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A modulus of smoothness on the unit sphere

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Abstract

For functions onS d−1 (the unit sphere inR d) and, in particular, forf∈L 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).

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References

  • [Ch-Di] W. Chen and Z. Ditzian,Best approximation and K-functionals, Acta Math. Hungar.,75 (1997), 165–208.

    Article  MATH  MathSciNet  Google Scholar 

  • [Di] Z. Ditzian,On the Marchaud-type inequality, Proc. Amer. Math. Soc.,103 (1988), 198–202.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ga] F. R. Gantmacher,The Theory of Matrices, Vol. 1, Chelsea Publ Co., New York, 1959.

    MATH  Google Scholar 

  • [Li-Tz] Y. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, Vols. I and II, Springer-Verlag, Berlin and New York, 1977, 1979.

    MATH  Google Scholar 

  • [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.

    MathSciNet  Google Scholar 

  • [Pe-Po] P. P. Petrushev and V. A. Popov,Rational Approximation of Real Functions, Cambridge University Press, 1987.

  • [St-We] E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.

    MATH  Google Scholar 

  • [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).

    MATH  Google Scholar 

  • [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).

    MathSciNet  Google Scholar 

  • [To] V. Totik,Sharp converse theorem of L p -polynomial approximation, Constr. Approx.4 (1988), 419–433.

    Article  MATH  MathSciNet  Google Scholar 

  • [Zy] A. Zygmund,A remark on the integral modulus of continuity, Univ. Nac. Tucumán Rev. Ser. A7 (1950), 259–269.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Alberta, T6G 2G1, Edmonton, Alberta, Canada

    Z. Ditzian

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  1. Z. Ditzian
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Supported by NSERC Grant A4816 of Canada.

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Ditzian, Z. A modulus of smoothness on the unit sphere. J. Anal. Math. 79, 189–200 (1999). https://doi.org/10.1007/BF02788240

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  • DOI: https://doi.org/10.1007/BF02788240

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