Multivariate Approximation Theory IV: Proceedings of the Conference at the Mathematical Research Institute at Oberwolfach, Black Forest, February 12–18, 1989

348 Pages·1989·7.5 MB·English

by  C. K. Chui

ISNM90:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
Serie internationale d'Analyse numerique
Vol. 90
Edited by
K.-H. Hoffmann, Augsburg; H. D. Mittelmann, Tempe;
J. Todd, Pasadena
Birkhauser Verlag
Basel . Boston . Berlin
Multivariate
Approximation
Theory IV
Proceedings of the Conference at the
Mathematical Research Institute at Oberwolfach,
Black Forest, February 12-18, 1989
Edited by
C. K. Chui
W. Schempp
K. Zeller
1989 Birkhauser Verlag
Basel . Boston . Berlin
Editors
C. K. Chui W. Schempp
Center for Approximation Theory Lehrstuhl fUr Mathematik I
Department of Mathematics Universitat Siegen
Texas A & M University Holderlinstrasse 3
College Station, Texas 77843 D-5900 Siegen
USA
K. Zeller
Mathematisches Institut der
UniversitatTiibingen
Auf der Morgenstelle 10
D-7400Tiibingen 1
CIP-Titelaufnahme der Deutschen Bibliothek
Multivariate approximation theory ... : proceedings of the
conference at the Mathematical Research Institute at
Oberwolfach, Black Forest. - Basel ; Boston ; Berlin :
Birkhauser, 1989
Teilw. mit d. Erscheinungsorten Basel, Boston, Stuttgart
NE: Mathematisches Forschungsinstitut <Oberwolfach>
4. February 12-18,1989. -1989
(International series of numerical mathematics; Vol. 90)
ISBN-13: 978-3-0348-7300-0 e-ISBN-13: 978-3-0348-7298-0
DOl: 10.1007/978-3-0348-7298-0
NE:GT
This work is subject to copyright. All rights are reserved, whether the whole or part
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made for other than private use a fee is payable to >>VerwertungsgesellschaftWort«,
Munich.
© 1989 Birkhauser Verlag Basel
Softcover reprint of the hardcover 1st edition 1989
ISBN-13: 978-3-0348-7300-0
Contents
A/feld P., Sirvent M.:
A Recursion Formula for the Dimension of Super Spline Spaces of Smoothness r
and Degree d > r2k .•.. . • . . . • . • . . . . . . . . • • . . . • • . • . • . . • • . . • • 1
Atteia M.:
Approximation with Barycentric Coordinates: The Hilbertian Case 9
Baszenski G., De/vas F.i.:
A Discrete Fourier Transform Scheme for Boolean Sums of Trigonometric
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
de Boor C:
A Local Basis for Certain Smooth Bivariate PP Spaces 25
de Boor C. Ron A.:
Polynomial Ideals and Multivariate Splines 31
BuhmannM.D.:
Cardinal Interpolation with Radial Basis Functions: An Integral Transform
Approach ............................................. 41
Carnicer i.M., Gasca M.:
On the Evaluation of Multivariate Lagrange Formulae ................ 65
Chen H.-L., Chui C.K., Micchelli CA.:
Asymptotically Optimal Sampling Schemes for Periodic Functions II: The
Multivariate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Chen T.:
Generalized Bochner-Riesz Means of Fourier Integrals . . . . . . . . . . . . . . . . 87
Ciesielski z.:
An Algorithm for Best Approximating Algebraic Polynomials in LP over a
Simplex .............................................. 95
Collatz L.:
Rational and Algebraic Approximation for Initial- and Boundary-Value-
Problems ............................................. 103
De/vas F.-i.:
R-th Order Blending Rectangle Rules 107
De/vas F.-i .• Nienhaus H.:
A Trivariate Boolean Cubature Scheme 115
Dinh-Dung:
Multivariate Band-Limited Functions: Sampling Representation and Appro-
ximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Feichtinger H.G., Grochenig K.:
Multidimensional Irregular Sampling of Band-Limited Functions in LP-Spaces. 135
VI
Feig E., Micchelli c.A.:
U-Synthesis by Ambiguity Functions 143
w.,
Freeden Gervens T.:
Vector Spherical Spline Interpolation 157
Gonska H H:
Simultaneous Approximation by Generalized n-th Order Blending Operators 173
Heindl G.:
Some Results on Quadratic Splines of Three (and more) Variables 181
HuY-S.:
On Iterates of Linear Variation Diminishing Operators and Characterization of
Bernstein-Type Polynomials ................................. 193
Jetter K., Koch P:
Methoden der Fourier-Transformation bei der kardinalen Interpolation perio-
discher Daten. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
JiaR.Q.:
Dual Bases Associated with Box Splines 209
Kirchhoff N., Nessel R.I.:
Some Pointwise Negative Results III Multivariate Approximation by Convo-
lution Processes of Fejer'sType . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 217
Lenze B.:
On Multidimensional Lebesgue-Stieltjes Convolution Operators 225
LightW.A.:
The Singularity of Distance Matrices 233
Madych W.R:
Cardinal Interpolation with Polyharmonic Splines 241
