Skip to main content
Log in

Strong boundedness, strong convergence and generalized variation

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all \({L^{p}}\) spaces, \({1\leq p < \infty}\). We obtain new results on strong convergence of Fourier series for functions of generalized bounded variation.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avdispahić M.: On the classes \({\Lambda BV}\) and \({V[\nu]}\),. Proc. Amer. Math. Soc., 95, 230–234 (1985)

    MathSciNet  MATH  Google Scholar 

  2. Avdispahić M.: Concepts of generalized bounded variation and the theory of Fourier series,. Int J. Math. Math. Sci., 9, 223–244 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avdispahić M.: On the strong convergence of trigonometric series of a special type,. Rad. mat., 3, 317–324 (1987)

    MathSciNet  MATH  Google Scholar 

  4. Avdispahić M.: Criteria for absolute and strong convergence of Fourier series,. Czechoslovak Math. J., 37(112), 547–550 (1987)

    MathSciNet  MATH  Google Scholar 

  5. M. Avdispahić and Z. Šabanac, Determination of a jump by Fourier and Fourier–Chebyshev series, Bull. Iranian Math. Soc. (to appear).

  6. N. K. Bari, Trigonometric Series, Fizmatgiz (Moscow, 1961) (in Russian).

  7. Chanturiya Z. A.: The modulus of variation of a function and its application in the theory of Fourier series,. Dokl. Akad. Nauk SSSR, 214, 63–66 (1974)

    MathSciNet  MATH  Google Scholar 

  8. Hyslop J. M.: The modulus of variation of a function and its application in the theory of Fourier series,. Dokl. Akad. Nauk SSSR, 214, 63–66 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Perelman, Functions of generalized variation, Fund. Math., 105 (1980), 199–211.

    MathSciNet  Google Scholar 

  10. Pippert R. E.: On the absolutely convergent trigonometric series,. Math Z., 85, 401–406 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  11. Prus-Wiśniowski F.: Separability of the space of continuous functions that are continuous in \({\Lambda}\) -variation,. J. Math. Anal. Appl., 344, 274–291 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sablin A.I.: Differential properties and Fourier coefficients of functions of \({\Lambda}\) -bounded variation,. Analysis Math., 11, 331–345 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Szalay I., Tanović-Miller N.: On Banach spaces of absolutely and strongly convergent Fourier series,. Acta Math. Hungar., 55, 149–160 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Szalay I., Tanović-Miller N.: On Banach spaces of absolutely and strongly convergent Fourier series II,. Acta Math. Hungar., 57, 137–149 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Szász O.: On the absolute convergence of trigonometric series,. Ann. Math. (2) 47, 213–220 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Tanović-Miller, On strong convergence of trigonometric and Fourier series, Acta Math. Hungar., 42 (1983), 35–43.

  17. Tanović-Miller N.: Strongly convergent trigonometric series as Fourier series,. Acta Math. Hungar., 47, 127–135 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tanović-Miller N.: On Banach spaces of strongly convergent trigonometric series,. J. Math. Anal. Appl., 146, 110–127 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang S.: Some properties of functions of \({\Lambda}\) -bounded variation,. Scientia Sinica Ser A, 25, 149–160 (1982)

    MathSciNet  Google Scholar 

  20. D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math., 44 (1972), 107-117; errata, ibid., 44 (1972), 651.

  21. Waterman D.: On the summability of Fourier series of functions of \({\Lambda}\) -bounded variation,. Studia Math., 55, 97–109 (1976)

    Google Scholar 

  22. Wiener N.: The quadratic variation of a function and its Fourier coefficients,. J. Math. Phys. MIT, 3, 72–94 (1924)

    Article  MATH  Google Scholar 

  23. C. Young, Sur une généralisation de la notion de variation de puissance p-ième borneé au sens de M. Wiener, et sur la convergence des séries de Fourier, C. R. Acad. Sci. Paris Sér. I Math., 204 (1937),470–472.

  24. Zygmund A, Trigonometric Series, Vols. I, II, Cambridge University Press (New York, 1968).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Avdispahić.

Additional information

To the memory of Naza Tanović-Miller

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Avdispahić, M., Šabanac, Z. Strong boundedness, strong convergence and generalized variation. Acta Math. Hungar. 152, 404–420 (2017). https://doi.org/10.1007/s10474-017-0717-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-017-0717-3

Key words and phrases

Mathematics Subject Classification

Navigation