Radial Basis Function Approximation: from Gridded Centers to Scattered Centers Radial Basis Function Approximation: from Gridded Centers to Scattered Centers

The paper studies L 1 (IR d)-norm approximations from a space spanned by a discrete set of translates of a basis function. Attention here is restricted to functions whose Fourier transform is smooth on IR d n0, 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 used for translating forms a lattice in IR d , and many optimal and quasi-optimal approximation schemes can already be found in the literature. In contrast, only few, mostly speciic, 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 schemes for uniform grids, an eeort is made to impose only a few mild conditions on the function , which still allow for a uniied 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.