On the Generalized Cesáro Summability of Trigonometric Fourier Series

Abstract

The class of generalized bounded variation is considered. For functions from this class, the deviation of the generalized Cesáro means of negative order in the norm of Lr, (1 ≤ r ≤ ∞) is estimated.

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References

  1. 1.

    C. Jordan, “Sur la série de Fourier,” C. R. Acad. Sci. Paris 92, 228–230 (1881).

    MATH  Google Scholar 

  2. 2.

    N. Wiener, “The quadratic variation of a function and its Fourier coefficients,” Massachusetts J. Math. 3 (2), 72–94 (1924).

    MATH  Google Scholar 

  3. 3.

    L. Young, “Sur une généralization de la notion de variation de puissance p-ième borneé au sens de N. Wiener, et sur la convergence des seriés de Fourier,” C. R. Acad. Sci. Paris 204 (1), 470–472 (1937).

    MATH  Google Scholar 

  4. 4.

    D. Waterman, “On the summability of Fourier series of functions of A-bounded variation,” Studia Math. 44 (2), 107–117 (1972).

    MathSciNet  Article  Google Scholar 

  5. 5.

    Z. Chanturia, “The modulus of variation of a function and its application in the theory of Fourier series,” Dokl. Akad. Nauk. SSSR 214 (1), 63–66 (1974).

    MathSciNet  Google Scholar 

  6. 6.

    T. Akhobadze, “Functions of generalized Wiener classes BV (p(n) ↑ ∞, φ) and their Fourier coefficients,” Georgian Math. J. 7 (3), 401–416 (2000).

    MathSciNet  Article  Google Scholar 

  7. 7.

    H. Kita and K. Yoneda, “A generalization of bounded variation,” Acta Math. Hungar. 56 (3–4), 229–238 (1990).

    MathSciNet  Article  Google Scholar 

  8. 8.

    H. Kita, “Convergence of Fourier series of a function of generalized Wiener’s class BV (p(n) ↑ p),” Acta Math. Hungar. 57 (3–4), 233–243 (1991).

    MathSciNet  Article  Google Scholar 

  9. 9.

    T. Akhobadze, “A generalization of bounded variation,” Acta Math. Hungar. 97 (3), 223–256 (2002).

    MathSciNet  Article  Google Scholar 

  10. 10.

    R. Siddiqi, “The order of Fourier coefficients of higher variation,” Proc. Japan Acad. 48 (7), 569–572 (1972).

    MathSciNet  Article  Google Scholar 

  11. 11.

    A. Zigmund, Trigonometric Series (Cambridge University Press, 2002), Vols. I, II.

  12. 12.

    I. B. Kaplan, “Cesáro means of variable order,” Izv. Vyssh. Uchebn. Zaved. Mat. 18 (5), 62–73 (1960) [in Russian].

    Google Scholar 

  13. 13.

    T. Akhobadze, “On the convergence of generalized Cesáro means of trigonometric Fourier series I,” Acta Math. Hungar. 115 (1–2), 59–78 (2007).

    MathSciNet  Article  Google Scholar 

  14. 14.

    T. Akhobadze, “On a theorem of M. Satô,” Acta Math. Hungar. 130 (3), 286–308 (2011).

    MathSciNet  Article  Google Scholar 

  15. 15.

    Sh. Tetunashvili, “On iterated summability of trigonometric Fourier series,” Proc. A. Razmadze Math. Inst. 139, 142–144 (2005).

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Sh. Tetunashvili, “On the summability of Fourier trigonometric series of variable order,” Proc. A. Razmadze Math. Inst. 145, 130–131 (2007).

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Sh. Tetunashvili, “On the summability method defined by matrix of functions,” Proc. A. Razmadze Math. Inst. 148, 141–145 (2008).

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Sh. Tetunashvili, “On the summability method depending on a parameter,” Proc. A. Razmadze Math. Inst. 150, 150–152 (2009).

    MathSciNet  Google Scholar 

  19. 19.

    Sh. Tetunashvili, “On divergence of Fourier trigonometric series by some methods of summability with variable orders,” Proc. A. Razmadze Math. Inst. 155, 130–131 (2011).

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Sh. Tetunashvili, “On divergence of Fourier series by some methods of summability,” Journal of Function Spaces and Applications 2012 Article ID 542607 (2010).

  21. 21.

    G. Fichtengolz, Course of Differential and Integral Calculus (Nauka, 1970), Vol II [in Russian].

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Acknowledgments

The authors are very grateful to the referee for the careful reading of the paper and helpful comments and remarks, which have allowed the authors to improve the quality of the paper.

Funding

This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) under grant no. FR-18-1599.

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Correspondence to T. I. Akhobadze or Sh. V. Zviadadze.

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The article was submitted by the authors for the English version of the journal.

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Akhobadze, T.I., Zviadadze, S.V. On the Generalized Cesáro Summability of Trigonometric Fourier Series. Math Notes 107, 898–906 (2020). https://doi.org/10.1134/S000143462005020X

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Keywords

  • bounded variation
  • trigonometric Fourier series
  • generalized Cesáro summability