Manfred Reimer - Böcker

Multivariate polynomials are a main tool in approximation. The book begins with an introduction to the general theory by presenting the most important facts on multivariate interpolation, quadrature, orthogonal projections and their summation, all treated under a constructive view, and embedded in the theory of positive linear operators. On this background, the book gives the first comprehensive introduction to the recently developped theory of generalized hyperinterpolation. As an application, the book gives a quick introduction to tomography. Several parts of the book are based on rotation principles, which are presented in the beginning of the book, together with all other basic facts needed.

These proceedings contain the main topics and results presented at the Fourth International Conference on Multivariate Approximation. The meeting took place during the week of September 24-29, 2000 at the now well-known "Haus Bommerholz", the guest-house of the University of Dortmund. It hosted 43 participants from 16 countries, and the program included 9 invited one-hour lectures, 21 contributed talks, and two problem sessions. The articles collected here are carefully peer-refereed and suitably edited for publication. Following the tradition of this series of conferences, the meeting was aimed at advancing selected topics of Multivariate Approximation Theory. These include approximation on compact sets (such as spheres, balls, or compact ho­ mogeneous manifolds), spherical designs and energy functionals, interpolation by radial basis functions and by splines, frame theory and Gabor analysis, re­ finable function systems and subdivision, properties of harmonic, polyharmonic and blending functions, sampling and data compression, among others. The editors would like to express their thanks to all who have given their sup­ port to the conference, and to the preparation of this book. In particular, our thanks go to the Deutsche Forschungsgemeinschaft for their funding of the con­ ference, to the University of Dortmund, and to the staff of Haus Bommerholz for their help running the conference, and to the Universities of Dortmund and Duisburg for the financial support of this proceedings volume.

Multivariate polynomials are a main tool in approximation. The book begins with an introduction to the general theory by presenting the most important facts on multivariate interpolation, quadrature, orthogonal projections and their summation, all treated under a constructive view, and embedded in the theory of positive linear operators. On this background, the book gives the first comprehensive introduction to the recently developped theory of generalized hyperinterpolation. As an application, the book gives a quick introduction to tomography. Several parts of the book are based on rotation principles, which are presented in the beginning of the book, together with all other basic facts needed.

This volume contains a selection of eighteen peer-reviewed articles that were presented at the 5th International Conference on Multivariate Approximation, held in Witten-Bommerholz in September 2002. The contributions cover recent developments of constructive approximation on manifolds, approximation by splines and kernels, subdivision techniques and wavelet methods. The main topics are:- applications of multivariate approximation in finance- approximation and stable reconstruction of images, data reduction- multivariate splines for Lagrange interpolation and quasi-interpolation- radial basis functions- spherical point sets- refinable function vectors and non-stationary subdivision- applications of adaptive wavelet methods- blending functions and cubature formulae- singularities of harmonic functionsThe book provides an overview of state-of-the-art developments in a highly relevant field of applied mathematics, with many links to computer science and geophysics.