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Best approximations of periodic functions in generalized lebesgue spaces

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Ukrainian Mathematical Journal Aims and scope

In generalized Lebesgue spaces with variable exponent, we determine the orders of the best approximations in the classes of (ψ; β)-differentiable 2π-periodic functions, deduce an analog of the well-known Bernstein inequality for the (ψ; β)-derivative, and apply this inequality to prove the inverse theorems of approximation theory in these classes.

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References

  1. I. I. Sharapudinov, “On the topology of the space L p(x)([0; 1]),” Mat. Zametki, 26, No. 4, 613–632 (1979).

    MathSciNet  Google Scholar 

  2. O. Kováčik and J. Rakósník, “On spaces L p(x) and W k,p(x),” Czech. Math. J., 41 (116), No. 4, 592–618 (1991).

    Google Scholar 

  3. J. Musielak, Orlicz Spaces and Modular Spaces, Springer, Berlin (1983).

    MATH  Google Scholar 

  4. W. Orlicz, “Über conjugierte Exponentenfolgen,” Studia Math., 3, 200–211 (1931).

    Google Scholar 

  5. H. Nakano, Topology of Linear Topological Spaces, Maruzen, Tokyo (1951).

    Google Scholar 

  6. S. G. Samko, “Differentiation and integration of variable order and the spaces L p(x),” in: Proc. of the Internat. Conf. on the Operator Theory and Complex and Hypercomplex Analysis (Mexico, December 12–17, 1994), Contemp. Math., 212, 203–219 (1994).

  7. I. I. Sharapudinov, “On the uniform boundedness of some families of convolution operators in L p, (p=p(x)),” Mat. Zametki, 59 (2), 291–302 (1996).

    Article  MathSciNet  Google Scholar 

  8. X. Fan and D. Zhao, “On the spaces L p(x) and W m,p(x),” J. Math. Anal. Appl., 263, No. 2, 424–446 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  9. I. I. Sharapudinov, “Some problems of approximation theory in the spaces L p(x)(E),” Anal. Math., 33, 135–153 (2007).

    Article  MathSciNet  Google Scholar 

  10. L. Diening and M. Ruzicka, Calderon–Zygmund Operators on Generalized Lebesgue Spaces L p(x) and Problems Related to Fluid Dynamics, Preprint 21/2002, 04.07.2002, Albert-Ludwings-University Freiburg (2002).

  11. M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin (2000).

    Book  MATH  Google Scholar 

  12. S. G. Samko, “On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators,” Integral Transforms Spec. Funct., 16, No. 5–6, 461–482 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 1, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).

    Google Scholar 

  14. A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 2, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).

    Google Scholar 

  15. R. Akgün, “Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent,” Ukr. Mat. Zh., 63, No. 1, 3–23 (2011); English translation: Ukr. Math. J., 63, No. 1, 1–26 (2011).

    Article  MATH  Google Scholar 

  16. A. I. Stepanets and A. K. Kushpel’, Best Approximations and Widths for the Classes of Periodic Functions [in Russian], Preprint No. 84.15, Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1984).

    Google Scholar 

  17. A. I. Stepanets and A. K. Kushpel’, “Convergence rate of Fourier series and best approximations in the space L p ,” Ukr. Mat. Zh., 39, No. 4, 483–492 (1987); English translation: Ukr. Math. J., 39, No. 4, 389–398 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  19. S. N. Bernstein, “On the best approximation of continuous functions by polynomials of a given degree (1912),” in: S. N. Bernstein, Collected Works [in Russian], Vol. 1, Academy of Sciences of the USSR, Moscow (1952), pp. 11–104.

    Google Scholar 

  20. N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).

    Google Scholar 

  21. N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  22. E. I. Zhukina, “Imbedding theorems,” in: Approximation of Periodic Functions in the Metric of the Space L p [in Russian], Preprint No. 87.47, Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1987), pp. 3–32.

    Google Scholar 

  23. A. I. Stepanets, “Inverse theorems on approximation of periodic functions,” Ukr. Mat. Zh., 47, No. 9, 1266–1273 (1995); English translation: Ukr. Math. J., 47, No. 9, 1441–1448 (1995).

    Article  MathSciNet  Google Scholar 

  24. A. A. Konyushkov, “Best approximations by trigonometric polynomials and Fourier coefficients,” Mat. Sb., 44 (86), No. 1, 53–84 (1958).

    Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 9, pp. 1249–1265, September, 2012.

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Chaichenko, S.O. Best approximations of periodic functions in generalized lebesgue spaces. Ukr Math J 64, 1421–1439 (2013). https://doi.org/10.1007/s11253-013-0725-6

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