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

作者：Wang Heping^*; Zhai Xuebo; Zhang Yanwei

来源：Journal of Complexity, 2009, 25(4): 362-376.

DOI：10.1016/j.jco.2009.03.002

• Best approximation
• Approximation by multiplier
• Average width
• Sobolev space
• Gaussian measure

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 < infinity) metric, and Fourier partial summation operators and Vallee-Poussin operators are the asymptotically optimal linear operators and are (modulo a constant) as good as optimal nonlinear operators in the L(q) (1 <= q < infinity) metric.

• 出版日期2009-8
• 单位首都师范大学