N-Widths of Multivariate Sobolev Spaces with Common Smoothness in Probabilistic and Average Settings in the S-q Norm

In this article, we give the sharp bounds of probabilistic Kolmogorov (N, delta)-widths and probabilistic linear (N, delta)-widths of the multivariate Sobolev space W-2(A) with common smoothness on a S-q norm equipped with the Gaussian measure mu, where A subset of R-d is a finite set. And we obtain the sharp bounds of average width from the results of the probabilistic widths. These results develop the theory of approximation of functions and play important roles in the research of related approximation algorithms for Sobolev spaces.