Multivariate Approximation Theory III: Proceedings of the Conference at the Mathematical Research Institute at Oberwolfach, Black Forest, January 20–26, 1985

395 Pages·1985·7.95 MB·English

by  Prof. Dr. Walter Schempp

ISNM75:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
Serie internationale d'A nalyse numerique
Vol. 75
Edited by
Ch. Blanc, Lausanne; R. Glowinski, Paris;
G. Golub, Stanford; P. Henrici, Ziirich;
H. O. Kreiss, Pasadena; A. Ostrowski, Montagnola;
J. Todd, Pasadena
Birkhiiuser Verlag
Basel· Boston . Stuttgart
Multivariate
Approximation
Theory III
Proceedings of the Conference at the
Mathematical Research Institute at Oberwolfach,
Black Forest, January 20-26, 1985
Edited by
Walter Schempp
KariZeUer
1985 Birkhiiuser Verlag
Basel . Boston . Stuttgart
Editors
Prof. Dr. Walter Schempp Prof. Dr. Karl Zeller
Lehrstuhl flir Mathematik I Mathematisches Institut
Universitiit Siegen Universitiit Tiibingen
HOiderlinstrasse 3 Auf der Morgenstelle 10
D-5900 Siegen (FRG) D-7400 Tiibingen (FRG)
Library of Congress Cataloging in Publication Data
Main entry under title:
Multivariate approximation theory III.
(International series of numerical mathematics
vol. 75)
Proceedings of the Fourth International Symposium on
Multivariate Approximation Theory.
English and German.
1. Approximation theory - - Congresses. 2. Functions
of several real variables - - Congresses. I. Schempp, W.
(Walter), 1938- II. Zeller, Karl.
III. International Symposium on Multivariate
Approximation Theory (4th : 1985 : Oberwolfach
Mathematical Research Institute) IV. Title: Multivariate
approximation theory 3. V. Title: Multivariate
approximation theory three. VI. Series.
QA297.5.M847 1985 511'.4 85-15645
CIP-Kurztitelaufnahme der Deutscben Bibliothek
Multivariate approximation theory. .. :
proceedings of the conference at the Math.
Research Inst. at Oberwolfach, Black Forest.
- Basel ; Boston ; Stuttgart : Birkhiiuser
3. January 20-26, 1985. - 1985.
(International series of numerical mathematics
Vol. 75)
ISBN-13: 978-3-0348-9995-6 e-ISBN-13: 978-3-0348-9321-3
DOl: 10.1007/978-3-0348-9321-3
NE: Mathematisches Forschungsinstitut
(Oberwolfach); GT
All rights reserved.
No part of this publication may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without the prior
permission of the copyright owner.
© 1985 Birkhiiuser Verlag Basel
Softcover reprint of the hardcover 1s t edition 1985
5
PREFACE
The Fourth International Symposium on Multivariate Approximation
Theory was held at the Oberwolfach Mathematical Research Insti
tute, Black Forest, W.-Germany, during the week of January 20 -
26, 1985. The preceding conferences on this topic were held in
1976, 1979, and 1982 * . We were pleased to have more than 50
mathematicians from 13 countries in attendance. The program in
cluded 40 lectures. These Proceedings form a record of most of
the papers presented at the Symposium.
The topics treated cover different problems on multivariate
approximation such as polynomial approximation on simplices,
multivariate splines (box-splines, dimension of spline spaces),
blending methods, multivariate Hermite interpolation, data
smoothing and surface representation, and multivariate summation
methods.
We would like to thank the director of the Oberwolfach Mathe
matical Research Institute, Prof. Dr. M. Barner, and his staff
for providing the facilities. Of the people who gave their time
to help make this conference a success, we would like to mention
in particular Prof. Dr. F.J. Delvos (Siegen), Dr. G. Baszenski
(College Station, Texas), and Dipl.-Math. H. Nienhaus (Siegen).
Finally, our thanks are due to Carl Einsele of Birkhauser
Publishers for his valuable cooperation.
May 1985 Walter Schempp Karl Zeller
Siegen Ttibingen
Editors
*"Constructive Theory of Functions of Several Variables", Lecture
Notes in Mathematics 571 (1977)
"Multivariate Approximation Theory", ISNM 51 (1979)
"Mul tivariate Approximation Theory II", ISNM 61 (1982)
7
CONTENTS
Zur Tagung................................................ 11
List of Participants...................................... 13
Program of the Sessions................................... 19
L. BAMBERGER: Interpolation in bivariate spline spaces.... 25
G. BASZENSKI: n-th order polynomial spline blending....... 35
C. DE BOOR, K. H6LLIG, S. RIEMENSCHNEIDER: The limits
of multivariate cardinal splines............. .......... 47
M. BOZZINI, L. LENARDUZZI: Local smoothing for
scattered and noisy data .............................. . 51
GENG-ZHE CHANG, J. HOSCHEK: Convexity and variation
diminishing property of Bernstein polynomials
over triangles......................................... 61
Y.S. CHOU, LO-YUNG SU, R.H. WANG: The dimensions
of bivariate spline spaces over triangulations......... 71
C.K. CHUI, M.J. LAI: On bivariate vertex splines.......... 84
Z. CIESIELSKI: Biorthogonal system of polynomials
on the standard simplex................................ 116
L. COLLATZ: Anwendung der Approximationstheorie auf
gewisse singulare Randwertaufgaben..................... 120
W. DAHMEN, C.A. MICCHELLI: combinatorial aspects of
multivariate splines................................... 130
8
F.J. DELVOS: Intermediate blending interpolation........... 138
H. ENGELS: Hermite-interpolation in N variables
