Skip to main content
Log in

On the mean convergence of Fourier–Jacobi series

  • Published:
Ukrainian Mathematical Journal Aims and scope

The convergence of Fourier–Jacobi series in the spaces L p,A,B is studied in the case where the Lebesgue constants are unbounded.

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.

Similar content being viewed by others

References

  1. H. Pollard, “The mean convergence of orthogonal series,” Trans. Amer. Math. Soc., 62, 387–403 (1947).

    MATH  MathSciNet  Google Scholar 

  2. H. Pollard, “The mean convergence of orthogonal series,” Duke Math. J., 16, No. 1, 189–191 (1949).

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Neumann and W Rudin, “Mean convergence of orthogonal series,” Proc. Amer. Math. Soc., 3, 219–222 (1952).

    Article  MathSciNet  Google Scholar 

  4. G. M. Wing, “The mean convergence of orthogonal series,” Amer. J. Math., 72, 792–807 (1950).

    Article  MathSciNet  Google Scholar 

  5. B. Muckenhoupt, “Mean convergence of Jacobi series,” Proc. Amer. Math. Soc., 23, No. 2, 306–310 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  6. N. M. Kazakova, On the Orders of the Lebesgue Constants of Fourier–Jacobi Sums in Spaces [in Russian], Dep. in VINITI, Sverdlovsk (1981).

  7. T. H. Gronwall, “Über die Laplacesche Reihe,” Math. Ann., 74, 213–270 (1913).

    Article  MATH  MathSciNet  Google Scholar 

  8. P. K. Suetin, “On representation of continuous and differentiable functions by Legendre polynomials,” Dokl. Akad. Nauk SSSR, 158, No. 6, 1275–1277 (1964).

    MathSciNet  Google Scholar 

  9. S. A. Agakhanov and G. I. Natanson, “Approximation of functions by Fourier–Jacobi sums,” Dokl. Akad. Nauk SSSR, 161, No. 1, 9–10 (1966).

    Google Scholar 

  10. S. A. Agakhanov and G. I. Natanson, “Lebesgue functions of Fourier–Jacobi sums,” Vestn. Leningr. Univ., Ser. Mat., 1, No. 1, 11–23 (1968).

  11. V. M. Badkov, “On approximation by Fourier–Jacobi sums,” Dokl. Akad. Nauk SSSR, 167, No. 4, 731–734 (1966).

    MathSciNet  Google Scholar 

  12. V. M. Badkov, “Approximation of functions by partial sums of Fourier series in generalized Jacobi polynomials,” Mat. Zametki, 3, No. 6, 671–682 (1968).

    MATH  MathSciNet  Google Scholar 

  13. V. M. Badkov, “Estimates for Lebesgue functions and the remainder of the Fourier–Jacobi series,” Sib. Mat. Zh., 9, No. 6, 285–295 (1968).

    MathSciNet  Google Scholar 

  14. A. M. Belen’kii, “On the expansion of functions in Fourier–Legendre series,” in: Constructive Theory of Functions and Theory of Mappings [in Russian], Kiev (1981), pp. 35–48.

  15. S. Z. Rafal’son, “On partial sums of Fourier series in orthogonal polynomials,” Dokl. Akad. Nauk SSSR, 237, No. 6, 1297–1300 (1977).

    MathSciNet  Google Scholar 

  16. V. P. Motornyi, “On the convergence of Fourier series in Legendre polynomials in the mean,” Dokl. Akad. Nauk SSSR, 204, No. 4, 788–790 (1972).

    MathSciNet  Google Scholar 

  17. V. P. Motornyi, “On the convergence of Fourier series in Legendre polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat., 37, No. 1, 135–147 (1973).

    MATH  MathSciNet  Google Scholar 

  18. V. P. Motornyi, “Approximation of functions by algebraic polynomials in the metric of L p ;” Izv. Akad. Nauk SSSR, Ser. Mat., 35, No. 4, 874–899 (1971).

    MathSciNet  Google Scholar 

  19. V. M. Badkov, “Approximation properties of Fourier series in orthogonal polynomials,” Usp. Mat. Nauk, 33, No. 4, 51–106 (1978).

    MATH  MathSciNet  Google Scholar 

  20. V. P. Motornyi, “Approximation of functions by Fourier–Legendre sums in the mean,” Dokl. Akad. Nauk SSSR, 259, No. 1, 39–42 (1981).

    MathSciNet  Google Scholar 

  21. L. B. Khodak, “Convergence of Fourier series in Jacobi polynomials in the mean,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 8, 28–31 (1982).

  22. G. Szegö, Orthogonal Polynomials [Russian translation], Fizmatgiz, Moscow (1962).

    MATH  Google Scholar 

  23. M. K. Potapov, “On structural characteristics of classes of functions with given order of the best approximation,” Tr. Mat. Inst. Akad. Nauk SSSR, 134, 260–277 (1975).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 6, pp. 814–828, June, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Motornyi, V.P., Goncharov, S.V. & Nitiema, P.K. On the mean convergence of Fourier–Jacobi series. Ukr Math J 62, 943–960 (2010). https://doi.org/10.1007/s11253-010-0402-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-010-0402-y

Keywords

Navigation