Approximation Theory, Spline Functions and Applications

481 Pages·1992·22.07 MB·English

by  E. W. Cheney (auth.)

Approximation Theory, Spline Functions
and Applications
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Series C: Mathematical and Physical Sciences - Vol. 356
Approximation Theory,
Sp ine Functions and Applications
edited by
S. P. Singh
Memorial University of Newfoundland,
St. John's, NF, Canada
with the assistance of
Antonio Carbone
Universitä della Calabria,
Arcavacata di Rende (CS), Italy
R. Charron and B. Watson
Memorial University of Newfoundland,
St. John's, NF, Canada
^^^^^^^^
Springer Science+Business Media, B.V.
Proceedings of the NATO Advanced Study Institute on
Approximation Theory, Spline Functions and Applications
Maratea, Italy
April 28-May9, 1991
Library of Congress Cataloging-in-Publication Data
Approximation theory, spline functions, and applications / edited by
S.P. Singh with the assistance of Antonio Carbone ... let al.l.
p. cm. — (NATO ASI series. Series C, Mathematical and
physical sciences : 356)
"Published 1n cooperation with NATO Scientific Affairs Division."
"Proceedings of the NATO Advanced Study Institute on Approximation
Theory, Spline Functions, and Applications held In the Hotel villa
del Mare, Maratea, Italy between April 28, 1991 and May 9, 1991"-
-Pref.
Includes bibliographical references and Index.
IISSBBNN 997788--9944--001100-5-511664-444- ISB ISBNN 997788--9944--001111-2-263643-422- ((eeBBooookk))
DDOOII 1100..11000077//997788-9-944-0-0111-12-623643-242-
1. Approximation theory—Congresses. 2. Spline theory-
-Congresses. I. Singh, S. P. (Sankatha Prasad), 1937-
II. North Atlantic Treaty Organization. Scientific Affairs
Division. III. Series: NATO ASI series. Series C, Mathematical and
physical sciences ; no. 356.
QA221.A68 1992
5ir.42~dc20 91-43975
ISBN 978-94-010-5164-4
Printed on acid-free paper
All Rights Reserved
©1992 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1992
Softcover reprint of the hardcover 1st edition 1992
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical, including photo
copying, recording or by any information storage and retrieval system, without written
permission from the copyright owner.
This volume is dedicated to the memory of
Kanhaya Lal Singh
(February lS, 1944 - November 22, 1990)
Table of Contents
Preface ix
List of Contributors xi
List of Participants xiii
Approximation by Functions of Nonclassical Form, E. W. Cheney 1
Wavelets- with Emphasis on Spline-Wavelets and Applications to Signal Analysis,
Charles K. Chui 19
Pade Approximation in one and more Variables, Annie Cuyt 41
Rational Hermite Interpolation in one and more variables, Annie Cuyt 69
The Method of Alternating Orthogonal Projections, Fronk Deutsch 105
Selections for Metric Projections, Frank Deutsch 123
Weighted Polynomials, M. v. Golitschek 139
Some Aspects of Radial Basis Function Approximation, W. A. Light 163
A Tutorial on Multivariate Vlavelet Decomposition,
Charles A. Micchelli -191
Using the Refinement Equation for the Construction of Pre-Wavelets VI:
Shift Invariant Subspaces, Char'les A. Micchelli 213
Error Estimates for Near-Minimax Approximations, G. M. Phillips 223
Different Metrics and Location Problems,
E. Casini and P. L. Papini 243
On the Effectiveness of Some Inversion Methods for Noisy Fourier Series,
L. De Michele, M. Di Natale, and D. Roux 255
A Generalization of N-Widths, A. G. Aksoy 269
The Equivalence of the Usual and Quotient Topologies for COO(E) when
E C an is Whitney p-Regular, L. P. Bos and P. D. Milman 279
Korovkin Theorems for Vector-Valued Continuous Functions,
Michele Campiti 293
On Modified Bojanic-Shisha Operators,
A. S. Cavaretta and S. S. Guo 303
viii
A property of zeros and Cotes numbers of Hermite and Laguerre
orthogonal polynomials, F. Costabile 311
Hermite-Fejer and Hermite Interpolation,
G. Criscuolo, B. Della Vecchia, and G. Mastroianni 317
New Results on Lagrange Interpolation,
G. Criscuolo and G. Mastroianni 333
Ambiguous Loci in Best Approximation Theory, F. S. De Blasi and J. Myjak 341
A Theorem on best approximations in topological vector spaces,
E. De Pascale and G. Trombetta 351
On the Characterization of Totally Positive Matrices, M. Gasca and J. M. Pena 357
Iterative Methods for the General Order Complementarity Problem, G. [sac 365
Wavelets, Splines, and Divergence-Free Vector Functions,
Pierre-Gilles Lemarie-Rieusset 381
An Approach to Meromorphic Approximation in a Stein Manifold,
Clement H. Lutterodt 391
Approximating Fixed Points for Nonexpansive Maps in Hilbert Spaces,
G. Marino 405
On Approximation and Interpolation of Convex Functions, Marian Neamtu 411
Convergence of Approximating Fixed Point Sets for Multivalued Nonexpansive
Mappings, Paola maria Pietramala 419
A Subdivision Algorithm for Non-Uniform B-Splines,
Ruibin Qu and John A. Gregory 423
Some Applications of an Approximation Theorem for Fixed Points of
Multi-valued Contractions, Biagio Ricceri 437
Geometrical Differentiation and High-Accuracy curve Interpolation,
