Table Of ContentTopics in Multivariate Approximation
Edited by
C. K. Chui
L. L. Schumaker
Center for Approximation Theory
Department of Mathematics
Texas A&M University
College Station, Texas
F.I. Utreras
Department of Mathematics and Computer Science
University of Chile
Santiago, Chile
ACADEMIC PRESS, INC.
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Library of Congress Cataloging-in-Publication Data
Topics in multivariate approximation.
Proceedings of an international workshop held
at the University of Chile in Santiago, Chile,
December 15-19, 1986.
Bibliography: p.
1. Approximation theory—Congresses. 2. Functions
of several real variables—Congresses. I. Chui, C. K.
II. Schumaker, Larry L., 1939- III. Utreras
F.I.
QA297.5.T66 1987 51Γ.4 87-17454
ISBN 0-12-174585-6
Printed in the United States of America
87 88 89 90 9 8 7 6 5 4 3 21
PREFACE
During the week of December 15 - 19, 1986, an international workshop on
multi vari ate approximation was held at the University of Chile in Santiago,
Chile. The purpose of the conference was to bring leading researchers in the
field together for an intensive discussion of several current problem areas. The
conference was organized by an international committee consisting of Mira
Bozzini (Italy), Charles Chui (USA), Kurt Jetter (Germany), Pierre-Jean
Laurent (France), Larry Schumaker (USA), and Florencio Utreras (Chile).
Twenty-four researchers from ten countries gave one-hour survey lectures.
The topics covered by the lectures (and summarized in the papers included
in this proceedings volume) included the following:
- multivariate splines
- fitting of scattered data
- tensor approximation methods
- multivariate polynomial approximation
- numerical grid generation
- finite element methods
- constrained interpolation and smoothing.
In addition to the survey papers, this volume includes a bibliography
with over 1100 entries. While the authors, R. Franke and L. Schumaker,
make no claim of completeness, we feel that this bibliography will be a very
useful tool for researchers interested in working in the areas of multivariate
approximation discussed here, as well as in related areas.
The conference was supported by grants from a number of international
scientific organizations. These included CNR (Italy), INS A (France), CON-
ICYT (Chile), DFG (Germany), NSF (USA), and PNUD-UNESCO (United
Nations). We would also like to acknowledge the support of the Departa-
mento de Relaciones Internacionales and the Facultad de Ciencias Fisicas y
Matemâticas of the University of Chile, as well as the extensive work of the lo
cal organizing committee which included Patricio Basso, Maria Cecilia Rivara,
and Maria Leonor Varas, all of the University of Chile.
The manuscript for this volume was prepared at the Center for Approx
imation Theory at Texas A&M University, College Station, Texas, using the
TßX typesetting system. In this connection we would like to thank Dr. Nor
man W. Naugle for his help with the T^Knical aspects of putting the book
together, and Mrs. Jan Want who assisted with the preparation of a number
of the papers.
April 15, 1987
vii
PARTICIPANTS
Raul Aguila, Universidad Católica de Valparaiso, Instituto de Matematica,
Blanco Viel 596, Cerro Barón, Valparaiso, Chile
Herman Alder, Universidad de Concepción, Departamento de Matematicas,
Concepción, Chile
Nélida Iris Auriol, Universidad Nacional de San Luis, Martin de Loyola 1835-
13° San Luis, Argent ina
Patricio Basso, Depto. Matematicas Aplicadas, Fac. de Ciencias Fisicas y
Matematicas, Universidad de Chile, Casilla 170-3, Correo 3, Santi
ago, Chile
Maria E. Canales Tapia, Avda. Angamos 601, Departamento Matematicas,
Universidad de Antofagasta, Antofagasta, Chile
Magdalena Cantizani, Universidad de San Luis, Estado de Israel 1436, San
Luis, Argent ina
Eduardo Carrizo, Dept. de Computación, Universidad de Buenos Aires,
Maipu 241 4° piso, Buenos Aires, Argentina
Norma Cenzola, Universidad Nacional de San Luis, Av. Figueroa 828, San
Luis, Argentina
E. W. Cheney, Department of Mathematics, University of Texas, Austin,
Texas, 78712
Charles K. Chui, Center for Approximation Theory, Texas A&M University,
College Station, Texas, 77843
Miguel Cifuentes, ASMAR, Talcahuano, Chile.
