Probabilistic and average linear widths of Sobolev space with Gaussian measure in \(L_\infty\)-norm
Let \(r\) be a real number satisfying \(r> 1/2\), and let the Sobolev space \(W^r_2\) be equipped with a Gaussian measure \(\mu\) whose mean is zero and whose correlation operator has the eigenfunctions \(\exp(ikt)\) and the eigenvalues \(|k|^{-s}(k\in \mathbb{Z}\setminus\{0\})\), where \(s> 1\). Furthermore, it is assumed that \(W^r_2\) is embedded in the normed space \(L_\infty\). By using discretization techniques the authors prove the following asymptotic equalities: \[ \begin{aligned} \lambda_{n,\delta}(W^r_2, \mu,L_\infty)\asymp n^{-\rho} \sqrt{\ln(n/\delta)}\quad &\text{whenever }\delta\in (0,1/2);\\ \lambda^{(a)}_n(W^r_2, L_\infty)p\asymp n^{-\rho} \sqrt{\ln n}\quad &\text{whenever }p\in (0,\infty).\end{aligned} \] Here \(\lambda_{n,\delta}(W^r_2, \mu,L_\infty)\) and \(\lambda^{(a)}_n(W^r_2, \mu,L_\infty)_p\) denote the probabilistic linear \((n,\delta)\)-width and the \(p\)-average linear \(n\)-width of \(W^r_2\), respectively, while \(\rho:= r+ (s-1)/2\).
scientific article; zbMATH DE number 2101903