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

Abstract

This article studies the problem of approximating functions belonging to a Hilbert space ${\mathcal{H}}_d$ with a reproducing kernel of the form $$\tilde K_d(\boldsymbol x,\boldsymbol t):=\prod_{\ell=1}^d \left(1-α_\ell^2+α_\ell^2K_{γ_\ell}(x_\ell,t_\ell)\right)\ \ \ \mbox{for all} \ \ \ \boldsymbol x,\boldsymbol t\in\mathbb R^d.$$ 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.

15 pages. arXiv admin note: text overlap with arXiv:1012.2605

References in corpus (1)

• Rate of Convergence and Tractability of the Radial Function Approximation Problem