  We consider a problem of approximation of $d$-variate functions defined on
$\mathbb{R}^d$ which belong to the Hilbert space with tensor product-type
reproducing Gaussian kernel with constant shape parameter. Within worst case
setting, we investigate the growth of the information complexity as
$d\to\infty$. The asymptotics are obtained for the case of fixed error
threshold and for the case when it goes to zero as $d\to\infty$.
