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\begin{document}
\underline{Abstracts of appeared papers:}

\bigskip
\underline{File: optimal-recovery.dvi.Z}
\begin{center}
 Optimal Recovery in \\ Translation--invariant Spaces of Functions
 R. Schaback \\
\end{center} 
{\bf Abstract:}  
Starting from optimal recovery (in the sense of Micchelli,
Rivlin, and Winograd) of functions in reproducing kernel Hilbert spaces from 
function values
at scattered data points, we show that any continuously embedded 
translation--invariant Hilbert subspace $H$ of $L_2(\R^d)$ that allows
continuous point evaluation is necessarily principal, i.e.\ it is the native
space of a positive definite function $\Phi$. If, in addition, $H$ is invariant
under orthogonal transformations of $\R^d$, the function $\Phi$ necessarily is a
positive definite radial basis function. This serves to show that positive
definite radial basis functions and their native spaces
arise very naturally and are by no means exotic theoretical constructions.

\bigskip 
\underline{ap1.dvi.Z}
\begin{center}
Approximation by \break Radial Basis Functions with Finitely Many Centers \\
R. Schaback
\end{center}  
{\bf Abstract:}  
 Interpolation by translates of ``radial'' basis
functions $\Phi$ is optimal in the sense that it minimizes the pointwise error
functional among all comparable quasi--interpolants on a certain ``native''
space of functions $\call F_\Phi$. Since these spaces are rather  small for
cases where $\Phi$ is smooth, we study the behavior of interpolants on larger
spaces of the form $\call F_{\Phi_0}$ for less smooth functions $\Phi_0$. It
turns out that interpolation by translates of $\Phi$ to mollifications of
functions $f$ from $\call F_{\Phi_0}$ yields approximations to $f$ that attain
the same asymptotic error bounds as (optimal) interpolation of $f$ by translates
of $\Phi_0$ on $\call F_{\Phi_0}$.

\smallskip
{\bf AMS Classification}: 
41A15, 41A25, 41A30, 41A63, 65D10

\smallskip
{\bf Keywords}: Radial Basis Functions, Multivariate Approximation,
Approximation Order


\bigskip
\underline{rs-wu-operators2.dvi.Z}
\begin{center}
Operators on Radial Functions \\
Robert Schaback and Z. Wu
\end{center} 
{\bf Abstract:}  
A general theory is provided that allows to write multivariate
Fourier transforms or convolutions of radial functions as very simple 
univariate operations.
As a byproduct, an interesting group of operators $\{ I_\alpha \}_{\alpha \in
\R}$ with $I_{\alpha +\beta} =I_\alpha  \circ I_\beta =I_\beta \circ I_\alpha
$ is defined. It contains the classical derivatives as $I_{-1} =\bruch
d{dr}$ and is intimately connected to the Fourier transform. Applications to the
construction of new positive definite radial functions and to new identities for
special functions are included.

\bigskip
\underline{File: boch1.dvi.Z}
\begin{center}
Characterization of
Positive Definite Radial Functions \\
Zongmin  Wu
\end{center}
{\bf Abstract:}  
Positive definite multivariate radial functions
with existing multivariate Fourier transforms are characterized by
multiple monotonicity of their univariate Fourier
transforms, when written as functions of $r^2$ instead of $r$.
This result allows to prove that  uniform B--splines are not
positive definite when used as radial multivariate functions.
Furthermore, there is a general construction scheme that 
allows to generate all positive definite radial functions
via multiply monotone functions by purely univariate operations.

\bigskip 
\underline{File: iske1.dvi.Z}
\begin{center}
Characterization of Function Spaces associated to \\
Conditionally Positive Definite Functions \\
A. Iske
\end{center}
{\bf Abstract:}  
We give a unified theory that yields three characterizations 
      for the native function spaces associated to conditionally 
      positive definite functions on an open subset $\Omega$ of the 
      $d$ dimensional Euclidean space. As a byproduct, we completely describe
      the dependence of the spaces on $\Omega$.

\bigskip
\underline{File: ulvik.tar.Z}
\begin{center}
Creating Surfaces from Scattered Data \\
Using Radial Basis Functions \\
R. Schaback
\end{center}
{\bf Abstract:}  This paper gives an introduction to certain techniques for the 
construction of
geometric objects from scattered data. Special emphasis is put on
interpolation methods using compactly supported
radial basis functions.

