Multivariate Approximation and Splines

328 Pages·1997·9.64 MB·English

by  V. F. Babenko, V. A. Kofanov, S. A. Pichugov (auth.), Günther Nürnberger, Jochen W. Schmidt, Guido Walz (eds.)

ISNM
International Series of Numerical Mathematics
Vol. 125
Managing Editors:
K.-H. Hoffmann, Mlinchen
D. Mittelmann, Tempe
Associate Editors:
R. E. Bank, La Jolla
H. Kawarada, Chiba
R. J. LeVeque, Seattle
c. Verdi, Milano
Honorary Editor:
J. Todd,Pasadena
Multivariate Approximation
and Splines
Edited by
Günther Nürnberger
Jochen W. Schmidt
Guido Walz
Springer Basel AG
Editors:
Günther Nürnberger Jochen W. Schmidt
Fakultät für Mathematik und Informatik Institut für Numerische Mathematik
Universität Mannheim Technische Universität Dresden
0-68131 Mannheim 0-01062 Dresden
Germany Germany
e-mail:
v
Contents
Preface ................................................................... vii
Multivariate Inequalities of Kolmogorov Type and Their Applications
V. F. Babenko, V. A. Kofanov, and S. A. Pichugov. . . . . . . . . . . . . . . . . . . . 1
Monotone Iterative Technique for Impulsive Differential-Difference Equations
with Variable Impulsive Perturbations
D. Bainov, A. Dishliev, and S. Hristova................................ 13
Multivariate Cosine Wavelets
K. Bittner, C. K. Chui, and J. Prestin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
On Almost Interpolation by Multivariate Splines
O. Davydov, M. Sommer, and H. Strauss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Locally Linearly Independent Systems and Almost Interpolation
O. Davydov, M. Sommer, and H. Strauss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Exponential-Type Approximation in Multivariate Harmonic Hilbert Spaces
F.-J. Delvos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Interpolation by Continuous Function Spaces
M. von Golitschek. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Discrete Characterization of Besov Spaces and Its Applications to Stochastics
A. Kamont............................................................ 89
One-Sided Approximation and Interpolation Operators Generating Hyper-
bolic Sigma-Pi Neural Networks
B. Lenze.............................................................. 99
Unconstrained Minimization of Quadratic Splines and Applications
W. Li ................................................................. 113
Interpolation by Translates of a Basis Function
W. Light .............................................................. 129
On the Sup-Norm Condition Number of the Multivariate Triangular Bernstein
Basis
T. Lyche and K. Scherer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 141
Integration Methods of Clenshaw-Curtis Type, Based on Four Kinds of Cheby-
shev Polynomials
J. C. Mason and E. Venturino......................................... 153
Tensor Products of Convex Cones
B. Mulansky. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167
vi Contents
The Curse of Dimension and a Universal Method for Numerical Integration
E. Novak and K. Ritter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 177
Interpolation by Bivariate Splines on Crosscut Partitions
G. Nurnberger, O. V. Davydov, G. Walz, and F. Zeilfelder ............. 189
Necessary and Sufficient Conditions for Orthonormality of Scaling Vectors
G. Plonka ............................................................. 205
Trigonometric Preconditioners for Block Toeplitz Systems
D. Potts, G. Steidl, and M. Tasche.................................... 219
The Average Size of Certain Gram-Determinants and Interpolation on Non-
Compact Sets
M. Reimer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 235
Radial Basis Functions Viewed From Cubic Splines
R. Schaback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 245
Wavelet Modelling of High Resolution Radar Imaging and Clinical Magnetic
Resonance Tomography
W. Schempp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 259
A New Interpretation of the Sampling Theorem and Its Extensions
G. Schmeisser and J. J. Voss .......................................... 275
Gridded Data Interpolation with Restrictions on the First Order Derivatives
J. W. Schmidt and M. Walther........................................ 289
Affine Frames and Multiresolution
J. StockIer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 307
List of Participants ....................................................... 321
vii
Preface
This book contains the refereed papers which were presented at the interna-
tional conference on "Multivariate Approximation and Splines" held in Mannheim,
Germany, on September 7-10,1996. Fifty experts from Bulgaria, England, France,
Israel, Netherlands, Norway, Poland, Switzerland, Ukraine, USA and Germany
participated in the symposium.
