Optimal Estimation in Approximation Theory
THE IBM RESEARCH SYMPOSIA SERIES Computational Methods in Band Theory Editors: P.M. Marcus, J.F. Janak, and A.R. Williams Computational Solid State Physics Editors: F. Herman, N.W. Dalton, and T.R. Koehler Sparse Matrices and Their Applications Editors: D.j. Rose and R.A. Willoughby Complexity of Computer Computations Editors: R.E. Miller and J.W. Thatcher Associate Editor: J.D. Bohlinger Computational Methods for Large Molecules and Localized States in Solids Editors: F. Herman, A.D. Mclean, and R.K. Nesbet lon Implantation in Semiconductors and Other Materials Editor: Billy L. Crowder Stiff Differential Systems Editor: Ralph A. Willoughby Optimal Estimation in Approximation Theory Editors: Charles A. Micchelli and Theodore J. Rivlin A Continuation Order Plan is available for this series. A continualion order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contacl the publisher.
Optimal Estimation in Approximation Theory Edited by Charles A. Micchelli and Theodore j. Rivlin IBM Yorktown Heights, New York SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging in Publication Data International Symposium on Optimal Estimation in Approximation Theory, Freudenstadt, Ger., 1976. Optimal estimation in approximation theory. (The IBM research symposia series) lncludes index. 1. Approximation theory-congresses. 2. Mathematical optimization-congresses. 3. Computational complexity-congresses.l. Micchelli, Charles A. 11. Rivlin, Theodore J., 1926- III. Title. IV. Series: International Business Machines Corporation. IBM research symposia series. QA297.5.157 1976 511 '.4 774329 ISBN 978-1-4684-2390-7 ISBN 978-1-4684-2388-4 (ebook) DOI 10.1007/978-1-4684-2388-4 Proceedings of an International Symposium on Optimal Estimation in Approximation Theory held in Freudenstadt, Federal Republic of Germany, September 27-29, 1976 1977 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1977 Softcover reprint of the hardcover 1 st edition 1977 Ali rights reserved No part of this book may be reproduced, stored in a retrieval system,or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilm ing, recording, or otherwise, without written permission from the Publisher
PREFACE The papers in this volume were presented at an International Symposium on Optimal Estimation in Approximation Theory which was held in Freudenstadt, Federal Republic of Germany, September 27-29, 1976. The symposium was sponsored by the IBM World Trade Europe/Middle East/Africa Corporation, Paris, and IBM Germany. On behalf of all the participants we wish to express our appreciation to the sponsors for their generous support. In the past few years the quantification of the notion of complexity for various important computational procedures (e.g. multiplication of numbers or matrices) has been widely studied. Some such concepts are necessary ingredients in the quest for optimal, or nearly optimal, algorithms. The purpose of this symposium was to present recent results of similar character in the field or approximation theory, as well as to describe the algorithms currently being used in important areas of application of approximation theory such as: crystallography, data transmission systems, cartography, reconstruction from x-rays, planning of radiation treatment, optical perception, analysis of decay processes and inertial navigation system control. It was the hope of the organizers that this confrontation of theory and practice would be of benefit to both groups. Whatever success th ~ symposium had is due, in no small part, to the generous and wise scientific counsel of Professor Helmut Werner, to whom the organizers are most grateful. Dr. T.J. Rivlin IBM T.J. Watson Research Center Yorktown Heights, N. Y. Symposium Chairman Dr. P. Schweitzer IBM Germany Scientific and Education Programs Symposium Manager Dr. C.A. Micchelli IBM T.J. Watson Research Center Yorktown Heights, N. Y. v
CONTENTS A Survey of Optimal Recovery. C. A. Micchelli and T. J. Rivlin The Setting, Basic Bounds and Relationship to Previous Work Optimal Estimation of Linear Functionals Examples of Optimal Recovery of Linear Functionals by Linear Methods Example of Optimal Recovery of a Function Optimal Recovery by Restricted Algorithms n-widths and Optimal Interpolation of Time- and Band-Limited Functions Avraham A. Melkman n-widths Optimal Interpolation Computational Aspects of Optimal Recovery Carl de Boor The Optimal Recovery Scheme of Micchelli, Rivlin, and Winograd The Envelope Construction The Construction of Norm Preserving Extensions to all of IL1 Construction of the Knots for the Optimal Recovery Scheme Construction of the Optimal Interpolant Interpolation Operators as Optimal Recovery Schemes for Classes of Analytic Functions Michael Golomb Interpolating ~ Splines Optimal Recovery Schemes for a Ball in bv. 1 55 69 93 vii
viii CONTENTS Spaces of Analytic Functions bf(dr)-splines from Interpolation Data bf(dr)-splines Satisfying Initial Conditions. Hyperoptimal J#'(AR) -Splines bf(ar),- bfd~b)- and bf(er)-splines Comparison with n-widths Stabilization of the Recovery Schemes Optimal Degree of Approximation by Splines... Karl Scherer The Problem Inverse Theorems Minimal Projections Carlo Franchetti Estimation Problems in Crystallography.... Robert Schaback and Problem Formulation Determination of Unit Cell Dimensions Estimation of the Relative Positions of Atoms Within the Unit Cell Estimation Problems in Data-Transmission Systems G. Ungerboeck General Solution by Means of a Single Likelihood Function Estimation of Timing Phase and Carrier Phase Adaptive Equalization Signal Detection Conclusion 139 151 159 181 Optimal Approximation in Automated Cartography....... 201 Wigand Weber Determination of Information Content A Model of Cartographic Generalization Evaluation of the Generalization Model Some Available Partial Solutions of Automated Cartographic Generalization Optimal Approximation in Other Domains of Automated Cartography Reconstruction from X-Rays... K. T. Smith, S. L. Wagner, and R. B. Guenther Mathematical Generalities Computerized Axial Tomography 215
CONTENTS ix Discrimination Between Cander and Fibrocystic Disease in the Breast Noninvasive Angiography Planning of Radiation Treatment Udo Ebert Foundations of Radiotherapy Model for Spatial Distribtuion of Fields Some Aspects of the Mathematics of Limulus K. P. Hadeler Facts from Biology The Model The Equilibrium States Oscillating Solutions The Vector System Related Problems from Ecology Analysis of Decay Processes and Approximation Dietrich Braess by Exponentials Optimal State Estimation and Its Application to Index W. Hofmann Inertial Navigation System Control Modelling of Physical and Random Disturbing States The Linear, Gaussian Estimation Problem State Estimation and Platform Error SUDDnary Angle Control of a Doppler-Aided Inertial Navigation System 229 241 257 267 297