\bigskip
\underline{File: er1.dvi.Z + er1-tabelle.dvi.Z}
\begin{center}
Error Estimates and Condition Numbers for Radial Basis Function 
Interpolation \\
Robert Schaback \\ 
\end{center}
{\bf Abstract}: For interpolation of scattered multivariate data by radial
basis functions, an ``uncertainty relation'' between the attainable error and
the condition of the interpolation matrices is proven. It states that the error
and the condition number cannot both be kept small. Bounds on the Lebesgue
constants are obtained as a byproduct. A variation of the Narcowich--Ward theory
of upper bounds on the norm of the inverse of the interpolation matrix is
presented in order to handle the whole set of radial basis functions that are
currently in use. 

\smallskip
{\bf Note: A table in oblong format is contained in file er1-tabelle.dvi.Z} 

\newpage
\underline{File: Qi.ps.Z} 

Journal of Approx. Theory and its Appl. 10:3, 1994, 1--7. Baltzer, Basel
\begin{center}
Limits of Bernstein--B\'ezier Curves for Periodic Control Nets \\
Dong-xu Qi and Robert Schaback \\ 
\end{center} 
{\bf Abstract:}
If $n$ given control points $b_0,\ldots,b_{n-1}\in \R^d$ are repeated 
periodically  by $b_{j+kn}=b_j$ for all $k\in \Z$, the uniform limit of the 
Bernstein--B\'ezier polynomial curves of degree $r$ with control points 
$b_0,\ldots,b_r$ for $r\to\infty$ is a Poisson curve
(after a suitable reparametrization). 
This fact reveals some interesting self--similar 
structures in case of regular $n$--gons in the plane.

\bigskip
\underline{File: wu1.dvi.Z}

IMA Journal of Numerical Analysis 13, 1993, 13--27. Academic Press, London
\begin{center}
Local Error Estimates for Radial Basis Function 
Interpolation of Scattered Data \\
 Zong--min Wu and Robert Schaback
\end{center}
{\bf Abstract}
We introduce  a suitable variational formulation for the local error of 
scattered data interpolation by radial basis functions $\phi(r)$.
 The error can be bounded
by a term depending on the Fourier transform of the interpolated function $f$
and  a certain ``Kriging function'', which allows a formulation
as an integral involving the Fourier transform of $\phi$.
The explicit construction of locally well--behaving admissible coefficient vectors
makes the Kriging function  bounded by some power of
the local density $h$ of data points. This leads to error estimates
for interpolation of functions $f$ whose Fourier transform $\hat f$ is
``dominated'' by the nonnegative Fourier transform $\hat\phi$ 
of the radial basis function $\phi$ in the sense $\int |\hat f|^2 \hat\phi^{-1}dt < \infty$.
Convergence orders are arbitrarily high for interpolation with Hardy
multiquadrics, inverse multiquadrics and Gaussian  kernels,
while interpolation with $\phi(r)=r^s$ or $\phi(r)=r^{2s}\log r$ are shown to have order
$h^{s/2}$ or $h^s$, respectively.

\smallskip
{\bf AMS classifications}: 41A05, 41A63, 41A25

\smallskip
{\bf Keywords}: Multivariate interpolation, radial basis functions, 
orders of convergence, Kriging, Multiquadrics.

\newpage
\underline{File: RSCAGD.dvi.Z}

Computer Aided--Geometric Design 6, 1989, 219--233. North--Holland, Amsterdam
\begin{center}
Interpolation with piecewise quadratic 
visually $C^2$ B\'ezier polynomials \\
 Robert Schaback \\
\end{center}
{\bf Abstract.} For data satisfying certain generalized convexity conditions,
the existence and uniqueness of a piecewise quadratic curvature continuous 
parametric spline interpolant in $\R^2$ is proved. Convexity is preserved
by the interpolation, and the numerical construction of the interpolant can be
carried out efficiently. 
The interpolation is 4--th order accurate, and
a number of examples shows both the applicability and the
limitations of this interpolation scheme.

\bigskip
\underline{File: RSCS.dvi.Z}

Approximation Theory VI, 1989, 581--584. Academic Press, New York
\begin{center}
Convergence of Planar Curve Interpolation Schemes \\ R. Schaback 
\end{center}
{\bf Abstract:} This note provides convergence orders for
a number of parametric interpolation schemes
for planar curves.
The methods use piecewise quadratic or
cubic polynomials and are globally $GC^1$ or $GC^2$. 
The data are either of Lagrange 
or of Hermite type;
convergence orders range between 4 and 6.

