Acta Mathematica Sinica, English Series

, Volume 35, Issue 2, pp 185–203 | Cite as

Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz Spaces

  • Sonia AcinasEmail author
  • Sergio Favier
  • Felipe Zó


In this paper we pursue the study of the best approximation operator extended from LΦ to Lφ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calderón–Zygmund class t m p (x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–Orlicz indices for the function φ.


Orlicz spaces extended best polynomial approximation pointwise and norm convergence weak and strong type inequalities Orlicz indices 

MR(2010) Subject Classification

05B05 05B25 20B25 


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The authors would like to thank the referee for such valuable comments and suggestions.


  1. [1]
    Acinas, S., Favier, S.: Maximal inequalities in Orlicz spaces. Int. J. Math. Anal. (Ruse), 6(41–44), 2179–2198 (2012)MathSciNetzbMATHGoogle Scholar
  2. [2]
    Acinas, S., Favier, S., Zó, F.: Extended best polynomial approximation operator in Orlicz spaces. Numer. Funct. Anal. Optim., 36(7), 817–829 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Brunk, H., Johansen, S.: A generalized Radon–Nikodym derivative. Pacific J. Math., 34, 585–617 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Calderón, A., Zygmund, A.: Local properties of solutions of elliptic partial differential equations. Studia Math., 20, 171–225 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Cuenya, H.: Extension of the operator of best polynomial approximation in L p(Ω). J. Math. Anal. Appl., 376(2), 565–575 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Cuenya, H., Favier, S., Zó, F.: Inequalities in L p-1 for the extended L p best approximation operator. J. Math. Anal. Appl., 393, 80–88 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Favier, S., Zó, F.: A Lebesgue type differentiation theorem for best approximations by constants in Orlicz spaces. Real Anal. Exchange, 30(1), 29–41 (2004–2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Favier, S., Zó, F.: Extension of the best approximation operator in Orlicz spaces and weak-type inequalities. Abstr. Appl. Anal., 6(2), 101–114 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Favier, S., Zó, F.: Maximal inequalities for a best approximation operator in Orlicz spaces. Comment. Math. Prace Mat., 5(1), 3–21 (2011)MathSciNetzbMATHGoogle Scholar
  10. [10]
    Fiorenza, A., Krbec, M.: Indices of Orlicz spaces and some applications. Comment. Math. Univ. Carolin., 38(3), 433–451 (1997)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Gustavsson, J., Peetre, J.: Interpolation of Orlicz spaces. Studia Math., 60(1), 33–59 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Kokilashvili, V., Krbec, M.: Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, Singapore, 1991CrossRefzbMATHGoogle Scholar
  13. [13]
    Krasnosel’skii, M. A., Rutickii, Ya. B.: Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen (The Netherlands), 1961Google Scholar
  14. [14]
    Landers, D., Rogge, L.: Best approximants in L φ-spaces. Z. Wahrsch. Verw. Gabiete, 51, 215–237 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Landers, D., Rogge, L.: Isotonic approximation in L s. J. Approx. Theory, 31(3), 199–223 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II, Vol. 97 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Springer-Verlag, Berlin–New York, 1979Google Scholar
  17. [17]
    Maligranda, L.: Orlicz Spaces and Interpolation, Seminarios de Matemática, Campinas SP (Brasil), 1989zbMATHGoogle Scholar
  18. [18]
    Mazzone, F., Cuenya, H.: Maximal inequalities and Lebesgue’s differentiation theorem for best approximant by constant over balls. J. Approx. Theory, 110(2), 171–179 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Mazzone, F., Cuenya, H.: On best local approximants in L 2(ℝn). Rev. Un. Mat. Argentina, 42(2), 51–56 (2001–2002)zbMATHGoogle Scholar
  20. [20]
    Rao, M., Ren, Z.: Theory of Orlicz Spaces, Marcel Dekker, New York, 1991zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática, Facultad de Ciencias y Exactas NaturalesUniversidad Nacional de La PampaSanta Rosa, La PampaArgentina
  2. 2.Instituto de Matemática Aplicada San Luis, IMASLUniversidad Nacional de San Luis and CONICETSan LuisArgentina
  3. 3.Departamento de MatemáticaUniversidad Nacional de San LuisSan LuisArgentina

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