Multivariate Polynomial Approximation

360 Pages·2003·26.3 MB·English

by  Manfred Reimer (auth.)

ISNM
International Series of Numerical Mathematics
Vol.144
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
Multivariate Polynomial
Approximation
Manfred Reimer
Springer Basel AG
Author:
Manfred Reimer
Fachbereich Mathematik
Universität Dortmund
44221 Dortmund
Germany
e-mail: [email protected]
2000 Mathematics Subject Classification 41-02; 41A10, 41A05, 41A36, 41A55, 41A63, 43A90, 42C05, 42C15, 44A12,
45Q05, 92C55, 41A05,41A36, 41A55, 41A63, 43A90, 42C05, 42C15, 44A12, 45Q05, 92C55
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 <http://dnb.ddb.de>.
ISBN 978-3-0348-9436-4 ISBN 978-3-0348-8095-4 (eBook)
DOI 10.1007/978-3-0348-8095-4
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
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© 2003 Springer Basel AG
Originally published by Birkhäuser Verlag, Basel, Switzerland in 2003
Member of the BertelsmannSpringer Publishing Group
Cover graphic: Spherical polynomial approximant of degree 160, in area preserving parametrisation (see Figure 6.3)
Printed on acid-free paper produced of chlorine-free pulp. TCF °o
Cover design: Heinz Hiltbrunner, Basel
Contents
Preface ix
I Introduction 1
1 Basic Principles and Facts 3
1.1 Preliminaries . . . . . . . . . . . . . . . . . . 3
1.2 Existence ofa Reproducing Kernel Function . 4
1.3 Rotation-Invariant Spaces ..... 5
1.4 Rotation Principles, T-Kernels .. 8
1.5 Averages and T-Kernel Projections 10
1.6 Reproducing Kernels in C(sr-l) 16
1.7 Problems . 18
2 Gegenbauer Polynomials 19
2.1 Generating Function 19
2.2 Differential Equation 23
2.3 Orthogonality .. 25
2.4 Bessel Functions . . 28
2.5 Asymptotics ..... 31
2.6 Asymptotics ofthe Gegenbauer Zeros 35
2.7 Problems . 37
II Approximation Means 39
3 Multivariate Polynomials 41
3.1 The Zoo ofMultivariate Polynomials 41
3.2 Polynomials on Subsets 64
3.3 Problems . 65
vi Contents
4 Polynomials on Sphere and Ball 67
4.1 The Rotation-Invariant Subspaces ofIPT(ST-l) 67
4.2 Biorthonormal Systems on the Sphere ..... 81
4.3 Biorthonormal Systems on the Ball . . . . . . . 98
4.4 The Image of G~+K(a'.) under T-Kernel Projections 104
4.5 Problems . 108
III Multivariate Approximation 109
5 Approximation Methods 111
5.1 Bounded Linear Operators . 111
5.2 Bernstein Polynomials and the Theorem ofWeierstrass. 119
5.3 Best Approximation and Projections . 122
5.4 Interpolatory Projections in C(D) . 130
5.5 Extremal Bases and Extremal Fundamental Systems 136
5.6 Quadrature . 149
5.7 Best Approximation in the Maximum Norm 158
5.8 Examples 164
5.9 Problems .... . . . . . 176
6 Approximation on the Sphere 179
6.1 Orthogonal Projections and Laplace Series. 179
6.2 Minimal Projection in the Uniform Norm 182
6.3 Interpolation on the Sphere . . . . . . . . . 192
6.4 Quadrature on the Sphere . . . . . . . . . . 203
6.5 Geometry of Nodes and Weights in a Positive Quadrature 217
6.6 Hyperinterpolation on the Sphere . 226
6.7 Summation ofLaplace Series . . . . . . 230
6.8 Generalized Hyperinterpolation . . . . . . . . . . . . 244
6.9 Moduli ofContinuity and the Approximation Order 247
6.10 Truncated Generalized Hyperinterpolation . 254
6.11 Problems . 262
7 Approximation on the Ball 263
7.1 Orthogonal Projections and Appell Series 263
7.2 Summation ofAppell Series . 265
7.3 Interpolation on the Ball . 269
7.4 Quadrature on Sphere and Ball are Related Topics 276
7.5 Hyperinterpolation and Generalized Hyperinterpolation 280
7.6 Evaluation ofMultivariate Orthogonal Expansions 280
7.7 Problems . 282
Contents vii
IV Applications 283
8 Tomography 285
8.1 Radon Transform . . . . . . . . . . 286
