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

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