Numerical Methods in Approximation Theory, Vol. 9

365 Pages·1992·9.83 MB·English

by  G. Baszenski

ISNM 105:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
Serie Internationale d'Analyse Numerique
Vol. 105
Edited by
K.-H. Hoffmann, München; H. D. Mittelmann, Tempe;
J. Todd, Pasadena
Springer Basel AG
N u m e r i c al M e t h o ds in
A p p r o x i m a t i on T h e o r y,
V o l .9
Edited by
D. Braess
L. L. Schumaker
Springer Basel AG
Editors
Prof. Dr. Dietrich Braess Prof. Dr. Larry L. Schumaker
Fakultät und Institut Stevenson Professor
für Mathematik Dept. of Mathematics
Ruhr-Universität Bochum Vanderbilt University
Universitätsstr. 150, Geb. NA 1326 Stevenson Center
D-W-4630 Bochum 1 Nashville, TN 37240
Germany USA
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Numerical methods of approximation theory. - Basel ; Boston ; Berlin : Birkhäuser.
Bis Vol. 8 mit dem Parallelt.: Numerische Methoden der Approximationstheorie.
Bis Bd. 4 u.d.T.: Numerische Methoden der Approximationstheorie
NE: Tagung über Numerische Methoden der Approximationstheorie;
Numerische Methoden der Approximationstheorie
Vol. 9. Ed. by D. Braess ; L. L. Schumaker. - 1992
(International series of numerical mathematics ; Vol. 105)
ISBN 978-3-0348-9702-0 ISBN 978-3-0348-8619-2 (eBook)
DOI 10.1007/978-3-0348-8619-2
NE: Braess, Dietrich [Hrsg.]; GT
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically those of translation, reprinting, re-use of
illustrations, broadcasting, reproduction by photocopying machine or similar means,
and storage in data banks. Under § 54 of the German Copyright Law where copies are
made for other than private use a fee is payable to »Verwertungsgesellschaft Wort«, Munich.
© 1992 Springer Basel AG
Originally published by Birkhäuser Verlag Basel in 1992
Softcover reprint of the hardcover 1st edition 1992
Printed from the authors' camera-ready manuscripts on acid-free paper
ISBN 978-3-0348-9702-0
Lothar Collatz 6.7. 1910 - 26.9.1990
CONTENTS
Preface IX
Contributors XllI
Blending Approximations with Sine Functions
G. Baszenski, F.-J. Delvos, and S. Jester. . . . . . . 1
Quasi-interpolation in the Absence of Polynomial Reproduction
R. K. Beatson and W. A. Light . ........... . 21
Estimating the Condition Number for Multivariate Interpolation
Problems
P. Binev and K. Jetter 41
Wavelets on a Bounded Interval
Charles K. Chui and Ewald Quak 53
Quasi-Kernel Polynomials and Convergence Results
for Quasi-Minimal Residual Iterations
Roland W. Freund . . . . . . . . . . . 77
Rate of Approximation of Weighted Derivatives
by Linear Combinations of SMD Operators
Margareta Heilmann ...... . 97
Approximation by Multivariate Splines:
an Application of Boolean Methods
Rong-Qing Jia ..... . . . . . . . . . . . . . . . . . 117
Lm,c,s-Splines in IRd
A. Le Mehaute and A. Bouhamidi .. 135
Constructive Multivariate Approximation via Sigmoidal Functions
with Applications to Neural Networks
Burkhard Lenze .................... . . . 155
Contents
Vlll
Spline-Wavelets of Minimal Support
T. Lyche and K. M¢rken. . . . ... 177
Necessary Conditions for Local Best Chebyshev Approximations
by Splines with Free Knots
Bernd Mulansky . . . . . . . . . . . . . . . . . . . . . . . 195
C1 Interpolation on Higher-Dimensional Analogs
of the 4-Direction Mesh
A. Neff and J. Peters . . . . . . . . . . . . . . . . . . . . . 207
Tabulation of Thin Plate Splines on a Very Fine
Two-Dimensional Grid
M. J. D. Powell ............ . . .. 221
The L -Approximation Orders of Principal Shift-Invariant Spaces
2
Generated by a Radial Basis Function
Amos Ron ......................... 245
A Multi-Parameter Method for Nonlinear Least-Squares
Approximation
R. Schaback ....... 269
Analog VLSI Networks
W. Schempp . ... . ..... 285
Converse Theorems for Approximation on Discrete Sets II
G. Schmeisser . . . . . . . . . . . . . . . . . . . . . . . . 301
A Dual Method for Smoothing Histograms using Nonnegative C1-Splines
Jochen W. Schmidt . . . . . . . . . . . . . . . . . . . . . . 317
Segment Approximation By Using Linear Functionals
Manfred Sommer . . . . . . . . . . . . . . . . ... 331
Construction of Monotone Extensions to Boundary Functions
Cornelis Traas . . . . . . . . . . . . . . . . . . . . . . . . 347
PREFACE
This book is the official proceedings of a conference on Numerical Methods
in Approximation Theory which was held at the Mathematisches Forschungs
institut in Oberwolfach during the week of November 24~30, 1991. It contains
refereed and edited papers by 20 of the 49 participants.
The book is dedicated to the memory of Prof. Lothar Collatz who main
