Advertisement

Journal d’Analyse Mathématique

, Volume 79, Issue 1, pp 189–200 | Cite as

A modulus of smoothness on the unit sphere

  • Z. Ditzian
Article

Abstract

For functions onS d−1 (the unit sphere inR d) and, in particular, forfL p(S d−1), we define new simple moduli of smoothness. We relate different orders of these moduli, and we also relate these moduli to best approximation by spherical harmonics of order smaller thann. Our new moduli lead to sharper results than those now available for the known moduli onL p(S d−1).

Keywords

Banach Space Unit Sphere Spherical Harmonic Triangle Inequality Orthogonal Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ch-Di] W. Chen and Z. Ditzian,Best approximation and K-functionals, Acta Math. Hungar.,75 (1997), 165–208.MATHCrossRefMathSciNetGoogle Scholar
  2. [Di] Z. Ditzian,On the Marchaud-type inequality, Proc. Amer. Math. Soc.,103 (1988), 198–202.MATHCrossRefMathSciNetGoogle Scholar
  3. [Ga] F. R. Gantmacher,The Theory of Matrices, Vol. 1, Chelsea Publ Co., New York, 1959.MATHGoogle Scholar
  4. [Li-Tz] Y. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, Vols. I and II, Springer-Verlag, Berlin and New York, 1977, 1979.MATHGoogle Scholar
  5. [Ni-Li] S. M. Nikolskii and P. I. Lizorkin,Function spaces on a sphere that are connected with approximation theory, Mat. Zametki41 (1987), 509–516. English transl. in Math. Notes41 (1987), 286–291.MathSciNetGoogle Scholar
  6. [Pe-Po] P. P. Petrushev and V. A. Popov,Rational Approximation of Real Functions, Cambridge University Press, 1987.Google Scholar
  7. [St-We] E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.MATHGoogle Scholar
  8. [A, Ti] A. F. Timan,Theory of Approximation of Functions of a Real Variable, Pergamon Press, Oxford, 1963 (translated from the Russian original which appeared in 1960 and was published by Fizmatgiz).MATHGoogle Scholar
  9. [M, Ti] M. F. Timan,Converse theorem of the cònstructive theory of functions in the spaces L p, Mat. Sb.44 (88) (1958), 125–132 (Russian).MathSciNetGoogle Scholar
  10. [To] V. Totik,Sharp converse theorem of L p -polynomial approximation, Constr. Approx.4 (1988), 419–433.MATHCrossRefMathSciNetGoogle Scholar
  11. [Zy] A. Zygmund,A remark on the integral modulus of continuity, Univ. Nac. Tucumán Rev. Ser. A7 (1950), 259–269.MATHMathSciNetGoogle Scholar

Copyright information

© The Magnes Press 1999

Authors and Affiliations

  • Z. Ditzian
    • 1
  1. 1.Department of Mathematical SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations