Advertisement

Journal of Mathematical Sciences

, Volume 239, Issue 1, pp 30–42 | Cite as

Approximative Characteristics of Modular Orlicz Spaces

  • Stanislav O. ChaichenkoEmail author
  • Andrii L. Shydlich
Article
  • 1 Downloads

Abstract

We obtain the exact values of the best approximations, basic widths and Kolmogorov widths for some sets of images of multipliers in the modular Orlicz spaces lM: We give a description of the space SM,N of all multipliers from the space lM to lN.

Keywords

Modular Orlicz spaces best approximation basic width Kolmogorov width multiplier 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Springer, Berlin, 1977.CrossRefzbMATHGoogle Scholar
  2. 2.
    W. Orlicz, “Über Räume (L M),” Bull. Intern. de l’Acad. Pol., Serie A, 93–107 (1936).Google Scholar
  3. 3.
    W. Orlicz, “Über konjugierte Exponentenfolgen,” Studia Math., No. 3, 200–211 (1931).Google Scholar
  4. 4.
    L. Diening and M. Ružička, “Calderon-Zygmund operators on generelized Lebesgue spaces L p(x) and problems related to fluid dynamics,” J. Reine Angew. Math., 563, 197–220 (2003).MathSciNetzbMATHGoogle Scholar
  5. 5.
    M. Ružička, Electroreological Fluids: Modeling and Mathematical Theory, Springer, Berlin, 2000.zbMATHGoogle Scholar
  6. 6.
    S. G. Samko, “On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators,” Integr. Transf. Spec. Funct., 16, Nos. 5–6, 461–482 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    P. Harjulehto, P. Hästö, and R. Klén, “Generalized Orlicz spaces and related PDE,” Nonlin. Anal.: Theory, Meth. Appl., 143, 155–173 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    P. Hästö, “The maximal operator on generalized Orlicz spaces,” J. Funct. Anal., 269, No. 12, 4038–4048 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    L. Diening, “Maximal function on generalized Lebesgue spaces L p(·),” Math. Inequal. Appl., 7, No. 2, 245–253 (2004).MathSciNetzbMATHGoogle Scholar
  10. 10.
    L. Pick and M. Ružička, “An example of a space L p(x) on which the Hardy––Littlewood maximal operator is not bounded,” Expo. Math., 19, 369–371 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Cruz-Uribe, A. Fiorenza, and C. Neugebauer, “The maximal function on variable L p spaces,” Ann. Acad. Sci. Fenn. Math., 28, 223–238 (2003); Ann. Acad. Sci. Fenn. Math., 29, 247–249 (2004).Google Scholar
  12. 12.
    A. Nekvinda, “Equivalence of l pn norms and shift operators,” Math. Inequal. Appl., 5, No. 4, 711–723 (2002).MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. Nekvinda, “A note on maximal operator on l pn and L p(x)(ℝ),” J. Funct. Spaces Appl., 5, No. 1, 49–88 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    A. Nekvinda, “Embeddings between discrete weighted Lebesgue spaces,” Math. Inequal. Appl., 10, No. 1, 165–172 (2007).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Yu. I. Gribanov, Nonlinear Operators in Orlicz Spaces [in Russian], [in Russian], Kazan Univ., Kazan, 1955.Google Scholar
  16. 16.
    Yu. I. Gribanov, To the Theory of Spaces l M [in Russian], Kazan Univ., Kazan, 1957.Google Scholar
  17. 17.
    P. B. Djakov and M. S. Ramanujan, “Multipliers between Orlicz sequence spaces,” Truk. J. Math., No. 24, 313–319 (2000).MathSciNetzbMATHGoogle Scholar
  18. 18.
    M. Aiyub, “On some seminormed sequence spaces defined by Orlicz function,” Proyecc. J. Math., 32, No. 3, 267–280 (2013).MathSciNetzbMATHGoogle Scholar
  19. 19.
    F.-Y. Maeda, Y. Mizuta, T. Ohno, and T. Shimomura, “Boundedness of maximal operators and Sobolev’s inequality on Musielak–Orlicz–Morrey spaces,” Bull. Sci. Math., 137, 76–96 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    F.-Y. Maeda, Y. Mizuta, T. Ohno, and T. Shimomura, “Approximate identities and Young type inequalities in Musielak–Orlicz spaces,” Czechoslovak Math. J., 63(138), No. 4, 933–948 (2013).Google Scholar
  21. 21.
    M. A. Krasnosel’skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.Google Scholar
  22. 22.
    L. Diening, P. Harjulehto, P. Hästö, and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin, 2011.CrossRefzbMATHGoogle Scholar
  23. 23.
    J. Musielak, Orlicz Spaces and Modular Spaces, Springer, Berlin, 1983.CrossRefzbMATHGoogle Scholar
  24. 24.
    A. L. Shydlich and S. O. Chaichenko, “Approximative characteristics of diagonal operators in the spaces l p,” in: Mathematical Problems of Mechanics and Computational Mathematics [in Ukrainian], Institute of Mathematics of the NAS of Ukraine, Kiev, 2014, pp. 399–412.Google Scholar
  25. 25.
    A. L. Shidlich and S. O. Chaichenko, “Approximative properties of diagonal operators in Orlicz spaces,” Numer. Funct. Analys. Optimiz., 36, No. 10, 1339–1352 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. I. Stepanets, “Problems of approximation theory in linear spaces,” Ukr. Mat. Zh., 58, No. 1, 47–92 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    V. M. Tikhomirov, Some Questions in Approximation Theory [in Russian], Moscow State Univ., Moscow, 1976.Google Scholar
  28. 28.
    A. I. Stepanets, “Approximative characteristics of spaces \( {S}_{\upvarphi}^{\mathrm{p}} \) ,” Ukr. Mat. Zh., 53, No. 3, 392–416 (2001).CrossRefGoogle Scholar
  29. 29.
    A. I. Stepanets, Methods of Approximation Theory [in Russian], Institute of Mathematics of the NAS of Ukraine, Kiev, 2002, Part 2.Google Scholar
  30. 30.
    A. Pinkus, n-Widths in Approximation Theory, Springer, Berlin, 1985.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Donbas State Pedagogical UniversitySlov’yanskUkraine
  2. 2.Institute of Mathematics of the NAS of UkraineKievUkraine

Personalised recommendations