Radial basis function approximation : from gridded centers to scattered centers

The paper studies L.(IRd)-norm approximations from a space spanned by a discrete set of translates of a basis function 4. Attention here is restricted to functions 0 whose Fourier transform is smooth on IRd\O, and has a singularity at the origin. Examples of such basis functions are the thin-plate splines and the multiquadrics, as well as other types of radial basis functions that are employed in Approximation Theory. The above approximation problem is well-understood in case the set of points E used for translating 0 forms a lattice in IRd, and many optimal and quasi-optimal approximation schemes can already be found in the literature. In contrast, only few, mostly specific, results are known for a set of scattered points. The main objective of this paper is to provide a general tool for extending approximation schemes that use integer translates of a basis function to the non-uniform case. We introduce a single, relatively simple, conversion method that preserves the approximation orders provided by a large number of schemes presently in the literature (more precisely, to almost all "stationary schemes"). In anticipation of future introduction of new schemefor uniform grids, an effort is made to impose only a few mild conditions on the function 0, which still allow for a unified error analysis to hold. In the course of the discussion here, the recent results of [BuDL] on scattered center approximation are reproduced and improved upon. AMS (MOS) Subject Classifications: primary 41A15, 41A25, 41A63 secondary 35E05 42B99