## 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

### MSC:

 41A05 Interpolation in approximation theory

distance matrix

Zbl 0607.00009