Approximation of functions on the Sobolev space on the sphere in the average case setting

  • Authors:

  • Heping Wang;Xuebo Zhai;Yanwei Zhang

  • Affiliations:

  • Department of Mathematics, Capital Normal University, Beijing 100037, People's Republic of China;Department of Mathematics, Capital Normal University, Beijing 100037, People's Republic of China;Department of Mathematics, Capital Normal University, Beijing 100037, People's Republic of China

  • Venue:
  • Journal of Complexity
  • Year:
  • 2009

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Abstract

In this paper, we discuss the best approximation of functions by spherical polynomials and the approximation by Fourier partial summation operators, Vallee-Poussin operators, Cesaro operators, and Abel operators, on the Sobolev space on the sphere with a Gaussian measure, and obtain the average error estimates. We also get the asymptotic values for the average Kolmogorov and linear widths of the Sobolev space on the sphere and show that, in the average case setting, the spherical polynomial subspaces are the asymptotically optimal subspaces in the L"q(1@?q