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Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent

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Ukrainian Mathematical Journal Aims and scope

We investigate the approximation properties of the trigonometric system in \( L_{2\pi }^{p\left( \cdot \right)} \). We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class.

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References

  1. R. Akgün and D. M. Israfilov, “Approximation and moduli of smoothness of fractional order in Smirnov–Orlicz spaces,” Glas. Mat. Ser. III, 42, No. 1, 121–136 (2008).

    Article  Google Scholar 

  2. R. Akgün and D. M. Israfilov, “Polynomial approximation in weighted Smirnov–Orlicz space,” Proc. A. Razmadze Math. Inst., 139, 89–92 (2005).

    MathSciNet  Google Scholar 

  3. R. Akgün and D. M. Israfilov, “Approximation by interpolating polynomials in Smirnov–Orlicz class,” J. Korean Math. Soc., 43, No. 2, 413–424 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Akgün and D. M. Israfilov, “Simultaneous and converse approximation theorems in weighted Orlicz spaces,” Bull. Belg. Math. Soc. Simon Stevin, 17, 13–28 (2010).

    MathSciNet  MATH  Google Scholar 

  5. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Vol. 1, Birkhäuser (1971).

  6. P. L. Butzer, H. Dyckoff, E. Görlicz, and R. L. Stens, “Best trigonometric approximation, fractional derivatives and Lipschitz classes,” Can. J. Math., 24, No. 4, 781–793 (1977).

    Article  Google Scholar 

  7. P. L. Butzer, R. L. Stens, and M.Wehrens, “Approximation by algebraic convolution integrals,” in: J. B. Prolla (editor), Approximation Theory and Functional Analysis, North Holland (1979), pp. 71–120.

  8. R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer (1993).

  9. L. Diening, “Maximal functions on generalized Lebesgue space L p(x),” Math. Ineq. Appl., 7, No. 2, 245–253 (2004).

    MathSciNet  MATH  Google Scholar 

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

  11. P. L. Duren, Theory of H p Spaces, Academic Press (1970).

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

    Article  MathSciNet  MATH  Google Scholar 

  13. E. A. Haciyeva, Investigation of Properties of Functions with Quasimonotone Fourier Coefficients in Generalized Nikol’skii–Besov Spaces [in Russian], Thesis, Tbilisi (1986).

  14. H. Hudzik, “On generalized Orlicz–Sobolev spaces,” Funct. Approxim. Comment. Mat., 4, 37–51 (1976).

    MathSciNet  MATH  Google Scholar 

  15. D. M. Israfilov and R. Akgün, “Approximation in weighted Smirnov–Orlicz classes,” J. Math. Kyoto Univ., 46, No. 4, 755–770 (2006).

    MathSciNet  MATH  Google Scholar 

  16. D. M. Israfilov and R. Akgün, “Approximation by polynomials and rational functions in weighted rearrangement invariant spaces,” J. Math. Anal. Appl., 346, 489–500 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  17. D. M. Israfilov, V. Kokilashvili, and S. Samko, “Approximation in weighted Lebesgue and Smirnov spaces with variable exponent,” Proc. A. Razmadze Math. Inst., 143, 45–55 (2007).

    MathSciNet  Google Scholar 

  18. D. M. Israfilov, B. Oktay, and R. Akgun, “Approximation in Smirnov–Orlicz classes,” Glas. Mat. Ser. III, 40, No. 1, 87–102 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Jackson, Über Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung, Thesis, Götingen (1911).

  20. I. Joó, “Saturation theorems for Hermite–Fourier series,” Acta Math. Acad. Sci. Hung., 57, 169–179 (1991).

    MATH  Google Scholar 

  21. V. Kokilashvili and V. Paatashvili, “On variable Hardy and Smirnov classes of analytic functions,” Georg. Int. J. Sci., 1, No 2, 181–195 (2008).

    MathSciNet  Google Scholar 

  22. V. Kokilashvili and S. G. Samko, “Singular integrals and potentials in some Banach spaces with variable exponent,” J. Funct. Spaces Appl., 1, No. 1, 45–59 (2003).

    MathSciNet  MATH  Google Scholar 

  23. N. P. Korneichuk, Exact Constants in Approximation Theory, Cambridge University Press (1991).

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

    Google Scholar 

  25. N. X. Ky, “An Alexits’s lemma and its applications in approximation theory,” in: L. Leindler, F. Schipp, and J. Szabados (editors), Functions, Series, Operators, Budapest (2002), pp. 287–296.

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

    MATH  Google Scholar 

  27. J. Musielak, “Approximation in modular function spaces,” Funct. Approxim., Comment. Mat., 25, 45–57 (1997).

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  29. P. P. Petrushev and V. A. Popov, Rational Approximation of Real Functions, Cambridge University Press (1987).

  30. E. S. Quade, “Trigonometric approximation in the mean,” Duke Math. J., 3, 529–543 (1937).

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer (2000).

  32. S. G. Samko, “Differentiation and integration of variable order and the spaces L p(x),” in: Operator Theory for Complex and Hypercomplex Analysis, Proceedings of the International Conference on Operator Theory and Complex and Hypercomplex Analysis, December 12–17, 1994, Mexico City, Mexico, American Mathematical Society (1998), pp. 203–219.

  33. 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 

  34. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach (1993).

  35. B. Sendov and V. A. Popov, The Averaged Moduli of Smoothness in Numerical Methods and Approximation,Wiley, New York (1988).

    MATH  Google Scholar 

  36. I. I. Sharapudinov, “Topology of the space L p(t)([0, 1]),” Math. Notes, 26, No. 3–4, 796–806 (1979).

    MathSciNet  MATH  Google Scholar 

  37. I. I. Sharapudinov, “Uniform boundedness in L p(p = p(x)) of some families of convolution operators,” Math. Notes, 59, No. 1–2, 205–212 (1996).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. S. B. Stechkin, “On the order of the best approximations of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 15, 219–242 (1951).

    MATH  Google Scholar 

  40. R. Taberski, “Two indirect approximation theorems,” Demonstr. Math., 9, No. 2, 243–255 (1976).

    MathSciNet  MATH  Google Scholar 

  41. A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press/MacMillan (1963).

  42. A. Zygmund, Trigonometric Series, Vols. 1, 2, Cambridge, (1959).

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Authors and Affiliations

  1. Balıkesir University, Balıkesir, Turkey

    R. Akgün

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 1, pp. 3–23, January, 2011.

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Akgün, R. Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent. Ukr Math J 63, 1–26 (2011). https://doi.org/10.1007/s11253-011-0485-0

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  • DOI: https://doi.org/10.1007/s11253-011-0485-0

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