It was the aim of the conference to give an overview of recent developments
in multivariate approximation with special emphasis on spline methods. The field
is characterized by rapidly developing branches such as approximation, data fit-
ting, interpolation, splines, radial basis functions, neural networks, computer aided
design methods, subdivision algorithms and wavelets.
The research has applications in areas like industrial production, visualization,
pattern recognition, image and signal processing, cognitive systems and modeling
in geology, physics, biology and medicine.
In the following, we briefly describe the contents of the papers.
Exact inequalities of Kolmogorov type which estimate the derivatives of mul-
tivariate periodic functions are derived in the paper of BABENKO, KOFANovand
PICHUGOV. These inequalities are applied to the approximation of classes of mul-
tivariate periodic functions and to the approximation by quasi-polynomials.
BAINOV, DISHLIEV and HRISTOVA investigate initial value problems for non-
linear impulse differential-difference equations which have many applications in
simulating real processes. By applying iterative techniques, sequences of lower and
upper solutions are constructed which converge to a solution of the initial value
problem.
The construction of bivariate biorthogonal cosine wavelets on certain rectan-
gular grids with bell functions not necessarily of tensor product type is the aim of
the paper by BITTNER, CHUI and PRESTIN. The biorthogonal system, the frame
and the Riesz basis conditions are given explicitly. A main tool are bivariate folding
operators.
DAVYDOV, SOMMER and STRAUSS give a survey of recent developments in
multivariate interpolation by functions from arbitrary finite-dimensional spaces. A
basic result says that almost interpolation sets are characterized by a Schoenberg-
Whitney type condition. Of special interest are spaces of generalized splines defined
on polyhedral partitions.
In a further paper, these authors describe methods for constructing almost
interpolation sets. This is done for spaces with locally independent systems of basis
functions. Several examples of such systems, including translates of box splines and
finite-element functions, are given.
viii Preface
DELVOS introduces the concept of harmonic Hilbert spaces in the multivari-
ate setting as an extension of periodic Hilbert spaces. Approximation methods for
these spaces are studied via Fourier partial integrals and exponential-type inter-
polation.
The classical results of Runge and Faber show that in general, interpolating
polynomials do not converge to the given function. In the univariate and multivari-
ate case, VON GOLITSCHEK investigates further interpolation operators for which
the operator norm grows with the dimension of the approximation spaces.
In the paper of KAMONT, the several characterizations of multivariate Besov
spaces are given, which only involves values of the functions on dyadic points.
The results are used to study the regularity of realizations of random fields such
as fractional Brownian motion, fractional Levi fields and fractional anisotropic
Wiener fields.
LENZE constructs three-layer feedforward neural networks which are used for
one-sided approximation and interpolation of regular gridded data. The concrete
networks are obtained in real-time by using a one-shot learning scheme. An appli-
cation of this strategy is discussed.
A review of unconstrained minimization of multivariate quadratic splines and
of some related problems is given by L1. In this context, the author discusses
numerical methods for solving such type of problems and error bounds which are
useful for analyzing the convergence of the algorithms.
Multivariate interpolation by translates of a given function, such as radial
basis function interpolation, has a strong connection with variational principles.
LIGHT describes a general approach to a variational principle, gives some applica-
tions and shows how the method is related to results of other authors.
LYCHE and SCHERER derive an upper bound for the condition number of the
multivariate Bernstein basis with respect to the uniform norm. It is shown that
the upper bound grows like (8 + 1)n and that it is independent of 8 for n :s: 8 + 1,
where 8 is the number of variables and n is the total degree of the polynomials.
