ISNM International Series of Numerical Mathematics Vol.142 Managing Editors: K.-H. Hoffmann, München 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 Advanced Problems in Constructive Approximation 3rd International Dortmund Meeting on Approximation Theory (IDoMAT) 2001 Edited by M.D. Buhmann D.H. Mache Birkhäuser Verlag Basel· Boston· Berlin Editors: Martin O. Buhmann Detlef H. Mache Justus-Liebig University University of Oortmund Mathematical Institute Institute of Applied Mathematics Amdtstr.2 Vogelpothsweg 87 0-35392 Giessen 0-442210ortmund Germany Germany e-mail: [email protected] e-mail: [email protected] 2000 Mathematics Subject Classification 30EIO, 33F05, 41A05, 42AIO, 42A15, 42A82, 47A58, 47B65, 65005, 65007, 65015, 65020, 65030, 65T60 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <hup:/Idnb.ddb.de>. ISBN 978-3-0348-7602-5 ISBN 978-3-0348-7600-1 (eBook) DOI 10.1007/978-3-0348-7600-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2003 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Softcover reprint of the hardcover 1s t edition 2003 Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced of chlorine-free pulp. TCF 00 Cover design: Heinz Hiltbrunner, Basel ISBN 978-3-0348-7602-5 987654321 www.birkhauser.eh Table of Contents Preface ................................................................... vii List of Participants ....................................................... IX Linear Perturbations of the Classical Orthogonal Polynomials which are Eigenfunctions of Linear Differential Operators Herman Bavinck ........ . ...................................... .. .... 1 (0,1) Peil-type Interpolation: A General Method for Regularity Marcel G. de Bruin and Detlef H. Mache ............................ 21 De la Vallee Poussin Means for the Hankel Transform Wolfgang zu Castell and Frank Filbir 27 Polynomial Bases on the Sphere Noemi Lain Femandez ............................................... 39 A Shooting Method for Symbolic Computation of Splines Christoph Fredebeul ................................................. . 53 Error Estimates for the Caratheodory-Fejer Method in Polynomial Approximation M anfred Hollenhorst ................................................. 63 Shape Preserving Widths of Weighted Sobolev-Type Classes V. N. Konovalov and Dany Leviatan ................................ 79 On Approximation Methods by Using Orthogonal Polynomial Expansions Rupert Lasser, Detlef H. Mache and loset Obermaier 95 vi Table oE Contents Curious q-Series as Counterexamples in Pade Approximation Doron S. Lubinsky ................................................... 109 On the Degree of Approximation in Multivariate Weighted Approximation Hrushikesh N. Mhaskar .............................................. 129 Semigroups Associated to Mache Operators Ioan Rasa ........................................................... 143 A Survey on Lagrange Interpolation Based on Equally Spaced N odes Michael Revers...................................................... 153 Multiresolution Analysis with Pulses Carl H. Rohwer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165 H-Splines and Quasi-interpolants on a Three Directional Mesh Paul Sablonniere .................................................... 187 Approximation by Positive Definite KerneIs Robert Schaback and Holger Wendland ............................... 203 Inequalities for Polynomials With Weights Having Infinitely many Zeros on the Real Line Joseph Szabados ..................................................... 223 Some Erdös-type Convergence Processes in Weighted Interpolation Laszl6 Szili and Peter vertesi .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 237 Absolute Continuity of Spectral Measure for Certain Unbounded Jacobi Matrices Ryszard Szwarc ...................................................... 255 Approximation on Compact Subsets of R Vilmos Totik ........................................................ 