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

By Xuan Zhou and Fred J. Hickernell


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.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1012.260

Topics: Mathematics - Numerical Analysis
Year: 2014
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