For our investigations on radial basis functions in the following chapters it is crucial to collect several results from different branches of mathematics. Hence, this chapter is a repository of such results. In particular, we are concerned with special functions such as Bessel functions and the Γ-function. We discuss the features of Fourier transforms and give an introduction to the aspects of measure theory that are relevant for our purposes. The reader who is not interested in these technical matters could skip this chapter and come back to it whenever necessary. Nonetheless, it is strongly recommended that one should at least have a look at the definition of Fourier transforms to become familiar with the notation we use. Because of the diversity of results presented here, we cannot give proofs in every case.

Bessel functions

Bessel functions will play an important role in what follows. Most of what we discuss here can be found in the fundamental book [187] by Watson.

The starting point for introducing Bessel functions is to remind the reader of the classical Γ-function and some of its features.