This book ends with a short chapter that has two goals. The first one is to catch up with some recent results on radial basis function approximation which have not been included in the main body of the book, so as to keep the book on a sufficiently introductory level and homogeneous as to the direction of the results. Nonetheless, there are many additional, new results which deserve to be mentioned in a monograph like this about radial basis functions. We take care of some of them by way of a short summary of results with a reference to a cited paper or book that is suitable for further study.

The second one is to mention several open problems that have, as far as the author is aware, not yet been solved at the time of writing this book and which are, partly, in close connection with the recent results in this chapter. (Since writing this book was a long term project, some very recent solutions may almost certainly have been overlooked by the author.) We hope to raise interest in the reader in those problems as subjects of general further, perhaps their own, research work.

Further results

This section enlists and briefly discusses a number of further results in the analysis of radial basis functions.

One of the very recent developments under much investigation now is the idea of the so-called radial basis function multilevel methods.