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 $(b_{\xi})_{\xi\in\Xi}$ that function as bases for kernel spaces $S(k,\Xi)$ so that each basis function is constructed using very few kernels. In other words, each function $b_{\zeta}(x) = \sum_{\xi\in\Xi} A_{\zeta,\xi} k(x,\xi)$ is a linear combination of samples of the kernel with few nonzero coefficients $A_{\zeta,\xi}$. 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=#\Xi). 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 $\Xi$.