficients for Lip α class functions. Acta Math. Hungar. 161, 327–340 (2020).

Download citation

Key words and phrases

  • orthonormal system
  • trigonometric system
  • Haar system
  • Rademacher system
  • Fourier coefficient
  • function class Lip α

Mathematics Subject Classification

  • 42C10