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![PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge, CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, VIC 3166, Australia
Ruiz de Alarcon 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.canibridge.org
© Cambridge University Press 2001
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2001
Printed in the United Kingdom at the University Press, Cambridge
Typeface Computer Modern 10/12pt System IATgX [UPH]
A catalogue record for this book is available from the British Library
Library of Congress Cataloguing in Publication data
ISBN 0 521 80023 4 hardback](https://image.slidesharecdn.com/39217-250703110241-a8396c43/75/Multivariate-Approximation-and-Applications-1st-Edition-N-Dyn-7-2048.jpg)
Multivariate Approximation and Applications 1st Edition N. Dyn
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- 5. Multivariate Approximation andApplications 1st Edition N. Dyn Digital Instant Download Author(s): N. Dyn, D. Leviatan, D. Levin, A. Pinkus ISBN(s): 9780521800235, 0521800234 Edition: 1st File Details: PDF, 11.09 MB Year: 2001 Language: english
- 6. Multivariate Approximation and Applications Editedby N. DYN Tel Aviv University D. LEVIATAN Tel Aviv University D. LEVIN Tel Aviv University A. PINKUS Technion-Israel Institute of Technology 1 CAMBRIDGE UNIVERSITY PRESS
- 7. PUBLISHED BY THEPRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge, CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, VIC 3166, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.canibridge.org © Cambridge University Press 2001 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2001 Printed in the United Kingdom at the University Press, Cambridge Typeface Computer Modern 10/12pt System IATgX [UPH] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data ISBN 0 521 80023 4 hardback
- 8. Contents List of contributorsVa 9e v Preface vii 1 Characterization and construction of radial basis functions R. Schaback and H. Wendland 1 2 Approximation and interpolation with radial functions M.D. Buhmann 25 3 Representing and analyzing scattered data on spheres H.N. Mhaskar, F.J. Narcowich and J.D. Ward 44 4 A survey on .^-approximation orders from shift-invariant spaces K. Jetter and G. Plonka 73 5 Introduction to shift-invariant spaces. Linear independence A. Ron 112 6 Theory and algorithms for nonuniform spline wavelets T. Lyche, K. M0rken and E. Quak 152 7 Applied and computational aspects of nonlinear wavelet approximation A. Cohen 188 8 Subdivision, multiresolution and the construction of scalable algorithms in computer graphics P. Schroder 213 9 Mathematical methods in reverse engineering J. Hoschek 252 Index 285 m
- 10. Contributors M.D. Buhmann Mathematical Institute,Justus-Liebig University, 35392 Gieflen, Germany email: Martin. Buhmann8@inath. uni-giessen. de A. Cohen Laboratoire a"Analyse Numerique, Universite Pierre et Marie Curie, Paris, Prance email: cohen@8ann.jussieu.fr J. Hoschek Department of Mathematics, Darmstadt University of Technology, 64289 Darm- stadt, Germany email: hoschek88mathematik.tu-darmstadt.de K. Jetter Institut fur Angewandte Mathematik und Statistik, Universitat Hohenheim, 70593 Stuttgart, Germany email: kjetter89uni-hohenheim.de T. Lyche Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, 0316 Oslo, Norway email: tom98ifi.uio.no H.N. Mhaskar Department of Mathematics, California State University, Los Angeles, CA 90032, USA email: hmhaska88calstatela.edu K. M0rken Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, 0316 Oslo, Norway email: knutm88ifi.uio.no F.J. Narcowich Department of Mathematics, Texas A&M University, College Station, TX 77843, USA email: fnarc88math.tamu.edu
- 11. vi Contributors G. Plonka FachbereichMathematik, Universitdt Duisburg, 47048 Duisburg, Germany email: plonka@@math. uni-duisburg. de E. Quak SINTEF Applied Mathematics, P.O. Box 124 Blindern, 0314 Oslo, Norway email: Ewald. QuakQSmath. s intef . no A. Ron Computer Sciences Department, 1210 West Dayton, University of Wisconsin- Madison, Madison, WI 57311, USA email: amos@9cs.wisc.edu R. Schaback Institut fur Numerische und Angewandte Mathematik, Universitat Gottingen, Lotzestrafie 16-18, 37083 Gottingen, Germany email: schabackQQmath. uni-goett ingen. de P. Schroder Department of Computer Science, California Institute of Technology, Pasadena, CA 91125, USA email: psS9cs.caltech.ed J.D. Ward Department of Mathematics, Texas A&M University, College Station, TX 77843, USA email: jward@9math.tamu.edu H. Wendland Institut fur Numerische und Angewandte Mathematik, Universitat Gottingen, Lotzestrafie 16-18, 37083 Gottingen, Germany email: wendland99math.uni-goettingen.de
- 12. Preface Multivariate approximation theoryis today an increasingly active research area. It deals with a multitude of problems in areas such as wavelets, multi- dimensional splines, and radial-basis functions, and applies them, for exam- ple, to problems in computer aided geometric design, geometric modeling, geodesic applications and image analysis. The field is both fascinating and intellectually stimulating since much of the mathematics of the classical univariate theory does not straightforwardly generalize to the multivariate setting which models many real-world problems; so new tools have had to be, and must continue to be, developed. This advanced introduction to multivariate approximation and related topics consists of nine chapters written by leading experts that survey many of the new ideas and tools and their applications. Each chapter introduces a particular topic, takes the reader to the forefront of research and ends with a comprehensive list of references. This book will serve as an ideal introduction for researchers and graduate students who wish to learn about the subject and see how it may be applied. A more detailed description of each chapter follows: Chapter 1: Characterization and construction of radial basis functions, by R. Schaback (Gottingen) and H. Wendland (Gottingen) This chapter introduces characterizations of (conditional) positive defi- niteness and shows how they apply to the theory of radial basis functions. Complete proofs of the (conditional) positive definiteness of practically all relevant basis functions are provided. Furthermore, it is shown how some of these characterizations may lead to construction tools for positive definite functions. Finally, a new construction technique is given which is based on discrete methods which leads to non-radial, even non-translation invariant, local basis functions.
