  This article studies the problem of approximating functions belonging to a
Hilbert space $\mathcal H_d$ with a reproducing kernel of the form $$\tilde
K_d(\boldsymbol x,\boldsymbol t):=\prod_{\ell=1}^d
\left(1-\alpha_\ell^2+\alpha_\ell^2K_{\gamma_\ell}(x_\ell,t_\ell)\right)\ \ \
\mbox{for all} \ \ \ \boldsymbol x,\boldsymbol t\in\mathbb R^d.$$ The
$\alpha_\ell\in[0,1]$ are scale parameters, and the $\gamma_\ell>0$ are
sometimes called shape parameters. The reproducing kernel $K_{\gamma}$
corresponds to some Hilbert space of functions defined on $\mathbb R$. The
kernel $\tilde K_d$ generalizes the anisotropic Gaussian reproducing kernel,
whose tractability properties have been established in the literature. We
present sufficient conditions on $\{\alpha_\ell
\gamma_\ell\}_{\ell=1}^{\infty}$ under which polynomial tractability holds for
function approximation problems on $\mathcal H_d$. The exponent of strong
polynomial tractability arises from bounds on the eigenvalues of a positive
definite linear operator.