Robert Schaback 445
On Best Simultaneous Approximation in Normed Linear Spaces,
V. M. Sehgal and S. P. Singh 463
Some Examples Concerning Projection Constants, Boris Shekhtman 471
Subject Index 477
PREFACE
These are the Proceedings of the NATO Advanced Study Institute on
Approximation Theory, Spline Functions and Applications held in the Hotel
villa del Mare, Maratea, Italy between April 28,1991 and May 9, 1991.
The principal aim of the Advanced Study Institute, as reflected in these
Proceedings, was to bring together recent and up-to-date developments of
the subject, and to give directions for future research. Amongst the main
topics covered during this Advanced Study Institute is the subject of uni
variate and multivariate wavelet decomposition over spline spaces. This is
a relatively new area in approximation theory and an increasingly impor
tant subject. The work involves key techniques in approximation theory
cardinal splines, B-splines, Euler-Frobenius polynomials, spline spaces with
non-uniform knot sequences. A number of scientific applications are also
highlighted, most notably applications to signal processing and digital im
age processing. Developments in the area of approximation of functions
examined in the course of our discussions include approximation of periodic
phenomena over irregular node distributions, scattered data interpolation,
Pade approximants in one and several variables, approximation properties of
weighted Chebyshev polynomials, minimax approximations, and the Strang
Fix conditions and their relation to radial functions.
I express my sincere thanks to the members of the Advisory Commit
tee, Professors B.Beauzamy, E.W. Cheney, J. Meinguet, D.Roux, and G.M.
Phillips. My sincere appreciation and thanks go to A. Carbone, E. DePas
cale, R. Charron, and B. Watson for their excellent organization and smooth
running of the Institute. I express my warm appreciation and thanks to Ivar
Massabo for his help, cooperation and encouragement from the initial plan
ning until the successful completion of the Institute.
I extend my thanks to my colleagues who helped me with the planning
of the Institute.
I take this opportunity to extend my sincere thanks to the NATO Sci
entific Affairs Division for the generous support for the Institute, to the
Universita della Calabria and Department of Tourism, Potenza, for the fi
nancial support, and to Memorial University of Newfoundland for secretarial
help. Special thanks to Mrs. Philomena French for her excellent typing of
parts of the manuscript.
Finally I thank the staff of Kluwer Academic Publishers for their under
standing and cooperation.
ix
x
This volume is dedicated to the memory of Kanhaya Lal Singh (February 15, 1944
-November 22, 1990) who always participated in and contributed to all our conferences.
S.P. Singh
St. lohn's. Newfoundland, Canada
September 26 , 91
List of Contributors
Aksoy, A. G., Department of Mathematics, Clarement McKenna College, Claremont, California,
91711, USA.
Bos, L., Department of Mathematics, University of Calgary, Calgary, Alberta, T2N IN4, CANADA.
Campiti, M., Dipartimento di Matematica, Universita di Bari, Traversa 200 Via Re David, 4,
70125 Bari, ITALY.
Casini, E., Department of Mathematics, University of Bologna, Piu.a Porta S. Donato 5, Bologna,
40127, ITALY.
CavareUa, A. S., Department of Mathematics and Computer Science, Kent State University,
Kent, Ohio 44242, USA.
Cheney, W., Department of Mathematics, University of Texas, Austin, Texas 78712, USA.
Chui, C. K., Department of Mathematics, Texas A & M University, College Station, Texas 77843,
USA.
Costabile, F., Dipartimento di Matematica, Universita della Calabria, 87036 Arcavacata di Rende
(Cosenla), ITALY.
Criscuolo, G., Istituto per Ie Applicasioni della Matematica - CNR Napoli, Via P. Castellino,
111, 80131 Napoli, ITALY.
Cuyt, A., Department of Mathematics & Computer Science, University of Antwerp (UIA), Uni
versitaetsplein 1, B-2610 WUrijk-Antwerp, BELGIUM.
De Blyi, F.S., Department of Mathematics, University of Roma 11,00133, Roma, ITALY.
Della Vecchia, B., Istituto per Ie Applicuioni della Matematica - CNR Napoli, Via P. Castellino,
111, 80131 Napoli, ITALY.
De Michelle, L., Dipartimento di Matematica, Universita di Milano, Via C. Saldini, 50, 20133
Milano, ITALY.
De Pascale, E., Dipartimento di Matematica, Universita della Calabria, 87036 Arcavacata di
Rende (Cosenla), ITALY.
Deutsch, F., Department of Mathematics, Penn State University, University Park, PA 16802,
USA.
Di Natale, M., Dipartimento di Matematica, Universita di Milano, Via C. Saldini, 50, 20133
Milano, ITALY.
Gasca, M.V., Dep. de MatematicaAplicada, Universidad Zarago.a, Fac. Ciencias, 50009 Zaragola,
SPAIN.
Golitschek, M., Institut fiir Anglewandte Mathematik und Statistik, Universitit Wiirlburg, 8700
Wiirlburg, GERMANY.
Gregory, J. A., Department of Mathematics, BruneI University, Uxbridge, Middlesex, UB8 3PH,
U.K.
Guo, Shu Shen, Department of Mathematics, Hebei Normal University, Shijiuhuang, Hebei,
050016, P.R.C. (CHINA).
xi