Probabilistic and average linear widths of weighted Sobolev spaces on the ball equipped with a Gaussian measure

Abstract

Let \(L_{q,\mu}\), \(1\leq q\leq\infty\), denotes the weighted \(L_q\) space of functions on the unit ball \(\Bbb B^d\) with respect to weight \((1-\|x\|_2^2)^{\mu-\frac12},\,\mu\ge 0\), and let \(W_{2,\mu}^r\) be the weighted Sobolev space on \(\Bbb B^d\) with a Gaussian measure \(\nu\). We investigate the probabilistic linear \((n,\delta)\)-widths \(\lambda_{n,\delta}(W_{2,\mu}^r,\nu,L_{q,\mu})\) and the \(p\)-average linear \(n\)-widths \(\lambda_n^{(a)}(W_{2,\mu}^r,\mu,L_{q,\mu})_p\), and obtain their asymptotic orders for all \(1\le q\le \infty\) and \(0<p<\infty\).