Best approximation of functions on the ball on the weighted Sobolev space equipped with a Gaussian measure
This paper contains two parts. In the first part, the authors obtain relations between the classical and \(p\)-average Kolmogorov widths for all \(p\), \(0 < p < \infty\), which is a generalization of the corresponding results of \textit{J. Creuzig} [J. Complexity 18, No. 1, 287--303 (2002; Zbl 1008.41020)]. The second part of the paper is devoted to the study of the best approximation by polynomial subspaces on Weighted Sobolev space of functions on the unit ball \({\mathcal B}^d\) with a Gaussian measure. The authors obtain average error estimates of the best approximation by polynomial subspaces in the weighted \(L_q\) space for \(1 \leq q < \infty\), and find that in the average case setting, whether or not the polynomial subspaces are the asymptotically optimal subspaces in the weighted \(L_q\) space for \(1 \leq q < \infty\) depends on the weight and is independent of the dimension \(d\).
scientific article; zbMATH DE number 5726581
scientific article