Average Case (s, t)-Weak Tractability of L2-Approximation with Weighted Covariance Kernels

We study the multivariate L2-approximation problem APPd defined over a Banach space in the average case setting. The space is equipped with a zero-mean Gaussian measure with a weighted covariance kernel, which depends on parameter sequences α={αj}j∈N and β={βj}j∈N with 1<α1≤α2≤⋯ and 1≥β1≥β2≥⋯>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 (s,t)-weak tractability of the L2-approximation problem APP={APPd}d∈N from a Banach space whose zero-mean Gaussian measure has the above two weighted covariance kernels for some positive numbers s and t in the average case setting. Here, (s,t)-weak tractability means how the information complexity behaves as a function of dt and ε−s for large dimension d and small threshold ε. In particular, for all s>0 and t∈(0,1), we find the matching sufficient and necessary condition on the parameter sequences α={αj}j∈N and β={βj}j∈N to obtain (s,t)-weak tractability under the absolute error criterion or the normalized error criterion in the average case setting. We describe (s,t)-weak tractability by the matching sufficient and necessary condition, which reflects symmetry.