Best approximation of functions on the ball on the weighted Sobolev space equipped with a Gaussian measure

Abstract

This paper contains two parts. In the first part, we obtain the relations between the classical and p -average Kolmogorov widths for all p , 0 < p < ∞ , which is a generalization of the corresponding results of J. Creutzig given in [J. Creutzig, Relations between classical, average and probabilistic Kolmogorov widths, J. Complexity 18 (2002) 287–303]. In the second part, we investigate the best approximation of functions on the weighted Sobolev space B W 2 , μ r ( B d ) equipped with a centered Gaussian measure by polynomial subspaces in the L q , μ metric for 1 ≤ q < ∞ , where L q , μ , 1 ≤ q < ∞ , denotes the weighted L q space of functions on the unit ball B d with respect to the weight ( 1 − | x | 2 ) μ − 1 2 , μ ≥ 0 . The asymptotic orders of the average error estimations are obtained. We find a striking fact that, in the average case setting, the polynomial subspaces are the asymptotically optimal linear subspaces in the L q , μ metric only for 1 ≤ q < 2 + 1 / μ , which means that 2 + 1 / μ is the critical value and is independent of dimension d .