\bigskip
\underline{File: RSOslo.dvi.Z}

Math. Methods in Computer Aided Geometric Design I, 1989, 539--547.
\begin{center}
On Global $GC^2$ Convexity Preserving Interpolation \\
of Planar Curves by Piecewise Bezier Polynomials \\
R. Schaback
\end{center} 
{\bf Abstract}
For sequences of planar data points whose piecewise linear interpolant
has the property that two successive direction changes add up to
at most an acute angle, there is a unique global $GC^2$ interpolant
consisting of convex quadratic polynomial pieces where the data
allow a convex interpolant. 
Cubic pieces are used where the data require
inflection points. Collinear data points are interpolated by
straight lines embedded on both sides by cubic pieces.
Not--a--knot--boundary conditions are possible.

\bigskip
\underline{File: adaprat.dvi.Z}

Constructive Approximation 6, 1990, 167--179. Springer, New York
\begin{center}
Adaptive Rational Splines  \\
Robert Schaback
\end{center} 
{\bf Abstract.} The classical interpolation problems for cubic and rational
splines are merged to get an ``adaptive'' rational interpolating spline which 
automatically
uses cubic pieces to model unavoidable inflection points and retains 
convexity/concavity elsewhere. An existence proof, a numerical method,
 and a series of examples are presented. Furthermore,
the two--dimensional case is discussed.

\smallskip
{\bf Keywords.}  Rational functions, convexity--preserving splines.

\newpage
\underline{File: biri.ps.Z}

Math. Methods in Computer Aided Geometric Design II, 1992, 517--535.
\begin{center}
Rational Geometric Curve Interpolation \\
R. Schaback
\end{center} 
{\bf Abstract:} This survey covers geometric 
(i.e. parametrization--independent) curve
interpolation methods by piecewise rational functions 
with geometric continuity.
It includes shape--preserving properties, orders of accuracy, and provides a
number of examples.

\bigskip 
\underline{File: bm.dvi.Z}

SIAM Journal of Computing 17, 1988, 648--658. Philadelphia
\begin{center}
On the Expected Sublinearity  of the Boyer--Moore Algorithm \\
R. Schaback
\end{center} 
{\bf Abstract.} This paper analyzes the expected performance 
of a simplified version $BM'$ of the Boyer--Moore string matching algorithm. 
A probabilistic automaton $A$ is set up which models 
the expected behavior of $BM'$ under the assumption that 
both text and pattern are generated 
by a source which emits independent and uncorrelated
symbols with an arbitrary distribution of probabilities.
Formal developments 
lead then to the conclusion that $A$ takes expected sublinear time 
in a variety of situations. The sublinear behavior can be 
quantitatively predicted
by simple formulae involving the pattern length $m$ and the 
alphabet's probabilistic properties.
Finally, empirical evidence
is provided which is in satisfactory accordance with the theory.

\smallskip
{\bf Keywords : }{\it String searching, Pattern matching,
Average case analysis of algorithms.}

\bigskip
\underline{File: ch1.dvi.Z}

Multivariate Approximation. From CAGD to Wavelets, 1993. World Scientific, 
London.
\begin{center}
Comparison of Radial Basis Function Interpolants \\
Robert Schaback
\end{center}
{\bf Abstract:} This paper compares radial basis function interpolants on 
different spaces.
The spaces are generated by other radial basis functions,
and comparison is done via an explicit representation
of the norm of the
error functional. The results pose some new questions for further
research. 

\bigskip
\underline{File: cut1-12.ps.Z und cut13-17.ps.Z}

Computer Aided--Geometric Design 10, 1993, 277--291. North--Holland, Amsterdam
\begin{center}
A Parallel Multistage Method  for Surface/surface Intersection \\
Heiko B\"urger and Robert Schaback
\end{center}
{\bf Abstract:}
A global divide--and--conquer method and a local marching method are 
combined into a massively parallel algorithm
for surface/surface intersection.
Special attention is given to the comparison of bounding boxes 
and to the safe and efficient calculation of all 
components of the intersection. 
The computational complexity of the algorithm is analyzed, and a series of
examples is provided for illustration of the theoretical results.

\smallskip
{\bf AMS classifications}: 65D17, 65Y05, 65Y25, 68U05, 68U07

\smallskip 
{\bf Keywords}: Marching algorithms, Divide--and--conquer methods,
bounding boxes, uniqueness. 

\newpage
\underline{File: di-rs-1.dvi.Z}

Journal of Optimization Theory and  Appl. 67, 1990, 87--107. Plenum Press, New York.
\begin{center}
An extended continuous Newton method \\
  I. Diener and
  R. Schaback
\end{center} 
{\bf Abstract:}
This paper describes a numerical realization
of an extended continuous Newton method defined by I. Diener.
It traces a connected set of locally one--dimensional
trajectories which contains all critical points of a smooth
function $f: \R^n \rightarrow \R$. The results
show that the method is effectively applicable.

