Radial basis function approximation: From gridded centres to scattered centres

The paper studies L_x(R^d)-norm approximations from a space spanned by a discrete set of translates of a basis function φ. Attention here is restricted to functions φ whose Fourier transform is smooth on R_d\0, 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 the case where the setof points Ξ used for translating φ forms a lattice in R_d, and many optimal and quasi-optimal approximation schemes can already be found in theliterature.In contrast, only a few, mostly specific, results are known for a set Ξ of scattered points.The main objective of this paper is to provide a general tool for extendingapproximationschemes 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(moreprecisely, to almost all ‘stationary schemes’).In anticipation of future introduction ofnewschemes for uniform grids, an effort is made to impose only a few mild conditions on the function φ, which still allow fora unified error analysis to hold.In the course ofthe discussion here, the recent results of Buhmann, Dyn, and Levin [9] on scattered centreapproximation are reproduced and improved upon.