We have seen in the introduction what a radial basis function is and what the general purposes of multivariate interpolation are, including several examples. The aim of this chapter is more specifically oriented to the mathematical analysis of radial basis functions and their properties in examples.

That is, in this chapter, we will demonstrate in what way radial basis function interpolation works and give several detailed examples of its mathematical, i.e. approximation, properties. In large parts of this chapter, we will concentrate on one particular example of a radial basis function, namely the multiquadric function, but discuss this example in much detail. In fact, many of the very typical properties of radial basis functions are already contained in this example which is indeed a nontrivial one, and therefore quite representative. We deliberately accept the risk of being somewhat repetitive here because several of the multivariate general techniques especially of Chapter 4 are similar, albeit more involved, to the ones used now. What is perhaps most important to us in this chapter, among all current radial basis functions, the multiquadric is the best-known one and best understood, and very often used. One reason for this is its versatility due to an adjustable parameter c which may sometimes be used to improve accuracy or stability of approximations with multiquadric functions.