Abstract: This paper investigated multivariate approximation problems APP_d(d∈N₊)of Banach spaces equipped with zero-mean Gaussian measures in the average case setting, where covariance kernels of the zero-mean Gaussian measures had non-negative weighted sequences {α_j} and {γ_j}. In particular, the paper introduced covariance kernels with two different weights. We approximated the multivariate problems APPd by the algorithms that used finitely many continuous linear functionals. This paper discussed(s,t)-weak tractability for s>0 and t≥1 of the L₂-approximation problems APP={APP_d}_d∈N+ from the Banach spaces with the above two weighted covariance kernels under the absolute error criterion and the normalized error criterion. As a result, by the real analyzing the sufficient and necessary condition for(s,1)-weak tractability of these two L₂-approximation problems APP could be obtained as follows: the weight sequence {γ_j} tendde to 0 as j tends to infinity.

Key words: L₂-approximation, (s,t)-weak tractability, average case setting, covariance kernels