and minimal cubature formulae........................... 154
w.
FREEDEN, P. HERMANN: Some reflections on multi-
dimensional Euler and Poisson summation formulas........ 166
R.H.J. GMELIG MEYLING, P.R. PFLUGER: On the dimension
of the spline space Si(6) in special cases .........•.... 180
M.v.GOLITSCHEK: Degree of best approximation by
blending functions ..................................... . 191
T.N.T. GOODMAN: Shape preserving approximation
by polyhedral splines ................................... 198
K. GURLEBECK, W. SPROSSIG, M. TASCHE: Numerical
realization of boundary collocation methods ............ . 206
H. HAKOPIAN: Interpolation by polynomials and
natural splines on normal lattices ...................... 218
W. HAUSSMANN, K. ZELLER: Bivariate approximation
procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 221
G. HEINDL: Construction and applications of Hermite
interpolating quadratic spline functions of two
and three variables ...•................................. 232
H.-B. KNOOP: Hermite-Fej~r and higher Hermite-Fej~r
interpolation with boundary conditions .................. 253
,
A. KROO: Some uniqueness problems in best Chebyshev
and mean multivariate approximation ..................... 262
9
W.A. LIGHT: Minimal projections in tensor product
spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 270
F. LOCHER: Convergence of Hermite-Fejer interpolation
via Korovkin s theorem ................................. . 277
I
E. LUIK: Cubature error bounds using degrees of
approxima tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 286
I
F. MORICZ: Square functions in the theory of
Cesaro summability of double orthogonal series .......... 298
H. NIENHAUS: Generalized Melkes interpolation .............. 309
A. QUARTERONI: Approximation theory and analysis of
spectral methods........................................ 322
H.-J. RACK: On multivariate polynomial L1-approximation
to zero and related coefficient inequalities ............ 332
M. REIMER: Abschatzungen von Lagrange-Quadratsummen flir
die Sphare mit Hilfe gewisser Eigenwerte ................ 343
W. SCHEMPP: On Gabor information cells ..................... 349
R. SCHMIDT: Ein Beitrag zur Flachenapproximation
liber unregelmaBig verteilten Daten ...................... 363
XIE-CHANG SHEN: The basis and moment problems of
some systems of analytic functions ...................... 370
B. SUNDERMANN: Normen von Projektionen in
mehreren Verander lichen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
G.A. WATSON: The solution of generalized least
squares problems........................................ 388
11
ZUR TAGUNG
Vom 20. bis 26. Januar 1985 fand im Mathematischen Forschungs
institut Oberwolfach eine Tagung tiber "Mehrdimensionale kon
struktive Funktionentheorie" statt. Sie wurde, wie schon die
Oberwolfach-Tagungen gleichen Themas der Jahre 1976, 1979 und
1982 von den Herausgebern geleitet. Es nahmen insgesamt 54
Mathematiker an der Tagung teil, die aus China, der Deutschen
Demokratischen Republik, England, Israel, Italien, den Nieder
landen, Norwegen, Polen, Schottland, Schweden, Ungarn, den Ver
einigten Staaten von Amerika und der Bundesrepublik Deutschland
kamen.
Das Vortragsprogramm bestand aus 40 Vortragen, in dessen Mittel
punkt Fragen der Darstellung, Approximation und numerischen Be
handlung von Funktionen mehrerer Variablen standen. Als Schwer
punkte der Vortrage sind zu nennen
- Polynomapproximation tiber Simplices,
- Multivariate Splines (Box-Splines, Dimension von Spline-
Raumen) ,
- Blending-Methoden,
- Mehrdimensionale Hermite-Interpolation,
- Glattung von Daten und Flachendarstellung,
- Mehrdimensionale Summierungsmethoden.
Die Tagung verlief in einer sehr freundlichen Atmosphare, zu der
die Gastfrendschaft und zuvorkommende Hilfe der Mitarbeiter des
Oberwolfacher Instituts wesentlich beigetragen haben. Ihnen, den
Vortragenden und Sitzungsleitern sei an dieser Stelle sehr
herzlich gedankt.
Walter Schempp Karl Zeller
Siegen Ttibingen
Tagungsleiter
13
LIST OF PARTICIPANTS
Lothar Bamberger, Mathematisches Institut der Universitat
Munchen, TheresienstraBe 39, D-8000 Munchen 2, Fed. Rep.
Germany
Gunter Baszenski, Department of Mathematics, Center for Approxi
mation Theory, Texas A & M University, College Station,
Texas 77843-3368, U.S.A.
Jan Boman, Matematiska Institutionen, Stockholms Universitet,
Box 6701, S-11385 Stockholm, Sweden
Bruno Brosowski, Fachbereich Mathematik der Johann-Wolfgang
Goethe-Universitat, Robert-Mayer-StraBe 6-10, D-6000 Frank
furt a. M. 1, Fed. Rep. Germany
Geng-zhe Chang, Department of Mathematics, University of Science
and Technology of Hefei, Anhui, People's Republic of China
Charles K. Chui, Department of Mathematics, Center for Approxi
mation Theory, Texas A & M University, College Station,
Texas 77843-3368, U.S.A.
Zbigniew Ciesielski, Instytut Matematyczny, Polskiej Akademii
Nauk, Oddzial w Gdansku, 81-825 Sopot, ul. Abrahama 18,
Poland
Lothar Collatz, Institut fur Angewandte Mathematik der Universi
tat Hamburg, Bundesstr. 55, D-2000 Hamburg 13, Fed. Rep.
Germany
Wolfgang Dahmen, Fakultat fur Mathematik der Universitat Biele
feld, UniversitatsstraBe, Postfach 8640, D-4800 Bielefeld 1,
Fed. Rep. Germany
Franz-Jurgen Delvos, Lehrstuhl fur Mathematik I der Universitat
Siegen, H6lderlinstraBe 3, D-5900 Siegen, Fed. Rep. Germany