Weighted Best Approximation by Polynomials

  • Feng Dai
  • Yuan Xu
Part of the Springer Monographs in Mathematics book series (SMM)


The structure of spherical harmonics allows us to develop a theory of best polynomial approximation on the sphere based essentially on multipliers, which is, historically, the first approach in this direction. It turns out that the entire framework based on multipliers can be established more generally for h-spherical harmonics associated with reflection-invariant weight functions developed in Chap. 7, which leads to a theory of weighted best approximation by polynomials that we shall present in its full generality.


  1. 13.
    Belinsky, E., Dai, F., Ditzian, Z.: Multivariate approximating averages. J. Approx. Theor. 125, 85–105 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 26.
    Butzer, P.L., Jansche, S.: Lipschitz spaces on compact manifolds. J. Funct. Anal. 7, 242–266 (1971)zbMATHCrossRefGoogle Scholar
  3. 40.
    Dai, F., Ditzian, Z.: Combinations of multivariate averages. J. Approx. Theor. 131, 268–283 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 41.
    Dai, F., Ditzian, Z.: Littlewood–Paley theory and a sharp Marchaud inequality. Acta Sci. Math. (Szeged) 71, 65–90 (2005)MathSciNetzbMATHGoogle Scholar
  5. 43.
    Dai, F., Ditzian, Z., Tikhonov, S.: Sharp Jackson inequalities. J. Approx. Theor. 151, 86–112 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 55.
    Ditzian, Z.: Fractional derivatives and best approximation. Acta Math. Hungar. 81, 323–348 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 60.
    Ditzian, Z., Prymak, A.: Convexity, moduli of smoothness and a Jackson-type inequality. Acta Math. Hungar. 130(3), 254–285 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 94.
    Kalybin, G.A.: On moduli of smoothness of functions given on the sphere. Soviet Math. Dokl. 35, 619–622 (1987)Google Scholar
  9. 103.
    Kušnirenko, G.G.: The approximation of functions defined on the unit sphere by finite spherical sums (Russian). Naučn. Dokl. Vysš. Skoly Fiz. Mat. Nauki 4, 47–53 (1958)Google Scholar
  10. 132.
    Nikolskii, S.M., Lizorkin, P.I.: Approximation of functions on the sphere. Izv. AN SSSR, Ser. Mat. 51(3), 635–651 (1987)Google Scholar
  11. 135.
    Pawelke, S.: Über Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen. Tôhoku Math. J. 24, 473–486 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 146.
    Rustamov, Kh.P.: On the best approximation of functions on the sphere in the metric of \(L_{p}({\mathbb{S}}^{n})\), 1 < p < . Anal. Math. 17, 333–348 (1991)Google Scholar
  13. 148.
    Rustamov, Kh.P.: On the approximation of functions on a sphere (Russian). Izv. Ross. Akad. Nauk Ser. Mat. 57, 127–148 (1993). Translation in Russian Acad. Sci. Izv. Math. 43(2), 311–329 (1994)Google Scholar
  14. 165.
    Timan, M.F.: Converse theorems of the constructive theory of functions in the spaces L p. Mat. Sborn. 46(88), 125–132 (1958)MathSciNetGoogle Scholar
  15. 166.
    Timan, M.F.: On Jackson’s theorem in L p spaces. Ukrain. Mat. Zh. 18(1), 134–137 (1966 in Russian)Google Scholar
  16. 174.
    Wang, K.Y., Li, L.Q.: Harmonic Analysis and Approximation on the Unit Sphere. Science Press, Beijing (2000)Google Scholar
  17. 189.
    Xu, Y.: Generalized translation operator and approximation in several variables. J. Comp. Appl. Math. 178, 489–512 (2005)zbMATHCrossRefGoogle Scholar
  18. 190.
    Xu, Y.: Weighted approximation of functions on the unit sphere. Constr. Approx. 21, 1–28 (2005)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Feng Dai
    • 1
  • Yuan Xu
    • 2
  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of MathematicsUniversity of OregonEugeneUSA

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