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.

Introduction to wavelets and prewavelets

Already in the previous chapter we have discussed in what cases L2-approximants or other smoothing methods such as quasi-interpolation or smoothing splines with radial basis functions are needed and suitable for approximation in practice, in particular when data or functions f underlying the data are at the beginning not very smooth or must be smoothed further during the computation. The so-called wavelet analysis that we will introduce now is a further development in the general context of L2-methods, and indeed everything we say here will concern L2-functions, convergence in the L2-norm etc. only. Many important books have been written on wavelets before, and since this is not at all a book on wavelets, we will be fairly short here. The reader who is interested in the specific theory of wavelets is directed to one of the excellent works on wavelets mentioned in the bibliography, for instance the books by Chui, Daubechies, Meyer and others. Here, our modest goal is to describe what wavelets may be considered as in the context of radial basis functions. The radial basis functions turn out to be useful additions to the theory of wavelets because of the versatility of the available radial basis functions.

Given a square-integrable function f on ℝ, say, the aim of wavelet analysis is to decompose it simultaneously into its time and its frequency components.

In this chapter we summarise very briefly some general methods other than radial basis functions for the approximation and especially interpolation of multivariate data. The goal of this summary is to put the radial basis function approach into the context of other methods for approximation and interpolation, whereby the advantages and some potential disadvantages are revealed. It is particularly important to compare them with spline methods because in one dimension, for example, the radial basis function approach with integral powers (i.e. φ(r) = r or φ (r) = r3 for instance) simplifies to nothing else than a polynomial spline method. This is why we will concentrate on polynomial and polynomial spline methods. They are the most important ones and related to radial basis functions, and we will only touch upon a few others which are non(-piecewise-)polynomial. For instance, we shall almost completely exclude the so-called local methods although they are quite popular. They are local in the sense that there is not one continuous function s defined over the whole domain, where the data are situated, through the method for approximating all data. Instead, there is, for every x in the domain, an approximation s(x) sought which depends just on a few, nearby data. Thus, as x varies, this s(x) may not even be continuous in x (it is in some constructions). Typical cases are ‘natural neighbour’ methods or methods that are not interpolating but compute local least-squares approximations.

One of the most important themes of this book is the implementation of radial basis function (interpolation) methods. Therefore, after four chapters on the theory of radial basis functions which we have investigated so far, we now turn to some more practical aspects. Concretely, in this chapter, we will focus on the numerical solution of the interpolation problems we considered here, i.e. the computation of the interpolation coefficients. In practice, interpolation methods such as radial basis functions are often required for approximations with very large numbers of data sites ξ, and this is where the numerical solution of the resulting linear systems becomes nontrivial in the face of rounding and other errors. Moreover, storage can also become a significant problem if |Ξ| is very large, even with the most modern workstations which often have gigabytes of main memory.

Several researchers have reported that the method provides high quality solutions to the scattered data interpolation problem. The adoption of the method in wider applications, e.g. in engineering and finance, where the number of data points is large, was hindered by the high computational cost, however, that is associated with the numerical solution of the interpolation equations and the evaluation of the resulting approximant.

In this chapter we shall summarise and explain a few results about the orders of convergence of least squares methods. These approximants are computed by minimising the sum of squares of the error on the Euclidean space over all choices of elements from a radial basis function space. The main differences in the various approaches presented here lie in the way in which ‘sum of squares of the error’ is precisely defined, i.e. whether the error is computed continuously over an interval – or the whole space – by an integral, or whether sums over measurements over discrete point sets are taken. In the event, it will be seen that, unsurprisingly, the same approximation orders are obtained as with interpolation, but an additional use of the results below is that orthogonal bases of radial basis function spaces are studied which are useful for implementations and are also in very close connection to work of the next chapter about wavelets.

Introduction to least squares

Interpolation was the method of choice so far in this book for approximation. This, however, is by no means the only approximation technique which is known and used in applications. Especially least squares techniques are highly important in practical usage. There is a variety of reasons for this fact. For one, data smoothing rather than interpolating is very frequently needed.

The radial basis function method for multivariate approximation is one of the most often applied approaches in modern approximation theory when the task is to approximate scattered data in several dimensions. Its development has lasted for about 25 years now and has accelerated fast during the last 10 years or so. It is now in order to step back and summarise the basic results comprehensively, so as to make them accessible to general audiences of mathematicians, engineers and scientists alike.

This is the main purpose of this book which aims to have included all necessary material to give a complete introduction into the theory and applications of radial basis functions and also has several of the more recent results included. Therefore it should also be suitable as a reference book to more experienced approximation theorists, although no specialised knowledge of the field is required. A basic mathematical education, preferably with a slight slant towards analysis in multiple dimensions, and an interest in multivariate approximation methods will be suitable for reading and hopefully enjoying this book.

Any monograph of this type should be self-contained and motivated and need not much further advance explanations, and this one is no exception to this rule.