Best approximation characterization of Sobolev spaces with common smoothness in the probabilistic setting and Sq norm

In this paper, we obtain the sharp bounds of probabilistic Gel’fand width of the multivariate Sobolev space [Formula: see text] with common smoothness equipped with the Gaussian measure in the [Formula: see text] norm by discretization method, where [Formula: see text] is a finite set. Then we obtain the sharp bounds of average Gel’fand width from the results of probabilistic width.