Multivariate Approximation
Recent Trends and Results
1997
Akademie-Verlag

edited by

Werner Haußmann,
Universität Duisburg
Kurt Jetter,
Universität Hohenheim
Manfred Reimer,
Universität Dortmund

Contents of Web Page:

• Preface
• Table of Contents
• Participants
• Scientific Programm of the conference
• Contacting the Akademie Verlag
• Contacting the Editors

Preface

This volume consists of the Proceedings of the 2nd International Conference on Multivariate Approximation Theory at Bommerholz, Germany. The conference took place during the week of September 29 through October 4, 1996, and was hosted by ''Haus Bommerholz'', the guest--house of the University of Dortmund, located in the beautiful countryside south of Dortmund. It was attended by 48 participants from 14 countries. The program included 10 invited key lectures and 23 contributed talks. Most of these lectures were written up in a form suitable for publication, and appear here in print after careful peer--refereeing. The scientific scope of this conference was directed towards the following topics which combine classical aspects of Approximation Theory with new and modern methods in Applied Mathematics: Multiscale Techniques, Approximation by Solutions of Differential Equations, and Geometric Design. We are happy that we were able to gather leading experts of these areas of Mathematics, as well as quite a number of promising young researchers. The idea of restricting talks to these subjects, with possible applications in science and engineering, proved a valuable stimulus to interesting discussions. Last but not least, the hospitality and the warm atmosphere of the house itself provided an excellent environment for lively interactions and scientific progress. The editors are grateful to all who have contributed to the success of the conference and the production of this Proceedings volume. In particular, we are deeply indebted to the Deutsche Forschungsgemeinschaft whose funding made the conference possible. Our thanks also go to the authors and referees for their efforts to guarantee publications of high scientific standard, to the Akademie--Verlag for their kind cooperation and for publishing these Proceedings in the Mathematical Research series, and to the sponsors of this volume which are listed separately. Finally, we would especially like to thank the conference secretary, Ms. Naujokat, and the staff of Haus Bommerholz, for all their help.
May 1, 1997

Werner Haußmann
Kurt Jetter
Manfred Reimer

Table of Contents

Weights and Angles of Quadrature Rules on the Sphere
S. Blunck 19

Local Box Dimension of Fractal Functions
Z. Ciesielski 27

Trigonometric Approximation in Multivariate Periodic Hilbert Spaces
F.-J. Delvos 35

Harmonic Approximation and Boundary Behaviour
S. J. Gardiner 45

Properties of Bivariate Refinable Spline Pairs
T. N. T. Goodman 63

Polar Cone Splines
R. Gormaz 83

Discrepancies of Point Sequences on the Sphere and Numerical Integration
P. J. Grabner, B. Klinger and R. F. Tichy 95

Regularity of Multivariate Hermite Interpolation
H. Hakopian 113

Extremizers for the Multivariate Landau-Kolmogorov Inequality
O. Kounchev 123

Nested Sequences of Triangular Finite Element Spaces
A. Le Méhauté 133

Charge Distribution of Points on the Sphere and Corresponding Cubature Formulae
U. Maier and J. Fliege 147

Solving of Algebraic Equations -- An Interplay of Symbolical and Numerical Methods
H. M. Möller 161

An Orthonormal Bivariate Algebraic Polynomial Basis for of Low Degree
J. Prestin and F. Sprengel 177

Tensor Product Splines on Refined Grids in S-Convex Interpolation
J. W. Schmidt and M. Walther 189

Quasi-Balayage and A Priori Estimates for the Laplace Operator I
H. S. Shapiro 203

Quasi-Balayage and A Priori Estimates for the Laplace Operator II
H. S. Shapiro 231

Interpolation and Hyperinterpolation on the Sphere
I. H. Sloan 255

Interpolation and Wavelets on Sparse Gauß-Chebyshev Grids
F. Sprengel 269

The General Structure of Multivariate Affine Frames
J. Stöckler 287

Spectral Sythesis with Wavelet Methods
G. Zimmermann 303

Participants