Better bases for kernel spaces

In this article we investigate the feasibility of constructing stable, local bases for computing with kernels. In particular, we are interested in constructing families $(b_ξ)_{ξ\inΞ}$ that function as bases for kernel spaces $S(k,Ξ)$ so that each basis function is constructed using very few kernels. In other words, each function $b_ζ(x) = \sum_{ξ\inΞ} A_{ζ,ξ} k(x,ξ)$ is a linear combination of samples of the kernel with few nonzero coefficients $A_{ζ,ξ}$. This is reminiscent of the construction of the B-spline basis from the family of truncated power functions. We demonstrate that for a large class of kernels (the Sobolev kernels as well as many kernels of polyharmonic and related type) such bases exist. In fact, the basis elements can be constructed using a combination of roughly $O(\log N)^d$ kernels, where $d$ is the local dimension of the manifold and $N$ is the dimension of the kernel space (i.e. $N=#Ξ$). Viewing this as a preprocessing step -- the construction of the basis has computational cost $O(N(\log N)^d)$. Furthermore, we prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of $Ξ$.

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