8.2 Adjoint Operator and the Inverse. 288
8.3 Reconstruction by Approximation 292
8.4 Complexity and Stability 297
8.5 k-Plane Transform 301
8.6 Problems . . . . . . . . . 303
Appendix 305
A Legendre Basis 307
B Zeros ofthe Kernel Function 311
C Newman-Shapiro Operators 313
D Reconstruction 317
E Solutions 323
Bibliography 345
Index 351
Meiner lieben Renate
Preface
Multivariatepolynomialsarea basic tool intheapproximationoffunctions. When
mymonographonConstructiveApproximationofMultivariateFunctionswaspub
lished, now more than ten years ago, I could not know that a period ofextremely
active investigation on this field was just ahead. An important new key-word was
given by Ian H. Sloan, hyperinterpolation, which is a discrete projection method,
realizing the growth order ofthe minimal uniform projection norm on the sphere
- whereas it is an open question, whether interpolatory projections do the same.
In the general case, hyperinterpolation is still missing convergence. But by sum
mation, generalized versions arise, which are convergent even at the best possible
order. Anyways, the new issue caused new interest in positive quadratures ofhigh
exactness, in particular in the geometry of their nodes, and in their weight dis
tribution. Using the new knowledge we estimate the approximation error even of
certain discrete positive polynomial operators by means of the modulus of con
tinuity. They are gained from the Newman-Shapiro operators, which attend also
our new interest.
Naturally, there exists a severe complexity problem in higher dimensional spaces.
We master it byacontrolledtruncationofthediscreteoperators, withoutdestroy
ing the approximation order.
Mostoftheresults, gained for thesphere, definealsoan approximation methodfor
the ballsoflowerdimensions. In the descriptionofthis process, the oldorthogonal
polynomials of Appell and Kampe de Feriet are very helpful. Another reason for
their necessary revival is their behaviour under certain rotations, by which they
become the welcome tool in the evaluation of some projections, defined by the
average over a rotation group - with an important application to the Radon-, or
moregenerally, to the k-plane transform. Wefinish by mentioning a highaccuracy
positive approximation method for the unknown density function. It constructs
the approximants from the Radon-images as the data, thus solving the inversion
problemoftomographybyanapproximation methodofpolynomialstabilityorder.
In view ofthis development, it will not be too surprising, that I decided to write
the monograph quite anew, saving what has to be saved, but rejecting what is
not fitting to the new concept. A large variety ofproblems is attached to help the
reader to become familiar with the multivariate theory. All problems are solved in
a separate section.
x Preface
Several parts of this book have been subject to advanced lectures which I hold
at the University ofDortmund, or have been subject to discussion or cooperation
with former members of my research group. I mention in particular Burkhard
Siindermann, Michael Rosier and Ulrike Maier. I am aware of the motivation I
owethem, and acknowlege this gratefully. Iam also indepted to H. Michael Moller
for valuable hints and remarks.
Finally I acknowledge the great motivation which I received at the Bommerholz
conferencesonmultivariateapproximation, whichIwereluckytoorganizetogether
with my friends Werner HauBmann and Kurt Jetter, the last conference also to
gether with Joachim StockIer. My hope is that these conferences will stimulate
and promote multivariate approximation also in future.
Manfred Reimer
February 2003
Part I
Introduction