The basic problem in multivariate approximation theory can be summarized thus: given a subset D of &RR;n, and a function f : D → &RR;, find a function g : &RR;n → &RR; satisfying the condition that g(d) = f(d) for all d ∈ D, where g belongs to some prescribed subspace W of the space of all functions from &RR;n to &RR;. For example, we may insist that W be (finitely) generated by polynomial functions, spline functions, and so on. In the case under consideration, we are interested in subspaces W generated by translates of a radially symmetric function, namely a function from &RR;n to &RR; of the form x → &phgr;(||x||), where ||x|| denotes the Euclidean norm of x, and &phgr; is a continuous function from &RR;+ to &RR; taken along the vectors in D. That is to say, W is generated by the functions of the form x → &phgr;(||x - d||), where d ∈ D. Some examples of functions &phgr; encountered in such contexts are &phgr; : r → r2 log(r), &phgr; : r → √(r2 +c2), and &phgr; : r → exp(-ar2).
In real-world applications, radial basis function techniques have turned out to be extremely useful, and such methods have become an important computational tool. This book provides a comprehensive introduction to the theory behind such methods, places them in the context of other methods in multivariate interpolation and approximation, analyzes their mathematical properties, and studies several applications and special cases. It also summarizes the most recent research in the area, and the directions of current and future research. The bibliography is extensive and comprehensive.
In all, this is definitely a must read for anyone making direct use of this tool, and a must browse for anyone interested in keeping up with the state of the art in multivariate approximation theory in general. Its only drawback is that portions of the text are unfortunately made unnecessarily obscure by the author’s unsure command of English.
