|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 nowin 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. Nonetheless we mention here that for illustration and motivation, we
have included in this book several examples of practical applications of the
methods at various stages, especially of course in the Introduction, to demonstrate
how very useful this new method is and where it has already attracted
attention in real life applications. Apart from such instances, the personal interests
of the author mean that the text is dominated by theoretical analysis.
Nonetheless, the importance of applications and practical methods is underlined
by the aforementioned examples and by the chapter on implementations.
Since the methods are usually applied in more than two or three dimensions,
pictures will unfortunately not help us here very much which explains theirabsence.