Modern Developments in Multivariate Approximation: 5th International Conference, Witten-Bommerholz (Germany), September 2002

310 Pages·2003·24.498 MB·English

by  Werner Haussmann, Kurt Jetter, Manfred Reimer, Joachim Stöckler

ISNM
International Series of Numerical Mathematics
Vol. 145
Managing Editors:
K.-H. Hoffmann, Bonn
D. Mittelmann, Tempe
Associate Editors:
R. E. Bank, La Jolla
H. Kawarada, Chiba
R. J. LeVeque, Seattle
C. Verdi, Milano
Honorary Editor:
J.Todd,Pasadena
Modern Developments in
Multivariate Approximation
5th International Conference, Witten-Bommerholz (Germany),
September 2002
Werner Haussmann
Kurt Jetter
Manfred Reimer
Joachim Stockier
Editors
Springer Basel AG
Editors:
Werner Haussmann Kurt Jetter
Mathematisches Institut Institut fiir Angewandte Mathematik und Statistik
Gerhard-Mercator- Universitiit Duisburg Universitat Hohenheim
47048 Duisburg 70593 Stuttgart
Germany Germany
haussmann @math.uni-duisburg.de [email protected]
Manfred Reimer Joachim Stockier
Fachbereich Mathematik Fachbereich Mathematik
Universitiit Dortmund Universitiit Dortmund
44221 Dortmund 44221 Dortmund
Germany Germany
[email protected] [email protected]
2000 Mathematics Subject Classification 41-06; 41A63, 49K20, 65D30
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
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© 2003 Springer Basel AG
Originally published by Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland in 2003
Softcover reprint of the hardcover 1st edition 2003
Member of the BertelsmannSpringer Publishing Group
Printed on acid-free paper produced of chlorine-free pulp. TCF oo
ISBN 978-3-0348-9427-2 ISBN 978-3-0348-8067-1 (eBook)
DOI 10.1007/978-3-0348-8067-1
987654321
Preface
These Proceedings contain the main topics and talks presented at the Fifth
International Conference on Multivariate Approximation. The meeting took
place during the week of September 22-27, 2002 at the well-known "Haus
Bommerholz", the guest-house of the University of Dortmund. It hosted 49
participants from 10 countries, and the program included 11 invited one-hour
lectures, 21 contributed talks, and a problem session. The articles collected
here are carefully peer-refereed and suitably edited for publication, and they
reveal the state of the art in the field.
Following the tradition of this series of conferences, the meeting was aimed
at advancing selected topics of Multivariate Approximation Theory. These
include refinable function vectors, vector subdivision, shift-invariant systems,
elliptic control problems, discrepancy of point sets, approximation and differ
ential equations, radial basic function approximation, N -term approximation
in Lp, and one-sided £ -approximation, among others.
1
One of our distinguished main lecturers was Prof. Will A. Light, University of
Leicester. His lecture on "Error Estimates for Radial Basic Function Approxi
mation" was a highlight of the conference. Nobody in the audience could know
that this was to be one of his last conference lectures. Less than three months
later we received the almost unbelievable message of his sudden death. We are
mourning a mathematician of high scientific merit, an inspiring lecturer, a fine
colleague and friend.
The editors would like to express their thanks to all who have given their
support to the conference, and to the preparation of this book. In particular,
our thanks go to the Deutsche Forschungsgemeinschaft for their funding of the
conference, to the University of Dortmund and the staff of Haus Bommerholz
for their help in running the conference, and to the University of Dortmund
for the financial support of this proceedings volume. We are grateful to all
the authors and referees who helped us maintain a high scientific standard. In
addition, we acknowledge the constant help of Dr. Hermann Hoch (Duisburg)
during the production of this proceedings volume. Last but not least, we would
like to thank the publisher for including this book in the International Series
of Numerical Mathematics.
June 2003 Werner HauBmann Kurt Jetter
Manfred Reimer Joachim Stockier
6
Acknowledgment
We are deeply indebted to the
Deutsche Forschungsgemeinschaft (DFG)
whose generous funding made this conference possible.
The production of this volume was sponsored by the
Universitat Dortmund
The Editors
Table of Contents
Participants .............................................................. 9
Scientific Program of the Conference ..................................... 17
Multivariate Approximation Methods for the Pricing of
Catastrophe-Linked Bonds
H. Albrecher, J. Hartinger, and R. F. Tichy ............................ 21
Note on a Generalized Invariance Principle and its Relevance
for Cap Discrepancy and Energy
J. S. Brauchart ........................................................ 41
On Properties of Nonstationary Divided Difference Vector
Subdivision Schemes
M. Charina and C. Conti .............................................. 57
Multivariate Balanced Vector-Valued Refinable Functions
C. K. Chui and Q. Jiang .............................................. 71
Transfinite Interpolation by Blending Functions, Best One-Sided
£1- Approximation, and Cubature Formulae
D. P. Dryanov ....................................................... 103
Refining Oscillatory Signals by Non-Stationary Subdivision Schemes
N. Dyn, D. Levin, and A. Luzzatto ................................... 125
Asymptotic Error Expansions for Schoenberg Type Operators
K. Dziedziul and K. Jetter ........................................... 143
On the Approximation Order and Numerical Stability of Local
Lagrange Interpolation by Polyharmonic Splines
A. Iske ............................................................... 153
Isolated Singularities of Harmonic Functions
L. Karp and H. S. Shapiro ............................................ 167
8 Table of Contents
Two Applications of Adaptive Wavelet Methods
A. Kunath ........................................................... 175
Almost Interpolation and Radial Basis Functions
A. Le Mebaute ....................................................... 203
Fundamental Splines on Triangulations
G. Nurnberger and F. Zeilfelder ...................................... 215
Integer DCT- II by Lifting Steps
G. Plonka and M. Tascbe ............................................. 235
Approximation of Density Functions and Reconstruction
of the Approximant
M. Reimer ........................................................... 253
On Some Multivariate Quadratic Spline Quasi-Interpolants
on Bounded Domains
P. Sablonniere ........................................................ 263
How to Generate Smoother Refinable Functions from Given Ones
T. Sauer ............................................................. 279
Reconstructing Multivariate Functions from Large Data Sets
H. Wendland ......................................................... 295
Three Scale Versus Matrix Refinement Equations
G. Zimmermann ..................................................... 309
Participants
Elena E. Berdysheva, Institut fiir Angewandte Mathematik und Statistik,
Universitiit Hohenheim, D-70593 Stuttgart, Germany
email: [email protected]
Laura Beutel, Institut fiir Angewandte Mathematik,
Universitiit Dortmund, D-44221 Dortmund, Germany
email: laura. beutel@math. uni-dortmund.de
Kai Bittner, Abteilung Numerik, Fakultiit fiir Mathematik und Wirtschafts
wissenschaften, Universitiit Ulm, D-89069 Ulm, Germany
email: bittner@mathematik. uni-ulm.de
Heribert Blum, Institut fiir Angewandte Mathematik,
Universitiit Dortmund, D-44221 Dortmund, Germany
email: heribert. blum@mathematik. uni-dortmund.de
Dietrich Braess, Institut fiir Mathematik,
Ruhr-Universitiit Bochum, D-44780 Bochum, Germany
email: [email protected]
Johann Brauchart, Institut fiir Mathematik A,
Technische Universitiit Graz, Steyrergasse 30, A-8010 Graz, Austria
email: [email protected]. tu-graz.ac. at
Maria Charina, Institut fiir Angewandte Mathematik,
Universitiit Dortmund, D-44221 Dortmund, Germany
email: maria.charina@math. uni-dortmund.de