Average Case $(s, T)$-Weak Tractability of $L_2$-Approximation with Weighted Covariance Kernels

We study multivariate $L_2$-approximation problem $\textrm{APP}_d$ defined over a Banach space in the average case setting. The space is equipped with a zero-mean Gaussian measure with weighted covariance kernel which depends on parameter sequences \texorpdfstring{$\a=\{\az_j\}_{j\in\N}$ and $\ba=\{\bz_j\}_{j\in\N}$}{} with $1 0$. In this paper two interesting weighted covariance kernels are considered, which model the importance of the covariance kernels. Under the absolute error criterion or the normalized error criterion, we discuss \texorpdfstring{$(s,t)$}{}-weak tractability of the \texorpdfstring{$L_2$}{}-approximation problem $\textrm{APP}=\{\textrm{APP}_d\}_{d\in\N}$ with the above two weighted covariance kernels for some positive numbers $s$ and $t$ in the average case setting, where \texorpdfstring{$(s,t)$}{}-weak tractability means that how the information complexity depends on $d$ and $\va^{-1}$ for large dimension $d$ and small threshold $\va$. In particular, for all $s>0$ and $t\in(0,1)$ we find the matching sufficient and necessary condition on the parameter sequences \texorpdfstring{$\a=\{\az_j\}_{j\in\N}$ and $\ba=\{\bz_j\}_{j\in\N}$}{} to obtain average case \texorpdfstring{$(s,t)$}{}-weak tractability under the absolute error criterion or the normalized error criterion.