Wolfgang Dahmen, Fakultät für Mathematik, Universität Bielefeld, D-4800
Bielefeld, West Germany
Guido E. Del Pino, Pontifìcia Universidad Católica de Chile, Departamento
de Estadistica, Santiago, Chile.
Franz-Jurgen Delvos, Lehrstuhl für Mathematik I, Universität Siegen, Hölder-
linstr. 3, D-5900 Siegen, West Germany
Nira Dyn, Department of Mathematics, Tel-Aviv University, Tel Aviv, Israel
Sergio Favier, Universidad Nacional de San Luis, Bolivar 1349, San Luis,
Argent ina
Adela Fernandez, Universidad Nacional de San Luis, Caseros 1065, San Luis,
Argentina
Osvaldo Ferreiro, Pontifìcia Universidad Católica de Chile, Departamento de
Estadistica, Santiago, Chile.
vin
Participants ix
Melitta Fiebig, Departamento de Matematicas, Universidad de Concepción,
Casilla 2017, Concepción, Chile
Ferruccio Fontanella, Dipartimento di Energetica, Via di S. Marta 3, 150139,
Firenze, Italia
Richard Franke, Department of Mathematics, Naval Postgraduate School,
Monterey, California 93943
Willi Freeden, Institut für Reine und Angewandte Mathematik, Rheinisch-
Westfälische, Technische Hochschule Aachen, Templersgraben 55,
D-5100 Aachen, West Germany
Marcos Guerrero, Depto. Matematicas Aplicadas, Fac. de Ciencias Fisicas y
Matematicas, Universidad de Chile, Casilla 170-3 Correo 3, Santiago,
Chile
Werner Haussman, Department of Mathematics, University of Duisburg,
Lotharstr. 65, D-4100 Duisburg, West Germany
Ivan Huerta, Facultad de Matematica, Universidad Católica de Chile, Casilla
114-D, Santiago, Chile
Kurt Jetter, FB Mathematik, Universität-GH-Duisburg, Lotharstr. D-4100,
West Germany
Alain J. Y. Le Méhauté, Laboratoire de Mathématiques, INS A, 35043 Rennes
Cedex, France
Rafael Leiva, Depto. Matematicas Aplicadas, Fac. de Ciencias Fisicas y
Matematicas, Universidad de Chile, Casilla 170-3 Correo 3, Santiago,
Chile
Jerónimo Lorente Pardo, Universidad de Granada, Dpto. Matemàtica
Aplicada, Facultad de Ciencias, 18071 Granada, Espana
George G. Lorentz, Department of Mathematics, University of Texas, Austin,
Texas 78712
Miguel A. Marano, Departamento de Matematicas, Universidad Nacional de
Rio Cuarto, 5800 Rio Cuarto, Argentina
Charles A. Micchelli, IBM Research, P. O. Box 218, Yorktown Heights, New
York, 10598
Joaquin Morales, Universidad de La Serena, Area Ingenieria Industrial,
Benavente 980, La Serena, Chile
Henni ter Morsche, Eindhoven University of Technology, P. O. Box 513, 5600
MV Eindhoven, The Netherlands
Gregory M. Nielson, Computer Science Department, Arizona State University,
Tempe, Arizona 85287
Carlos Obreque, ASM AR, Talcahuano, Chile.
Fernando Paredes Cajas, Universidad Catolica de Valparaiso, Instituto de
Matematica, Blanco Viel 596, Cerro Barón, Valparaiso, Chile
Joäo Prolla, Departamento de Matematica, IMECC-UNICAMP, 13100 Camp
inas, SP, Brasil
X Participants
Victoriano Ramirez G., Universidad de Granada, Dpto. Matematica
Aplicada, Facultad de Ciencias, 18071 Granada, Espana
Maria Cecilia Rivara Z., Depto. de Matemàticas, Fac. de Ciencias Fisicas y
Matemàticas, Universidad de Chile, Casilla 170/3 Correo 3, Santi
ago, Chile
Oscar Rojo, Universidad del Norte, Latorre 3174, Antofagasta, Chile.
