Linear widths of a multivariate function space equipped with a Gaussian measure
This paper continues the investigation carried out in [J. Complexity 20, No. 6, 858--875 (2004; Zbl 1064.41018)]. Here the authors determine asymptotic orders of linear probabilistic \((N, \delta)\)-widths \(\lambda_{N,\delta}\) and linear \(p\)-average widths \(\lambda_N^{(a)}\) of the set \(MW_2^r({\mathbb T}^d)\) of multivariate functions with bounded mixed derivative equipped with the Gaussian measure \(\mu\) in \(L_q({\mathbb T}^d)\). If \(1<q<\infty\) and \(r=(r_1, \ldots , r_d)\), \(1/2<r_1=\ldots =r_\nu<r_{\nu+1}\leq \ldots \leq r_d\), then \[ \lambda_N^{(a)}\left ( MW_2^r({\mathbb T}^d), \mu , L_q({\mathbb T}^d) \right )\asymp (N^{-1}\ln^{\nu-1}N)^{r_1+(\rho-1/2)}\left (\ln^{(\nu-1)/2} N \right ), \] where \(\rho>1\) depends only on the eigenvalues of the correlation operator of the measure \(\mu\). A similar equivalence is established for \((N, \delta)\)-widths.
scientific article; zbMATH DE number 2136635
scientific article