Radial basis functions for multivariable interpolation: a review. (English) Zbl 0638.41001

Algorithms for approximation, Proc. IMA Conf., Shrivenham/Engl. 1985, Inst. Math. Appl. Conf. Ser., New Ser. 10, 143-167 (1987).
[For the entire collection see Zbl 0607.00009.]
Let \(\{x_ i:\) \(1\leq i\leq m\}\) be m distinct points in \(R^ n\), and let \(\{y_ i:\) \(1\leq i\leq m\}\) be m real values. The author studies the problem of finding functions \(s: R^ n\to R\) such that, if \(\phi: [0,\infty)\to R\), then s is a linear combination of functions of the form \(\phi(\| x-x_ i\|)\) plus a low order polynomial and \(s(x_ i)=y_ i\), \(1\leq i\leq m\). The main results involve conditions on \(\phi\) that are sufficient to define s. For a fixed \(\phi\), let \(A_{ij}=\phi (\| x_ i-x_ j\|)\) and let the matrix \(A=(A_{ij})\) be called the distance matrix. The non-singularity of the distance matrix is sufficient for the function s to exist. The author reviews known results for the distance matrix for functions of the form \(\phi (r)=r\) k, where k is a positive integer. In addition, a new proof is given for a result of C. A. Micchelli [Constructive Approximation (to appear)] stating that A is non-singular when \(A_{ij}=\| x_ i-x_ j\|_ 2\), \(1\leq i\), \(j\leq m\).
Reviewer: P.Lappan


41A05 Interpolation in approximation theory


distance matrix


Zbl 0607.00009