263 Preface 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 Birkhäuser 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. The recent meeting in August 2001 was particular in that it was held at the time of Professor Manfred Müller's retirement. It was therefore both acelebration of approximation theory and of the many mathematical contributions Professor Müller made to approximation theory, as well as his friendship with many of the meeting's delegates, including, of course, the editors of this volume. We are grateful for the fine contributions that were delivered at the time of the conference, several of which are now included in this book. It is meant not only as a proceedings of the IDoMAT meeting, but also as a Festschrift in honour of Manfred Müller. This volume is therefore in its entirety dedicated to hirn. At this point we also thank the Deutsche Forschungsgemeinschaft (Bonn) for providing the majority of the financial support of this conference and the publisher for accepting the proceedings into its International Series of N umerical Mathematici:i. Also we would like to thank all participants for their efforts towards making this a successful meeting. viii Preface Leading experts and colleagues in approximation theory and quite a number of young researchers made the conference a stimulating event, with interesting discussions and scientific interactions to support and initiate future research. In this sense the success of the IDoMAT conferences in the years 1995, 1998 & 2001 and the positive resonance will encourage us to continue in future this series of In ternational Meetings with new developments in approximation theory and applied mathematics in Witten-Bommerholz. Witten-Bommerholz, August 2002 Martin D. Buhmann Detlef H. Mache 3rd International Meeting on Approximation Theory Witten .. Haus Bommerholz (Germany) August 20-24, 2001 List of Participants 1. Bavinck, Herman, Delft University of Technology, Faculty of Information Technology and Systems, Mekelweg 4 NL-2628 CD Delft, The Netherlands; e-mail: [email protected] 2. Beatson, Rick, University of Canterbury, Department of Mathematics and Statistics, Private Bag 4800 Christchurch, New Zealand; e-mail: [email protected] 3. Berdysheva, Elena E., Universität Erlangen-Nürnberg, Mathematisches Institut, Bismarckstr. 1 1/2 D-91054 Erlangen, Germany; e-mail: [email protected] 4. Berens, Hubert, Universität Erlangen-Nürnberg, Mathematisches Institut, Bismarckstr. 1 1/2 D-91054 Erlangen, Germany; e-mail: [email protected] 5. Blum, Heribert, Universität Dortmund, Institut für Angewandte Mathematik, Lehrstuhl X, Vogelpothsweg 87 D-44221 Dortmund, Germany: heribert. blum@math. uni-dortm und. de 6. Braess, Dietrich, Ruhr-Universität Bochum, Fakultät für Mathematik, Universitätsstr. 150 D-44780 Bochum, Germany: e-mail: [email protected] x List oE Participants 7. Bruin, Marcel G. de, Delft University of Technology, Faculty of Information Technology and Systems, Department of Applied Mathematical Analysis, Mekelweg 4 2628 CD Delft, The Netherlands; e-mail: [email protected] 8. Buhmann, Martin D., Justus-Liebig Universität Giessen, Mathematisches Institut, Arndtstr. 2 D-35392 Giessen, Germany e-mail: [email protected] 9. Butzer, Paul L., RWTH Aachen, Lehrstuhl A für Mathematik, Templergraben 55 D-52056 Aachen, Germany; e-mail: [email protected] 10. Davydov, Oleg, Universität Giessen, Mathematisches Institut, Arndtstr. 2 D-35392 Giessen, Germany e-mail: [email protected] 11. Delvos, Franz J., Universität-Gesamthochschule-Siegen, Fachbereich Mathematik, Hölderlinstr. 3 D-57068 Siegen, Germany; delvos@mathematik. uni-siegen. de 12. Feiten, Michael, FernUniversität-GHS-Hagen, LG Numerische Mathematik, Lützowstr. 125 D-58084 Hagen, Germany; e-mail: [email protected] 13. Fermindez, Noemi Lain, Medizinische Universität Lübeck, Institut für Mathematik, Wallstr. 40 D-23560 Lübeck, Germany; [email protected] 14. Filbir, Frank, Medizinische Universität Lübeck, Institut für Mathematik, Wallstr. 40 D-23560 Lübeck, Germany; [email protected] 15. Fredebeul, Christoph, Universität Dortmund, Institut für Angewandte Mathematik, Lehrstuhl VIII, Vogelpothsweg 87 D-44221 Dortmund, Germany; e-mail: [email protected]