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

TL;DR

This paper studies the asymptotic behavior of probabilistic and average linear widths of weighted Sobolev spaces on the unit ball with Gaussian measures, providing new insights into their approximation properties.

Contribution

It introduces the asymptotic orders of probabilistic and average linear widths for weighted Sobolev spaces on the ball with Gaussian measures, extending understanding of their approximation complexity.

Findings

Asymptotic orders of probabilistic linear widths are established.

Asymptotic orders of average linear widths are derived.

Results hold for all q in [1, ∞) and p in (0, ∞).

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 $0<p<\infty$.