Radial Basis Function Approximation: From Gridded Centers to Scattered Centers
Abstract
The paper studies L-infinity (IR d)-norm approximations from a space spanned by a discrete set of translates of a basis function theta. Attention here is restricted to functions theta whose Fourier transform is smooth on IRd/ 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 case the set of points used for translating theta 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 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 schemes for uniform grids, an effort is made to impose only a few mild conditions on the function theta, 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1993
- Accession Number
- ADA274184
Entities
People
- Amos Ron
- Nira Dyn
Organizations
- University of Wisconsin–Madison