TY - JOUR

T1 - Radial basis function approximation

T2 - From gridded centres to scattered centres

AU - Dyn, N.

AU - Ron, A.

N1 - Funding Information:
Research supported in part by the Israel-U.S. Binational Science Foundation the U.S. Army (Contract DAAL03-G-90-0090), and by the National Science DMS-9000053, DMS-9102857, DMS-9224748).

PY - 1995/7

Y1 - 1995/7

N2 - The paper studies Lx(Rd)-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 Rd\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 Rd, 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.

AB - The paper studies Lx(Rd)-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 Rd\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 Rd, 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.

UR - http://www.scopus.com/inward/record.url?scp=0002679899&partnerID=8YFLogxK

U2 - 10.1112/plms/s3-71.1.76

DO - 10.1112/plms/s3-71.1.76

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AN - SCOPUS:0002679899

SN - 0024-6115

VL - s3-71

SP - 76

EP - 108

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 1

ER -