Multivariate approximation theory is today an increasingly active research area. It deals with a multitude of problems in areas such as wavelets, multi-dimensional splines, and radial-basis functions, and applies them, for example, to problems in computer aided geometric design, geometric modeling, geodesic applications and image analysis. The field is both fascinating and intellectually stimulating since much of the mathematics of the classical univariate theory does not straightforwardly generalize to the multivariate setting which models many real-world problems; so new tools have had to be, and must continue to be, developed.

This advanced introduction to multivariate approximation and related topics consists of nine chapters written by leading experts that survey many of the new ideas and tools and their applications. Each chapter introduces a particular topic, takes the reader to the forefront of research and ends with a comprehensive list of references.

This book will serve as an ideal introduction for researchers and graduate students who wish to learn about the subject and see how it may be applied.

A more detailed description of each chapter follows:

Chapter 1: Characterization and construction of radial basis functions, by R. Schaback (Göttingen) and H. Wendland (Göttingen)

This chapter introduces characterizations of (conditional) positive definiteness and shows how they apply to the theory of radial basis functions. Complete proofs of the (conditional) positive definiteness of practically all relevant basis functions are provided.