Introduction

This chapter deals with radial basis functions that are compactly supported, quite in contrast with everything else that we have encountered before. In fact the constructions, concepts and results developed in this chapter are closely related to the piecewise polynomial B- and box-splines of Chapter 3 and the finite elements of the well-known numerical methods for the numerical solution of (elliptic) differential equations. The radial basis functions we shall study now are particularly interesting for those applications. Compactly supported radial basis functions are particularly appealing amongst practitioners.

They can be used to provide a useful, mesh-free and computationally efficient alternative to the commonly used finite element methods for the numerical solution of partial differential equations.

All of the radial basis functions that we have considered so far have global support, and in fact many of them do not even have isolated zeros, such as the multiquadric function for positive c. Moreover, they are usually increasing with growing argument, so that square-integrability and especially absolute integrability are immediately ruled out. In most cases, this poses no severe restrictions since, according to the theory of Chapter 5, we can always interpolate with these functions. We do, however, run into problems when we address the numerical treatment of the linear systems that stem from the interpolation conditions, as we have seen in the discussion of condition numbers in the previous two chapters and as we shall see further on in Chapter 7.