- 13. viii Preface Chapter 2:Approximation and interpolation with radial Junctions, by M.D. Buhmann (Giessen) This chapter provides a short, up-to-date survey of some of the recent developments in the research of radial basis functions. Among these new developments are results on convergence rates of interpolation with radial basis functions, recent contributions concerning approximation on spheres, and computations of interpolants with Krylov space methods. Chapter 3: Representing and analyzing scattered data on spheres, by H.N. Mhaskar (California State Univ. at Los Angeles), F.J. Narcowich (Texas A & M) and J.D. Ward (Texas A k M) Geophysical or meteorological data collected over the surface of the earth via satellites or ground stations will invariably come from scattered sites. There are two extremes in the problems one faces when handling such data. The first is representing sparse data by fitting a surface to it. The second is analyzing dense data to extract features of interest. In this chapter various aspects of fitting surfaces to scattered data are reviewed. Analyzing data is a more recent problem that is currently being addressed via various spherical wavelet schemes, which are discussed along with multilevel schemes. Finally quadrature methods, which arise in many of the wavelet schemes as well as some interpolation methods, are touched upon. Chapter 4: A survey on L2-approximation orders from shift-invariant spaces, by K. Jetter (Hohenheim) and G. Plonka (Duisburg) The aim of this chapter is to provide a self-contained introduction to notions and results connected with the /^-approximation order of finitely generated shift-invariant spaces. Special attention is given to the principal shift-invariant case, where the shift-invariant space is generated from the multi-integer translates of a single generator. This case is of special interest because of its possible applications in wavelet methods. The general finitely generated shift-invariant space case is considered subject to a stability con- dition being satisfied, and the recent results on so-called superfunctions are developed. For the case of a refinable system of generators, the sum rules for the matrix mask and the zero condition for the mask symbol, as well as invariance properties of the associated subdivision and transfer operators, are discussed. Chapter 5: Introduction to shift-invariant spaces. Linear independence, by A. Ron (Madison) Shift-invariant spaces play an increasingly important role in various areas of mathematical analysis and its applications. They appear either implicitly or explicitly in studies of wavelets, splines, radial basis function approxima- tion, regular sampling, Gabor systems, uniform subdivision schemes, and
- 14. Preface ix perhaps insome other areas. One must keep in mind, however, that the shift-invariant system explored in one of the above-mentioned areas might be very different from those investigated in others. The theory of shift- invariant spaces attempts to provide a uniform platform for all these dif- ferent investigations. The two main pillars of that theory are the study of the approximation properties of shift-invariant spaces, and the study of gen- erating sets for such spaces. Chapter 4 had already provided an excellent up-to-date account of the first topic. The present chapter is devoted to the second topic, and its goal is to provide the reader with an easy and friendly introduction to the basic principles of that topic. The core of the presenta- tion is devoted to the study of local principal shift-invariant spaces, while the more general cases are treated as extensions of that basic setup. Chapter 6: Theory and algorithms for nonuniform spline wavelets, by T. Lyche (Oslo), K. M0rken (Oslo), E. Quak (Oslo) This chapter discusses mutually orthogonal spline wavelet spaces on non- uniform partitions of a bounded interval, addressing the existence, unique- ness and construction of bases of minimally supported spline wavelets. The relevant algorithms for decomposition and reconstruction are considered as well as some questions related to stability. In addition, a brief review is given of the bivariate case for tensor products and arbitrary triangulations. The chapter concludes with a discussion of some special cases. Chapter 7: Applied and computational aspects of nonlinear wavelet app- roximation, by A. Cohen (Paris) Nonlinear approximation is recently being applied to computational ap- plications such as data compression, statistical estimation and adaptive schemes for partial differential and integral equations, especially through the development of wavelet-based methods. The goal of this chapter is to provide a short survey of nonlinear wavelet approximation from the per- spective of these applications, as well as to highlight some remaining open questions. Chapter 8: Subdivision, multiresolution and the construction of scalable algorithms in computer graphics, by P. Schroder (Caltech) Multiresolution representations are a critical tool in addressing complexity issues (time and memory) for the large scenes typically found in computer graphics applications. Many of these techniques are based on classical sub- division techniques and their generalizations. In this chapter we review two exemplary applications from this area: multiresolution surface editing and semi-regular remeshing. The former is directed towards building algorithms which are fast enough for interactive manipulation of complex surfaces of arbitrary topology. The latter is concerned with constructing smooth pa-
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