This book contains the refereed papers which were presented at the second In ternational Dortmund Meeting on Approximation Theory (IDoMAT 98) at Haus Bommerholz, the conference center of Dortmund University, during the week of February 23-27,1998. At this conference 52 researchers and specialists from Bul garia, China, France, Great Britain, Hungary, Israel, Italy, Roumania, South Africa and Germany participated and described new developments in the fields of uni variate and multivariate approximation theory. The papers cover topics such as radial basis functions, bivariate spline interpolation, multilevel interpolation, mul tivariate triangular Bernstein bases, Pade approximation, comonotone polynomial approximation, weighted and unweighted polynomial approximation, adaptive ap proximation, approximation operators of binomial type, quasi interpolants, gen eralized convexity and Peano kernel techniques. This research has applications in areas such as computer aided geometric design, as applied in engineering and medical technology (e. g. computerised tomography). Again this international conference was wholly organized by the Dortmund Lehrstuhl VIII for Approximation Theory. The organizers attached great impor tance to inviting not only well-known researchers but also young talented math ematicians. IDoMAT 98 gave an excellent opportunity for talks and discussions between researchers from different fields of Approximation Theory. In this way the conference was characterized by a warm and cordial atmosphere. The success of IDoMAT 98 was above all due to everyone of the participants."

The current form of modern approximation theory is shaped by many new de velopments which are the subject of this series of conferences. The International Meetings on Approximation Theory attempt to keep track in particular of fun damental advances in the theory of function approximation, for example by (or thogonal) polynomials, (weighted) interpolation, multivariate quasi-interpolation, splines, radial basis functions and several others. This includes both approxima tion order and error estimates, as well as constructions of function systems for approximation of functions on Euclidean spaces and spheres. It is a piece of very good fortune that at all of the IDoMAT meetings, col leagues and friends from all over Europe, and indeed some count ries outside Europe and as far away as China, New Zealand, South Africa and U.S.A. came and dis cussed mathematics at IDoMAT conference facility in Witten-Bommerholz. The conference was, as always, held in a friendly and congenial atmosphere. After each meeting, the delegat es were invited to contribute to the proceed ing's volume, the previous one being published in the same Birkhauser series as this one. The editors were pleased about the quality of the contributions which could be solicited for the book. They are refereed and we should mention our gratitude to the referees and their work."

In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a powerful tool which work well in very general circumstances and so are becoming of widespread use as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.

It is necessary to estimate parameters by approximation and interpolation in many areas-from computer graphics to inverse methods to signal processing. Radial basis functions are modern, powerful tools which are being used more widely as the limitations of other methods become apparent. Martin Buhmann provides a complete analysis of radial basic functions from the theoretical and practical implementation viewpoints. He also includes a comprehensive bibliography.