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On Exponential Summability of Rectangular Partial Sums of Double Trigonometric Fourier Series

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Abstract

In this paper, we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of functions in L logL.

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References

  1. L. Fejér, “Untersuchungen über Fouriersche Reihen,” Math. Ann. 58, 51–69 (1904).

    Article  MATH  Google Scholar 

  2. H. Lebesgue, “Recherches sur la sommabilitéforte des series de Fourier,” Math. Ann. 61, 251–280 (1905).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. H. Hardy and J. E. Littlewood, “Sur la séries de Fourier d’une fonction à carre sommable,” Comptes Rendus (Paris) 156, 1307–1309 (1913).

    MATH  Google Scholar 

  4. J. Marcinkiewicz, “Sur unemethode remarquable de sommation des séries doublées de Fourier,” Ann. Scuola Norm. Sup. Pisa 8, 149–160 (1939).

    MathSciNet  MATH  Google Scholar 

  5. A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959).

    MATH  Google Scholar 

  6. K. I. Oskolkov, “On strong summability of Fourier series,” in Trudy Mat. Inst. Steklov Vol. 172: Studies in the Theory of Functions of Several Real Variables and Approximation of Functions, Collection of papers (Nauka, Moscow, 1985), pp. 280–290 [in Russian] [Proc. Steklov Inst.Math. 172, 303–314 (1987)].

    Google Scholar 

  7. V. A. Rodin, “The space BMO and strong means of Fourier series,” Anal.Math. 16 (4), 291–302 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  8. G. A. Karagulyan, “On the divergence of strong F-means of Fourier series,” J. Contemp.Math. Anal. 26 (2), 66–69 (1991).

    MathSciNet  Google Scholar 

  9. G. A. Karagulyan, “Everywhere divergent F-means of Fourier series,” Mat. Zametki 80 (1), 50–59 (2006) [Math. Notes 80 (1), 47–56 (2006)].

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Jessen, J. Marcinkiewicz, and A. Zygmund, “Note on the differentiability of multiple integrals,” Fund. Math. 25, 217–234 (1935).

    Article  MATH  Google Scholar 

  11. L. D. Gogoladze, “On strong summability almost everywhere,” Mat. Sb. 135 (177) (2), 158–168 (1988) [Math. USSR-Sb. 63 (1), 153–164 (1989)].

    MATH  Google Scholar 

  12. F. Schipp, “Über die starke Summation von Walsh–Fourier Reihen,” Acta Sci. Math. (Szeged) 30, 77–87 (1969).

    MathSciNet  MATH  Google Scholar 

  13. F. Schipp, “On strong approximation ofWalsh–Fourier series,” Magyar Tud. Akad.Mat. Fiz.Oszt. Közl. 19, 101–111 (1969).

    Google Scholar 

  14. F. Schipp and N. X. Ky, “On strong summability of polynomial expansions,” Anal. Math. 12 (2), 115–127 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Leindler, “On the strong approximation of Fourier series,” Acta Sci.Math. (Szeged) 38, 317–324 (1976).

    MathSciNet  MATH  Google Scholar 

  16. L. Leindler, “Strong approximation and classes of functions,” Mitt.Math. Sem. Giessen 132, 29–38 (1978).

    MathSciNet  MATH  Google Scholar 

  17. L. Leindler, Strong Approximation by Fourier Series (Akademiai Kiado, Budapest, 1985).

    MATH  Google Scholar 

  18. L. Leindler, “Über die Approximation im starken Sinne,” Acta Math. Acad. Sci. Hungar. 16, 255–262 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. Totik, “On the strong approximation of Fourier series,” ActaMath. Acad. Sci.Hungar. 35 (1-2), 151–172 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  20. V. Totik, “On the strong approximation of Fourier series,” ActaMath. Acad. Sci.Hungar. 35 (1-2), 151–172 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  21. V. Totik, “On the generalization of Fejér’s summation theorem,” in Functions, Series, Operators, Colloq. Math. Soc. János Bolyai (North Holland, Amsterdam–Oxford–New York, 1983), Vol. 35, pp. 1195–1199.

    MATH  Google Scholar 

  22. V. Totik, “Notes on Fourier series: strong approximation,” J. Approx. Theory 43, 105–111 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  23. U. Goginava and L. Gogoladze, “Strong approximation by Marcinkiewicz means of double Walsh–Fourier series,” Constr. Approx. 35 (1), 1–19 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  24. U. Goginava and L. Gogoladze, “Strong approximation of doubleWalsh–Fourier series,” Studia Sci. Math. Hungar. 49 (2), 170–188 (2012).

    MathSciNet  MATH  Google Scholar 

  25. U. Goginava, L. Gogoladze, and G. Karagulyan, “BMO-estimation and almost everywhere exponential summability of quadratic partial sums of double Fourier series,” Constr. Approx. 40 (1), 105–120 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Gát, U. Goginava, and G. Karagulyan, “Almost everywhere strong summability of Marcinkiewicz means of doubleWalsh–Fourier series,” Anal.Math. 40 (4), 243–266 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Gát, U. Goginava, and G. Karagulyan, “On everywhere divergence of the strong F-means of Walsh–Fourier series,” J. Math. Anal. Appl. 421 (1), 206–214 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  28. F. Weisz, “Strong summability of more-dimensional Ciesielski–Fourier series,” East J. Approx. 10 (3), 333–354 (2004).

    MathSciNet  MATH  Google Scholar 

  29. F. Weisz, “Lebesgue points of double Fourier series and strong summability,” J. Math. Anal. Appl. 432 (1), 441–462 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Weisz, “Lebesgue points of two-dimensional Fourier transforms and strong summability,” J. Fourier Anal. Appl. 21 (4), 885–914 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  31. F. Weisz, “Strong summability of Fourier transforms at Lebesgue points and Wiener amalgam spaces,” J. Funct. Spaces, No. 420750 (2015).

  32. O. D. Gabisoniya, “Points of strong summability of Fourier series,” Mat. Zametki 14 (5), 615–626 (1973) [Math. Notes 14 (5), 913–918 (1973)].

    MathSciNet  Google Scholar 

  33. F. Schipp, “On the strong summability of Walsh series,” Publ.Math. Debrecen 52 (3-4), 611–633 (1998).

    MathSciNet  MATH  Google Scholar 

  34. M. A. Krasnosel’skii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces (Fizmatgiz, Moscow, 1958) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Iv. Javakhishvili Tbilisi State University, Tbilisi, 0179, Georgia

    U. Goginava

  2. Yerevan State University, Yerevan, 0025, Armenia

    G. Karagulyan

Authors
  1. U. Goginava
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  2. G. Karagulyan
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Correspondence to U. Goginava.

Additional information

Original Russian Text © U. Goginava, G. Karagulyan, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 5, pp. 667–679.

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Goginava, U., Karagulyan, G. On Exponential Summability of Rectangular Partial Sums of Double Trigonometric Fourier Series. Math Notes 104, 655–665 (2018). https://doi.org/10.1134/S0001434618110056

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  • DOI: https://doi.org/10.1134/S0001434618110056

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