Radial basis functions: theory and implementations. (English) Zbl 1038.41001

Buhmann, M. D.

Cambridge Monographs on Applied and Computational Mathematics 12. Cambridge: Cambridge University Press (ISBN 0-521-63338-9/hbk; 978-0-511-54324-1/ebook). x, 259 p. (2003).

This first book on approximation and interpolation with radial basis functions is a work of great merit. In many areas of mathematics, science and engineering it is necessary to approximate multivariate functions. Radial basis functions are a powerful tool in multivariate approximation.
In this interesting book, the author gives a comprehensive treatment of radial basis functions from both the theoretical and practical implementation viewpoints. Many positive features of radial basis functions are emphasized such as the unique solvability of the interpolation problem, the fast computation of interpolants, their smoothness and convergence.
This book is divided into 10 chapters. The first two chapters have introductory character, where a summary of methods and applications is given. Chapter 3 puts radial basis functions in the general context of multivariate approximation. The important special case of radial basis functions on regularly spaced grids is discussed in Chapter 4. Many of the results are best possible convergence results.
Chapter 5 is the core of this hook. Here the author generalizes the results of Chapter 4 to scattered data interpolation. The convergence of these interpolation methods is of utmost importance in applications, since radial basis function interpolation of scattered data is a frequently used method for multivariate data fitting. In Chapter 6, radial basis functions with compact support are constructed. Such radial basis functions are suitable for solving partial differential equations by Galerkin methods, since they allow meshless approximation. Chapter 7 describes several iterative methods to compute radial basis function interpolants efficiently. Chapter 8 is devoted to least squares methods. Chapter 9 presents wavelet methods with radial basis functions.
This well written book ends with a short chapter on further results and open problems. A comprehensive bibliography is given. This book is of great interest to anyone working in radial basis function approximation. Doubtless this very valuable work will stimulate the further research in this field.

Reviewer: Manfred Tasche (Rostock)

MSC:

41-02	Research exposition (monographs, survey articles) pertaining to approximations and expansions
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
65D05	Numerical interpolation
41A30	Approximation by other special function classes
41A05	Interpolation in approximation theory
41A63	Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)

Keywords:

radial basis functions; multivariate approximation; radial basis function interpolation; interpolation of scattered data; least squares method; wavelets; convergence analysis; norm estimates; implementation; numerical computation