Tractability of the function approximation problem in terms of the kernel's shape and scale parameters

This article studies the problem of approximating functions belonging to a Hilbert space ${\mathcal{H}}_d$ with a reproducing kernel of the form $${\tilde{K}}_d(\mathit{x},\mathit{t}):=\prod _{\mathrm{\ell }=1}^d(1-{\alpha }_{\mathrm{\ell }}^2+{\alpha }_{\mathrm{\ell }}^2{K}_{{\gamma }_{\mathrm{\ell }}}(x_{\mathrm{\ell }},t_{\mathrm{\ell }}))forall\mathit{x},\mathit{t}\in {\mathbb{R}}^d\mathrm{.}$$ The ${\alpha }_{\mathrm{\ell }}\in [0,1]$ are scale parameters, and the ${\gamma }_{\mathrm{\ell }}>0$ 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 $\{{\alpha }_{\mathrm{\ell }}{\gamma }_{\mathrm{\ell }}{\}}_{\mathrm{\ell }=1}^{\mathrm{\infty }}$ 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.