tained a long and active interest in numerical approximation. It is the ninth
in a series of volumes published by Birkhiiuser resulting from conferences on
the subject held at Oberwolfach, and co-organized by Prof. Collatz.
We now briefly describe the contents of the book. The paper of BASZEN
SKI, DELVOS and JESTER deals with blending using sine double series expan
sions of functions defined on the unit square. In addition to giving explicit
error estimates for partial sums and for interpolating sine polynomials, they
also show that Boolean sums yield almost the same asymptotic error estimates
as the conventional tensor-product approach, but with a reduced number of
terms.
The paper of BEATSON and LIGHT discusses approximation by quasi
interpolants which are sums of scaled translates of a one-parameter family
of functions. They do not require reproduction of low degree polynomials,
but nevertheless are able to give error bounds and analyze quasi-interpolation
based on Gaussians and exponentials.
BINEV and JETTER deal with multivariate interpolation using shifts of a
single basis function. They treat both gridded data and scattered data. As
examples, they consider box splines and certain radial basis functions.
The subject of the paper by CHUI and QUAK is the currently very hot
topic of wavelets. The aim of their paper is to present two different approaches
to multiresolution analysis on a bounded interval. In particular, they explicitly
construct certain (non-orthogonal) wavelets in this case.
The contribution of FREUND is a continuation of his earlier work on quasi
kernel polynomials and their connection with the so-called quasi-minimal
residual algorithm (QMR) for solving general nonsingular non-Hermitian lin
ear systems. In particular, he derives bounds on the norms of such polynomi-
x Preface
als, and uses them to obtain convergence theorems for the QMR method and
certain of its variants.
The Durrmeyer modification of the Szasz-Mirakjan operators is the sub
ject of the paper by HEILMANN. The rate of simultaneous approximation by
linear combinations is related to the Ditzian-Totik modulus of smoothness,
and both direct and inverse theorems are established.
JIA investigates multivariate smooth splines on nonuniform rectangular
grids, and develops a general theory of Boolean methods in such a way that
it can be applied to noncommutative operators. Based on this theory, an
explicit quasi-interpolant is constructed so that it gives rise to an efficient
scheme of approximation by multivariate smooth splines. This scheme is
shown to achieve the optimal rate of approximation.
LE MEHAUTE and BOUHAMIDI introduce a space of splines which are
a natural generalization of the thin plate splines of Duchon, and investigate
their properties. As an application, they characterize thin plate splines under
tension.
In the paper of LENZE, sigmoidal functions are used to generate approx
imation operators for multivariate functions of bounded variation. He starts
with Lebesgue-Stieltjes type convolution operators, and then uses numerical
quadrature to pass to point-evaluation operators, and to give local and global
approximation results for them. He also treats an interesting application to
neural networks with one hidden layer consisting of so-called sigma-pi units.
The paper of LYCHE and M0RKEN is intimately connected with wavelets.
In particular, they show how to determine a basis for the orthogonal comple
ment of certain spline spaces in related larger ones.
Chebyshev approximation of real continuous functions from the class of
polynomial splines of degree n with at most k free knots is discussed in the
paper of MULANSKy. The analysis is based on introducing the notion of an
extended tangent cone which also contains discontinuous splines. An improved
necessary condition (which can also be formulated as an alternant criterion)
for local best approximations is derived.
NEFF and PETERS give sharp necessary and sufficient conditions on data
at the vertices of a certain 4-direction mesh which allow interpolation of data
+
by m-variate, Cl piecewise polynomials of degree m 1. They also show that
+
for degree m 2 and higher, values and normals at the vertices can be stably
interpolated, and exhibit a unit-norm C2 Lagrange function for each vertex.
In his paper, POWELL presents an algorithm for evaluating a thin plate
spline at all lattice points of a very fine square grid (with as many as 100 mil
lion points). He shows that the amount of work required by the algorithm is
bounded by a small constant multiple of the number of grid points plus a con-