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

Abstract

Let $L_{q,\mu }$, $1\le q\le \infty$, denotes the weighted $L_q$ space of functions on the unit ball ${\mathbb{B}}^d$ with respect to weight $(1-\Vert x{\Vert }_2^2{)}^{\mu -\frac{1}{2}},\mu \ge 0$, and let $W_{2,\mu }^r$ be the weighted Sobolev space on ${\mathbb{B}}^d$ 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 $0<p<\infty$.