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

Abstract

In this paper, we discuss the best approximation of functions by spherical polynomials and the approximation by Fourier partial summation operators, Vallée–Poussin operators, Cesàro 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 < ∞ ) metric, and Fourier partial summation operators and Vallée–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 < ∞ ) metric.