Approximation of functions on the sphere on a Sobolev space with a Gaussian measure in the probabilistic case setting

Abstract

In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vallée-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vallée-Poussin operators are the order optimal linear operators in the Lq space for 1 ≤ q ≤ ∞, but the spherical polynomial spaces are not order optimal in the Lq space for 2 < q ≤ ∞. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the Lq space for 1 ≤ q < ∞. Also, in the Lq space for 1 ≤ q ≤ ∞, worst-case order optimal subspaces are also order optimal in the probabilistic case setting.

Probabilistic linear widths of Sobolev space with Jacobi weights on [ − 1 , 1 ] $[-1,1]$

“…The probabilistic linear -width is defined by where runs through all possible ν -measurable subsets of W with measure . Compared with the classical case analysis (see [2] or [6]), the probabilistic case analysis, which reflects the intrinsic structure of the class, can be understood as the ν -distribution of the approximation on all subsets of W by n -dimensional subspaces and linear operators with rank n .…”