Moller HM.:
Multivariate Rational Interpolation 249
ter Morsche H G.:
On the Role of the Exponential Eigen Splines in Translation Invariant Periodic
Spline Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Prolla J. B. :
Uniform Closure of Tensor Product of Linear Subspaces 263
von Radziewski K.:
On Periodic Hermite-Birkhoff Interpolation by Translation 271
Reimer M.:
Problems and Results in the Calculation of Extremal Fundamental Systems for
Sphere and Ball ......................................... 279
VII
Sablonniere P.:
Bernstein Quasi-Interpolants on [0,1] 287
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
SchemppW:
Elementary Holograms, Artificial Neural Networks, and Theta-NullValues 295
x.-c.:
Shen
Some Recent Results on Complex Interpolatory Approximation 313
Steidl Go, Tasche M.:
IndexTransforms for Multidimensional Discrete FourierTransforms 321
Stockier Jo:
Minimal Properties of Periodic Box-Spline Interpolation on a Three Direction
Mesh 329
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Wang K.-Y.:
Strong Uniform Approximation by Bochner-Riesz Means 337
0
IX
PREFACE
Multivariate Approximation Theory forms a rapidly evolving field in Applied
Mathematics. The reason for its particular current interest lies in its
impact on Computer Aided Geometric Design (CAGD), Image Processing, Pattern
Recogni tion, and Mult idimensional Signal Processing. Mul ti var iate Bernstein
polynomials and box splines, for example, play an important role in CAGD.
Conversely, the highly important filter bank design problem of signal
processing, for instance, gives rise to a new family of multivariate
approximating functions, the Gabor wavelets, with interesting technological
and biological applications.
The conferences on Multivariate Approximation Theory held at the Mathematical
Research Institute at Oberwolfach, Black Forest, in 1976, 1979, 1982, 1985
and 1989 ref lect the progress made in this area and related fie Ids. The
present volume which is a continuation of the preceding volumes
Constructive Theory of Functions of Several Variables, Lecture Notes in
Mathematics 571 (1977)
Multivariate Approximation Theory, ISNM 51 (1979)
Multivariate Approximation Theory II, ISNM 61 (1982)
Multivariate Approximation Theory III, ISNM 75 (1985)
is based on the conference held on February 12-18, 1989. It includes most of
the lectures presented at the Oberwolfach meeting and reveals the wide
spectrum of activities in the field of multivariate approximation.
The organizers are grateful to the Director of the Oberwolfach Mathematical
Research Institute, Professor Dr. M. Barner, and his staff for providing the
facili ties, and to Dr. G. Baszenski, Professor Dr. F. J. Delvos, Dr. H.
Nienhaus, and Dr. K. von Radziewski for their valuable cooperation during the
preparation of the meeting.
July 1989 C.K. Chui (College Station, Texas)
W. Schempp (Siegen)
K. Zeller (Tlibingen)
International Series of
Numerical Mathematics, Vol. 90
© 1989 BirkhiiuserVerlag Basel
A RECURSION FORMULA FOR THE DIMENSION OF SUPER SPLINE SPACES
OF SMOOTHNESS r AND DEGREE d > r2k .
Peter Alfeld and Mari tza Sirvent
Department of Mathemat ics, Uni versi ty of Utah
Abstract
We consider super splines of global degree of smoothness r, polynomial
degree d, in a general number k of independent variables, defined on a k
dimensional triangulation T of a suitable domain Sl, which are r2k- m- 1_
times differentiable across every m-face (m = 0 ... k - 1) of a simplex in T.
For the case d > r2k we give a recursion formula for the dimension of these
super spline spaces.
1. Introduction
Let V C IRk be a given set of N distinct points, let Sl denote the convex
hull of V and let T be a set of k simplices defining a triangulation of Sl,
note that a p,-face of a simplex in T is itself a p,-dimensional simplex. We
denote by S" the set of all p,-faces of the simplices in T (p, = 0 ... k - 1)
and let S = U~:~ S". On the triangulation T we define a multivariate spline
space Sd(Sl) as usual by
Sd(Sl) = {s E CT(Sl) : sir E pj "IT E T}
(k d)
+
where -rn dk ·1 S th ed -d1· menS.1 0na1 l·ln ear space of all k-variate polyno-
mials of total degree less than or equal to d.
The area of multivariate spline spaces has seen a great deal of research
activity in recent years, culminating in a book [5J devoted to the subject.
Much of the activity has been centered around the surprisingly difficult
problem of calculating the dimension of bivariate spaces where k = 2 (see
e. g., [2J and the references quoted there), and significant progress has
been made. The dimension (and a basis) is known if the polynomial degree d