Scattered data approximation is a recent, fast growing research area. It deals with the problem of reconstructing an unknown function from given scattered data. Naturally, it has many applications, such as terrain modeling, surface reconstruction, fluid-structure interaction, the numerical solution of partial differential equations, kernel learning, and parameter estimation, to name a few. Moreover, these applications come from such different fields as applied mathematics, computer science, geology, biology, engineering, and even business studies.

This book is designed to give a thorough, self-contained introduction to the field of multivariate scattered data approximation without neglecting the most recent results.

Having the above-mentioned applications in mind, it immediately follows that any competing method has to be capable of dealing with a very large number of data points in an arbitrary number of space dimensions, which might bear no regularity at all and which might even change position with time.

Hence, in my personal opinion a true scattered data method has to be meshless. This is an assumption that might be challenged but it will be the fundamental assumption throughout this book. Consequently, certain methods, that generally require a mesh, such as those using wavelets, multivariate splines, finite elements, box splines, etc. are immediately ruled out. This does not at all mean that such methods cannot sometimes be used successfully in the context of scattered data approximation; on the contrary, it just explains why these methods are not discussed in this book.