  We study d-variate approximation problems in the average case setting with
respect to a zero-mean Gaussian measure. Our interest is focused on measures
having a structure of non-homogeneous linear tensor product, where covariance
kernel is a product of univariate kernels. We consider the normalized average
error of algorithms that use finitely many evaluations of arbitrary linear
functionals. The information complexity is defined as the minimal number n(h,d)
of such evaluations for error in the d-variate case to be at most h. The growth
of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the
covariance operator and determines whether a problem is tractable or not. Four
types of tractability are studied and for each of them we find the necessary
and sufficient conditions in terms of the eigenvalues of univariate kernels. We
illustrate our results by considering approximation problems related to the
product of Korobov kernels characterized by a weights g_k and smoothnesses r_k.
We assume that weights are non-increasing and smoothness parameters are
non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for
some non-increasing function g. In particular, we show that approximation
problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d
and 0<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k
>1. For other types of tractability we also show necessary and sufficient
conditions in terms of the sequences g_k and r_k.
