Approximation Theory and Spline Functions

485 Pages·1984·25.34 MB·English

by  Bernard Beauzamy (auth.)

Approximation Theory and Spline Functions
NATO ASI Series
Advanced Science Institutes Series
A series presenting the results of activities sponsored by the NATO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The series is published by an international board of publishers in conjunction with the
NATO Scientific Affairs Divisicn
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical D. Reidel Publishing Company
and Physical Sciences Dordrecht, Boston and Lancaster
D Behavioural and Social Sciences Martinus Nijhoff Publishers
E Engineering and The Hague, Boston and Lancaster
Materials Sciences
F Computer and Systems Sciences Springer-Verlag
G Ecological Sciences Berlin, Heidelberg, New York and Tokyo
Series C: Mathematical and Physical Sciences Vol. 136
Approximation Theory
and Spline Functions
edited by
S.P. Singh
J.W. H. Burry
and
B.Watson
Department of Mathematics and Statistics,
Memorial University, St. John's, Newfoundland, Canada
D. Reidel Publishing Company
Dordrecht / Boston / Lancaster
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on
Approximation Theory and Spline Functions
SI. John's, Newfoundland, Canada
August 22-September 2, 1983
library of Congress Cataloging in Publication Data
NATO Adllanced Study Institutll on Approximation Theory lind Spline Functions 11983: St.
John's, Nfld.l
Approximetion theory and spline functions.
INA TO ASI series. Series C, Mathematical and physical sciences; \/01. 136)
"Proceedings of the NATO Advanced Study Innitute on Approximation Theory lind
Spline Functionl, St. John's, Newfoundland, Canada, Augulit 22-September 2, 1983"-
T.p.llerso.
"Published in cooperation with NATO Scientific Affairs Oillision."
Bibliographv: p.
IncludltS index.
1. Approximation theory-Conllresses. 2. Spline theory-CongraSS8s. I. Singh, S. P.
(Sankatha Prasad), 1937- II. Burry, J. H. (John H.), 1938- III. Watson, 6"
1946- IV. North Atlantic Treaty Orllilnization. Scientific Affairs Oillision. V. Title.
VI. Series: NATO ASI series. Serin C, Mathematical and physical sciences; no. 136.
QA221.N36 1963 511'.4 B4- 15136
ISBN-13: 978-94-009-6468-6 e-ISBN-13: 973-94-00S-6466-2
DOl: 10.10071973-94-009-6466-2
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TABLE OF CONTENTS
PREFACE ix
B. Beauzamy:
PRODUCTS OF POLYNOMIALS
H. P. Blatt:
EXCHANGE ALGORITHMS, ERROR ESTIMATIONS AND STRONG
UNICITY IN CONVEX PROGRAMMING AND CHEBYSHEV
APPROXIMATION 23
E. W. Cheney:
FOUR LECTURES ON APPROXIMATION 65
~ruLTIVARIATE
D. M. E. Foster and G. M. Phillips:
THE APPROXIMATION OF CERTAIN FUNCTIONS BY
COMPOUND MEANS 89
J. Meinguet:
A PRACTICAL METHOD FOR OBTAINING A PRIORI ERROR
BOUNDS IN POIN1WISE AND MEAN-SQUARE APPROXIMATION
PROBLEMS 97
J. Meinguet:
SURFACE SPLINE INTERPOLATION: BASIC THEORY AND
COMPUTATIONAL ASPECTS 127
C. A. Micchelli:
INTERPOLATION OF SCATTERED DATA: DISTANCE
MATRICES AND CONDITIONALLY POSITIVE DEFINITE
FUNCTIONS 143
G. M. Phillips and P. J. Taylor:
SEMI-NORMS IN POLYNOMIAL APPROXIMATION 147
1. L. Schumaker:
ON SPACES OF PIECEWISE POLYNOMIALS IN 11'10
VARIABLES 151
A. Sharma:
BIRKHOFF INTERPOLATION ON lHE ROOTS OF UNITY 199
vi TABLE OF CONTENTS
J. Todd:
APPLICATIONS OF TRANSFORMATION THEORY: A LEGACY
FROM ZOLOTAREV (1847-1878) 207
J. M. Borwein and P. B. Borwein:
EXPLICIT ALGEBRAIC Nth ORDER APPROXIMATIONS TO PI 247
M. Brannigan:
SOLVING INTEGRAL EQUATIONS OF NUCLEAR SCATTERING
BY SPLINES 257
M. Branni gan :
H-SETS FOR NON-LINEAR CONSTRAINED APPROXIMATION 265
A. A. M. Cuyt:
OPERATOR PADE APPROXIMANTS: SOME IDEAS BEHIND THE
THEORY AND A NUMERICAL ILLUSTRATION 271
M. Goldstein:
HARMONIC APPROXIMATION 289
M. Goldstein, W. Haussman and K. Jetter:
BEST HARMONIC Ll APPROXIMATION TO SUBHARMONIC
FUNCTIONS 293
T. N. T. Goodman and S. L. Lee:
B-SPLINES ON THE CIRCLE AND TRIGONOMETRIC B-SPLINES 297
F. B. Guenard:
ON REDUCING THE COMPUTATIONAL ERROR IN THE
SUCCESSIVE APPROXIMATIONS METHOD 327
M. S. Henry:
LEBESGUE CONSTANTS DETERMINED BY EXTREMAL SETS 339
P. E. Koch:
ERROR BOUNDS FOR INTERPOLATION BY FOURTH ORDER
TRIGONOMETRIC SPLINES 349
A. Le Mehaute:
APPROXIMATION OF DERIVATIVES IN Rn APPLICATION:
CONSTRUCTION OF SURFACES IN R2 361
C. H. Lutterodt:
MEROMORPHIC FUNCTIONS, MAPS AND THEIR RATIONAL
APPROXIMANTS IN Cn. 379
S. P. Norsett:
SPLINES AND COLLOCATION FOR ORDINARY INITIAL
VALUE PROBLEMS 397