MASON and VENTURINO show that discrete orthogonality formulae hold for
four kinds of Chebyshev polynomials. Each formula yields a general quadrature
formula of Clenshaw-Curtis type for integrating weighted functions. For weights of
Jacobi type, the Clenshaw-Curtis formula reduces to a Gauss quadrature formula.
Aspects of error estimates are discussed.
MULANSKY derives properties of tensor products of convex cones from finite-
dimensional spaces. It is shown that cones arising in shape preserving interpolation
by tensor products are the intersection of injective cones. Therefore, sufficient
conditions for multivariate shape constraints can be derived from the univariate
conditions.
In general, the computational cost of solving numerical problems grows ex-
ponentially with the number of variables. NOVAK and RITTER study numerical
integration and show that by applying non-classical methods, a high number of
Preface ix
vari&bles can be compensated by a high degree of smoothness of the underlying
functions or by a favorable structure of the problem.
NURNBERGER, DAVYDOV, WALZ and ZEILFELDER give a survey of recently
developed methods for constructing interpolation points for spaces of splines of
arbitrary degree and smoothness on general crosscut partitions. For certain regular
type partitions, the approximation order of the corresponding interpolating splines
is given.
In the paper of PLONKA , conditions for the orthonormality of scaling vectors
in terms of its two-scale symbol and the corresponding transfer operator are in-
vestigated. In particular, it is shown that well-known conditions for the two-scale
symbol and criteria for the transfer operator are equivalent.
The numerical solution of systems of linear equations with positive defi-
nite, double symmetric block-Toeplitz matrices is subject of the paper of POTTS,
STEIDL and TASCHE. For such matrices, the authors construct optimal and strong
type preconditioners by using the Fejer and Fourier sum of the generating func-
tion of the Wiener class. An estimate of the steps needed for the preconditioned
conjugate gradient method is given.
REIMER proves results on the size of Gram determinants which are defined via
reproducing kernels. As a consequence, it is shown that there exist interpolation
points such that the uniform norm of the corresponding Lagrange functions is
small. These results also hold for functions on non-compact sets.
The paper of SCHABACK discusses some problems arising in the error analysis
of interpolation by radial basis functions. This is done by applying the general
theory to the case of natural cubic splines. By applying an optimality principle
for quasi-interpolants which reproduce polynomials, the author obtains improved
local error bounds for interpolation by natural cubic splines.
The modeling of high resolution radar imaging and clinical magnetic reso-
nance tomography with the aid of coherent wavelets is the subject of the paper
by SCHEMPP. It is shown that the construction of matched filter banks depends
on Kepler's spatiotemporal strategy applied to quantum holography and can be
described by Fourier analysis of the Heisenberg nilpotent Lie group.
Various versions of sampling theorems are discussed by SCHMEISSER and
VOSS. It is shown that there is an equivalence between the sampling of signals
and the sampling of entire harmonic functions. In this way, a uniqueness result on
entire harmonic functions of exponential type is obtained.
SCHMIDT and WALTHER investigate interpolation of data by biquadratic and
biquartic splines on rectangular grids. Conditions are developed under which the
first partial derivatives of the interpolating splines satisfy certain restrictions.
Moreover, it is shown that the interpolation problem is solvable if additional knots
are added to the original grid in a suitable way.
By using generalized Laurent operators, STOCKLER investigates multivariate
affine frames which are generated by multiresolution with a single scaling function.
x Preface
The relations to the transfer operator are described, a new representation of the
lifting scheme is given and the connections to generalized Toeplitz operators are
discussed.
In conclusion, the editors would like to thank Deutsche Forschungsgemein-
schaft and the University of Mannheim for their support, and Birkhauser-Verlag
for agreeing to publish the proceedings in the ISNM series.
Gunther Nurnberger Jochen W. Schmidt Guido Walz
Mannheim Dresden Mannheim
Summer 1997