Convergence properties of radial basis functions

Abstract

The multivariate interpolation problem is that of choosing a functions fromR“ toR that satisfies the interpolation conditions

$$\{ s(x_i ) = f_i :i = 1,2,...,m\}$$

atm different data points x _i . Employing the radial basis functionϕ:R ⁺→R, means thats is from the linear space spanned by the functions

$$\{ \varphi (\left\| {x - x_i } \right\|_2 ):x \in R^n :i = 1,2,...,m\} ,$$

except that low-order polynomials are sometimes added and further conditions imposed ons that take up the extra degrees of freedom. Micchelli [6] shows that for some choices ofϕ these conditions uniquely defines. Now, however, the more basic question of the suitability of the spaces is considered. Sufficient conditions are given for members of the linear space to converge locally uniformly to a prescribed continuous function on a bounded domain, as the data points {x _i :i=1,2,...} become dense. For the caseϕ(r)=r they are found to hold when n is odd but not whenn is even.