Paul Sablonnière, INSA de Rennes, 20, Avenue des Buttes de Coesmes, 35043
Rennes Cedex, France
Oscar Schnake, Depto. Matemàticas Aplicadas, Fac. de Ciencias Fisicas y
Matemàticas, Universidad de Chile, Casilla 170-3 Correo 3, Santiago,
Chile
Larry L. Schumaker, Center for Approximation Theory, Texas A&M Univer
sity, College Station, Texas, 77843
Ledya Spencer, Depto. Matemàticas Aplicadas, Fac. de Ciencias Fisicas y
Matemàticas, Universidad de Chile, Casilla 170-3 Correo 3, Santiago,
Chile
Florencio Utreras, Depto. de Matemàticas, Fac. de Ciencias Fisicas y Mate
màticas, Universidad de Chile, Casilla 170-3 Correo 3, Santiago,
Chile
Maria Leonor Varas, Depto. Matemàticas Aplicadas, Fac. de Ciencias Fisi
cas y Matemàticas, Universidad de Chile, Casilla 170-3 Correo 3,
Santiago, Chile
Grace Wahba, Department of Statistics, University of Wisconsin-Madison,
Madison, Wisconsin 53706
Joseph D. Ward, Department of Mathematics, Texas A&M University,
College Station, Texas 77843
Antonella Zambrana, Universidad Mayor de San Andres, Jaén No. 283 esq.
Velasco Galvarro, Oruro, Bolivia.
Felipe Zó, Universidad Nacional de San Luis, Escuela de Matemàticas, 5700
San Luis, Argentina
BOOLEAN METHODS IN FOURIER APPROXIMATION
by
G. Baszenski and F.-J. Del vos
Abstract
It is the objective of this paper to apply Boolean methods of
approximation in combination with the theory of right invertible op
erators to bivariate Fourier expansions. We construct the operator
of Fourier operational calculus and relate its spectral properties to
the construction of Korobov spaces. We will derive error estimates
for Fourier product approximation, Fourier blending approximation,
Fourier hyperbolic approximation, and the related Krylov-Lanczos
approximation in these spaces.
1. The Operational Taylor Formula
Let X be a linear space, X\ a linear subspace of X, and R : X -+ X &
linear and injective map such that
range(i?) C Χ. (1)
λ
Assume that P is a linear projector on Χχ with
ker(P) = range(Ä). (2)
Then for any / G X there is a unique g = B(f) such that
f = P(f) + R(B(f)). (3)
B is a linear operator in X with dom(i?) = Xi defined by R and P. R is a
right inverse of J5, i.e.,
BR = /, (4)
and the relation
ker(B) = range(P) (5)
Topics in Multivariate Approximation 1 Copyright © 1987 by Academic Press, Inc.
All rights of reproduction in any form reserved.
ISBN 0-12-174585-6
2 G. Baszenski and F.-J. Delvos
holds. The operational Taylor formula follows from relation (3) by an iterative
application:
m — 1
/ = Σ RiPBJ(f) + Rm(Bmf) (/ G dom(Bm)). (6)
j=0
It can be shown that Rm is injective on X with
range(Äm) C dom(jBm),
and that
m —1
P = Σ WPB*
m
j=0
is a projector on dom(jBm) satisfying
ker(P ) = range(Pm) (7)
m
[1, 4, 13, 14, 15]. For instance, we consider the space C m(J), J = [0,2π].
C(J) = C°(J) is a unitary space with inner product
1 r2*
(/,#) = 2^ / f{x)g{x)dx
and orthonormal basis ek(x) = exo(ikx), k G Z. Any / G C 77^/), m G IN, has
a convergent Fourier series which represents the periodic extension of /. An
important example is the Bernoulli function 6 , m G IN, which is given by
m
M*) = Σ W'meikx
\k\>0
We define R by
R(f) = h */ + (/,eo),
where / * g is the convolution of the Fourier series of / and g. R is injective
and satisfies range(i?) Ç C1(J). The projector P is defined on Cl(J) by
P(/) = -(D/, e )6i = ± (/(0) - /(27Γ)) 6 .
