In numerical analysis, the concept of locally supported basis functions is of general importance. Several function spaces used for approximation possess locally supported basis functions. The most prominent examples in the one-dimensional case are the well-known B-splines. The general advantages of compactly supported basis functions are a sparse interpolation matrix on the one hand, and the possibility of a fast evaluation of the interpolant on the other.

Thus, it seems to be natural to look for locally supported functions also in the context of radial basis function interpolation and we will give an introduction to this field in this chapter.

At the outset, though, we want to point out one crucial difference from classical spline theory. While the support radius of the B-splines can be chosen proportional to the maximal distance between two neighboring centers, something similar will not lead to a convergent scheme in the theory of radial basis functions. The correct choice of the support radius is a very delicate question, which we will address in a later chapter on numerical methods.

General remarks

Gaussians, (inverse) multiquadrics, powers, and thin-plate splines share two joint features. They are all globally supported and are positive definite on every ℝdx. The truncated powers from Theorem 6.20, however, are compactly supported but are also restricted to a finite number of space dimensions.

We will see that the two features are connected. But let us first comment on conditionally positive definite functions.