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On the Convergence of Fourier Series in the Space L1

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Abstract

We establish necessary and sufficient conditions for the convergence in the mean of trigonometric series whose coefficients satisfy the Boas–Telyakovskii conditions.

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REFERENCES

  1. N. K. Bari, Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  2. W. H. Young, “On the Fourier series of bounded functions,” Proc. London Math. Soc., 12, No. 1, 41–70 (1913).

    Google Scholar 

  3. A. Zygmund, Trigonometric Series [Russian translation], GONTI, Moscow (1939).

    Google Scholar 

  4. A. N. Kolmogorov, “Sur l'ordre be grandeur des coefficients de la serie be Fourier-Lebesgue,” Bull. Acad. Pol. Sci. (A), 83–86 (1923).

  5. E. Hille and J. D. Tamarkin, “On the summability of Fourier series. II,” Ann. Math., 34, No. 2, 329–348 (1933).

    Google Scholar 

  6. S. A. Telyakovskii, “On some estimates for trigonometric series with quasiconvex coefficients,” Mat. Sb., 63, No. 3, 426–444 (1964).

    Google Scholar 

  7. S. A. Telyakovskii, “On the convergence of Fourier series in the metric of L,” Mat. Zametki, 1, No. 1, 91–98 (1967).

    Google Scholar 

  8. S. A. Telyakovskii, “On the Sidon sufficient condition for the integrability of trigonometric series,” Mat. Zametki, 14, No. 3, 317–328 (1973).

    Google Scholar 

  9. S. Sidon, “Hinreichende Bedingungen fur den Fourier - Charakter einer trigonometrischen Reihen,” J. London Math. Soc., 14, No. 2, 158–160 (1939).

    Google Scholar 

  10. G. A. Fomin, “On one class of trigonometric series,” Mat. Zametki, 23, No. 2, 213–222 (1978).

    Google Scholar 

  11. N. Tanovic-Miller, “On a paper of Bojanic and Stanojevic,” Rend. Circ. Math. Palermo, Ser. II, 34, No. 2, 310–324 (1985).

    Google Scholar 

  12. Chand-Pao Chen, “L 1-convergence of Fourier series,” J. Austral. Math. Soc., 41, No. 3, 376–390 (1986).

    Google Scholar 

  13. C. V. Stanojevic and V. B. Stanojevic, “Generalizations of the Sidon-Teljakovskii theorem,” Proc. Amer. Math. Soc., 101, No. 4, 679–684 (1987).

    Google Scholar 

  14. M. Buntinas and N. Tanovic-Miller, “New integrability and L 1-convergence classes for even trigonometric series,” Rad. Mat., 6, 149–170 (1990).

    Google Scholar 

  15. G. A. Fomin, “On convergence of Fourier series in the mean,” Mat. Sb., 110, No. 2, 251–265 (1979).

    Google Scholar 

  16. S. A. Telyakovskii, “Conditions for the integrability of trigonometric series and their application to the investigation of linear summation methods for Fourier series,” Izv. Akad. Nauk SSSR, Ser. Mat., 28, No. 6, 1209–1236 (1964).

    Google Scholar 

  17. P. V. Zaderei, “On conditions for the integrability of multiple trigonometric series,” Ukr. Mat. Zh., 44, No. 3, 340–365 (1992).

    Google Scholar 

  18. S. A. Telyakovskii, “Estimation of the norm of a function in terms of its Fourier coefficients convenient for problems of approximation theory,” Tr. Mat. Inst. Akad. Nauk SSSR, 109, 65–97 (1971).

    Google Scholar 

  19. P. V. Zaderei, “On convergence of Fourier series in the mean,” Ukr. Mat. Zh., 41, No. 4, 562–568 (1989).

    Google Scholar 

  20. A. S. Belov, “On conditions for the convergence (boundedness) of partial sums of trigonometric series in the mean,” in: Metric Theory of Functions and Related Problems of Analysis [in Russian], AFTs, Moscow (1999).

    Google Scholar 

  21. A. Zygmund, Trigonometric Series [Russian translation], Vol. 1, Mir, Moscow (1965).

    Google Scholar 

  22. A. Zygmund, Trigonometric Series [Russian translation], Vol. 2, Mir, Moscow (1965).

    Google Scholar 

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Zaderei, P.V., Smal', B.A. On the Convergence of Fourier Series in the Space L1. Ukrainian Mathematical Journal 54, 785–794 (2002). https://doi.org/10.1023/A:1021683413740

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