0 X
The operator 5 defined by R and P is given by
S(/) = D/+(/-D/,eo).
Moreover, we have the relations
Rm(f) = b *f + (f,e),
m 0
Boolean Methods in Fourier Approximation
Bm(f) = Omf + (f-Omf,e ),
0
and
ra—1 1
i=o
Proposition 1. For any f G Cm(J), the following Lanzcos decomposition
holds for 0 < x < 2π:
m —1 -
/(*) = Σ 2Î(D,/(0) - ^'/(2'))&ί+ι(ϊ)
j'=0
(8)
+ ^jf"/M**
2π
Γ
+ ^[nbm(x-u)Drnf(u)du.
Proof: Relation (8) follows immediately from the operational Taylor formula
(6) and the representation
BT(Bmf) = ^Jn /(«) du + ~tJn M* - u)Dm/(u) du (9)
of the remainder projector J — P [11, 14, 16]. It follows from the well-known
m
properties of the Bernoulli functions that
kei(P ) = CZT1(J)nCm(J),
rn
where C^^l{J) is the subspace of functions / G Cm(J) with Dj/(0) =
Ό·7(2π) (0<j<m). ■
Corollary 1. For any f e Οψ~λ Π Cm( J), the representation
/»2π i /»2ir
/(*) = -^ / π /(«) du + ^ y" * 6 (x - «)Dm/(«) du (10)
m
holds.
We give an important application to Fourier approximation. Let λ^ =
(i/c)-1 (k φ 0) and λ = 1. The univariate Korobov space Ea(J), a > 1, is
0
given by
Ea(J) = {/ € L2(J) : |(/,e )| = 0(\X\a) (\k\ — oo)}
fe k
(cf. [7, 8, 10]). It follows from (8) that
CHJJÇÊ'W· (H)
Using (10) and (11) we obtain
CZT^J) ncra+1(J) ç Em+1(j). (12)
For n e IN and / e Ea(J), let
F (f)= X)(/,e )e .
n fc fe
\k\<n
F is the Fourier partial sum projector on Ea(J).
n
4 G. Baszenski and F.-J. Delvos
Proposition 2. Let f G Ea(J), a>\. Then
11/ - ^n(/)||oo = 0(n"a+1) (n —> oo). (13)
Proof: Since a > 1 we get
|/(x)-F (/)(x)|< £ l(/,e)l = ö(E^i=0(n_a+1)· "
n fc
|fc|>n \k>n /
The Krylov-Lanczos-approximant K (f) is defined for functions / G
n
Cm+1(J) by
^n(/) = Pm(/)+i ?,n(/-Pm(/)).
Proposition 3. Let / G C m+1(J). Then the asymptotic error relation
ΙΙ/-·Μ/)ΙΙ~ = σ ( ^) (n—»oo) (14)
holds.
Proof: Since
||/ - 2f (/)||oo = ||(/ - Pm(f)) - F (f - P (/))||oo,
n n m
relation (14) follows from (13), (12), (8) and (9). ■
2. The Bivariate Lanczos Decomposition
Let Cm,m(J2), m G Z+, denote the linear space of functions / G C(J 2) =
C°'°{J2) satisfying OjOkf G C(J2) (j, fc = 0,..., m). C ^'m(J2) is the linear
x y 2
subspace of functions / G Cm,m(J2) with
DÌDj;/(0,-)=DÌD|;/(2^.)
DÌDj/(·, 0) =DÌDj/(·, 2ττ) Ü, * = 0,..., m).
C(J2) is a linear subspace of the unitary space L2(J2) with inner product
-| /»27Γ /»27Γ
(/^) = τ^Λ2 / / }{x,y)g{x,y)dxdy
and orthonormal basis
e ,(rr, y) = e (a;)e(y) (A;, r G Z).
fcr fc r
Clearly the algebraic tensor product spaces Cm(J) <g> Cm(J) and C^W (8)
Cm(J) are subspaces of Cm^{J2) and C ™'m(J2) respectively. Let U be a
2
