Some Aspects of Radial Basis Function Approximation

This paper deals with three basic aspects of radial basis approximation. A typical example of such an approximation is the following. A function f in C (ℝn) is to be approximated by a linear combination of ‘easily computable’ functions g 1,…, g m For these functions the simplest choice in the radial basis context is to define g i by x ↦ ∥x − x i∥2 for x ∈ ℝn and i = l,2,…,m. Here ∥ · ∥2 is the usual Euclidean norm on ℝn. These functions are certainly easily computable, but do they form a flexible approximating set? There are various ways of posing the question of flexibility, and we consider here three possible criteria by which the effectiveness of such approximations may be judged. These criteria are labelled density, interpolation and order of convergence in the exposition.