TABLE OF CONTENTS vii
J. Prasad and A. K. Varma:
DEGREE OF APPROXIMATION OF QUASI-HERMITE-FEJER
INTERPOLATION BASED ON JACOBI ABSCISSAS Pn(a,a) (x) 419
B. E. Rhoades:
USING INCLUSION THEOREMS TO ESTABLISH THE
SUMMABILITY OF ORTHOGONAL SERIES 441
B. Shekhtman:
ON PROJECfIONS IN APPROXIMATION THEORY 455
J. L. Ullman:
A SURVEY OF EXTERIOR ASYMPTOTICS FOR ORTHOGONAL
POLYNOMIALS ASSOCIATED WITH A FINITE INTERVAL
AND A STUDY OF THE CASE OF THE GENERAL WEIGHT
467
MEASURES
LIST OF PARTICIPANTS 479
SUBJECT INDEX 483
PREFACE
A NATO Advanced Study Institute on Approximation Theory
and Spline Functions was held at Memorial University of
Newfoundland during August 22-September 2, 1983. This volume
consists of the Proceedings of that Institute.
These Proceedings include the main invited talks and
contributed papers given during the Institute. The aim of
these lectures was to bring together Mathematicians, Physicists
and Engineers working in the field. The lectures covered a
wide range including ~1ultivariate Approximation, Spline
Functions, Rational Approximation, Applications of Elliptic
Integrals and Functions in the Theory of Approximation, and
Pade Approximation.
We express our sincere thanks to Professors E. W. Cheney,
J. Meinguet, J. M. Phillips and H. Werner, members of the
International Advisory Committee. We also extend our thanks
to the main speakers and the invi ted speakers, whose contri
butions made these Proceedings complete.
The Advanced Study Institute was financed by the NATO
Scientific Affairs Division. We express our thanks for the
generous support.
We wish to thank members of the Department of Mathematics
and Statistics at MeMorial University who willingly helped with
the planning and organizing of the Institute.
Special thanks go to Mrs. Mary Pike who helped immensely
in the planning and organizing of the Institute, and to
Miss Rosalind Genge for her careful and excellent typing of
the manuscript of these Proceedings.
St. John's, Newfoundland, Canada S.P. Singh
April 1984 J.H.l'I. Burry
B. "latson
ix
PRODUCTS OF POLYNOMIALS
Bernard Beauzamy
In this survey paper, we shall present several results con
cerning estimates for products of polynomials, in one or in
several variables.
The results concerning polynomials in one variable are
taken from a joint paper of Per Enflo and the author [1]; the
results dealing with polynomials in many variables are due to
Per Enflo [2]. Though they were proved earlier, it seemed pre
ferable to put them into the last section, since they are tech
nically more complicated. However, the methods of proofs are of
different nature, and there is no interdependence between the
case of a single variable and the case of several variables.
In the following pages, only outlines of proofs will be
given: we refer the reader to [1] and to [2] for detailed proofs.
What we are looking for is estimates of the following type:
IlpQ11 Alipil • IIQII
~ (1)
where "." is some norm on the space of polynomials (in one or
in many variables), and A is a constant, depending only on the
II· II,
choice of and on the choices of the classes Cl and
C2, in which we will take P and Q respectively.
Let us first deal with polynomials in one variable. There
are many norms which are commonly used. Let us mention some of
them.
S. P. Singh et al. (eds.), Approximation Theory and Spline Functions, 1-22.
Ii:) 1984 by D. Reidel Publishing Company.
2 B. BEAUZAMY
N
Put Pen) = a + a.X + ••• + aNx • Then, we define:
0 1
II pill f:n I P(eiS) I ds
2n '
. 2 d 1/2
( (2n I
IIpII2 I P(elS) ~2n)
Jo
IIplla> = max IP(eiS) I
0:S;S:s;2n
For these three norms, P is considered as a (continuous) funct
ion on the Torus IT (that is, the interv.al [0, 2n], mod 2n).
dS
So these norms are just the norms of the spaces Ll(IT, 2n)'
ds de
L2(IT, 2n)' La> (IT, 2n)·
Another type of norms is obtained the following way. Put:
N
I pi 1 L la·1
o
J
N 2 1/2
IpI2 (L I a·1 )
o
J
I pi a> max I a·1
O:S;j:S;N J
This time, these norms can be viewed as norms on sequences spaces:
the first one is the norm in ~l' the second in ~2' the third
in ~a>' when P is identified with the sequence (aO' aI' ... ,
~).
I \I \I
The norm P 11 is sometimes written P A(IT), that is,
the norm in the algebra A(IT), of functions with absolutely
summable Fourier series.
The norm Ipla> is also written IIpll PM' norm in the space
of pseudo-measures (distributions on IT, the Fourier coefficients