Average case tractability of multivariate approximation with Gaussian kernels

Abstract

We consider d-variate approximation problems in the average case setting in the weighted L2 space with the standard Gaussian weight. This space is equipped with a zero-mean Gaussian measure whose covariance kernel is a product of univariate Gaussian kernels with non-increasing positive shape parameters γj2 for j=1,2,…,d. We study the average case error of algorithms that use finitely many arbitrary continuous linear functionals. We define the information complexity n(ε,d) to be the minimal number of linear functionals which are needed to find an algorithm whose average case error is at most ε. We investigate different notions of tractability or exponential convergence-tractability (EC-tractability), and get necessary and sufficient conditions in terms of the shape parameters. In particular, for fixed s>0 and t∈(0,1) we obtain that the sufficient and necessary condition on γj2 for which limd+ε−1→∞n(ε,d)ε−s+dt=0holds is limj→∞j1−tγj2ln+γj−2=0,where ln+x=max(1,lnx).