\smallskip
{\bf Key Words.} Global optimization, continuous Newton method, trajectory 
method, multiple solutions, critical points.


\bigskip
\underline{File: maratea.ps}

NATO ASI Series 356, 1992. Kluwer Academic Publ., Dordrecht.
\begin{center}
Geometrical differentiation and 
high--accuracy curve interpolation  \\
 Robert Schaback
\end{center} 
{\bf Abstract:} Let $f$ be a smooth curve in $\R^d$, parametrized by
arclength. If a large sample of data points $p_i=f(t_i)$ at unknown 
parameter values $t_i<t\ipl$ is given, one can use local $n$--th degree
polynomial interpolation at parameters $s_i=\| P_i-P_\ell \| 
\hbox{sgn}(i-\ell)$ 
of data
points $P_i$ around a fixed point $P_\ell$ to calculate approximations to the
derivatives $f^{(j)}(t_\ell)$ with accuracy $\calo(h^{n+1-j})$, where $h:=\max
(t_i-t\imi)$ and $0\leq j\leq k-1 \leq n$. Using these as data for properly
parametrized Hermite interpolation problems for polynomials of degree $\leq
2k-1 \leq n$ between successive data points, one can construct $GC^{k-1}$
interpolants of $f$ with accuracy $\calo(h^{2k})$.


\smallskip
{\bf Keywords:} Interpolation of Curves, Geometric Differentiation, 
Reparametrization, Computer--Aided--Design, Spline Functions

\bigskip
\underline{File: ow.ps.Z}

Numerical Methods of Approximation Theory, Proceedings, 105, 1992, 269--283.
\begin{center}
A Multi--Parameter Method for Nonlinear Least--Squares Approximation  \\
R. Schaback
\end{center} 
{\bf Abstract:} For discrete nonlinear least-squares approximation problems
$\sum_{j=1}^mf_j^2(x)\to\min$ for $m$ smooth functions 
$f_j : \R^n\to\R$ a numerical method is proposed 
which first minimizes each $f_j$ separately and then applies
a penalty strategy to gradually force the different minimizers to
coalesce.
Though the auxiliary nonlinear least-squares method works on
$\R^{n\cdot m}$, it is shown that the additional computational
requirements consist of $m-1$ gradient evaluations plus 
${\cal O}(m\cdot n)$ operations. The application to discrete rational 
approximation is discussed and numerical examples are given.

\newpage
\underline{File: rat.ps.Z}

Constructive Approximation 9, 1993, 373--389. North--Holland, Amsterdam.
\begin{center}
Planar Curve Interpolation by Piecewise Conics of Arbitrary Type \\
Robert Schaback
\end{center}
{\bf Abstract:}
Five points in general position in $\R^2$ always lie on a unique conic, and
three points plus two tangents also have a unique interpolating conic, the type
of which depends on the data. These well--known facts from projective geometry
are generalized: an odd number $2n+1\geq 5$ of points in $\R^2$, if they can be
interpolated at all by a smooth curve with nonvanishing curvature, will have a
unique $GC^2$ interpolant consisting of pieces of conics of varying type.
This interpolation process reproduces conics of arbitrary type
and preserves  strict convexity. Under weak additional 
assumptions its approximation order is
$\calo(h^5)$, where $h$ is the maximal distance
of adjacent data points $f(t_i)$ sampled from a smooth and regular
planar curve $f$ with nonvanishing curvature. 
Two algorithms for the construction of the interpolant
are suggested, and some examples are presented.

\smallskip
{\bf Keywords:} Geometric rational curve interpolation, 
shape preservation, convexity, parametric splines, conics,
approximation order.

\smallskip
{\bf MSC Classification:} 41A05, 41A15, 41A20, 65D05, 65D07, 65D17, 68U07

\bigskip
\underline{File: sd.ps.Z}

Computer Aided--Geometric Design 10, 1993, 57--66. North--Holland, Amsterdam
\begin{center}
Error Estimates for  Approximations from Control Nets \\
Robert Schaback
\end{center} 
{\bf Abstract:} 
This paper provides numerically accessible bounds for the error 
of certain simple approximations to functions defined by a control net
and a partition of unity. This generalizes ``flatness tests'' 
for the results of refinement algorithms like knot insertion and subdivision.
The order of the error terms may be more than quadratic with respect to
the domain size. A large number of examples is given, including 
rational representations.

\smallskip
{\bf Keywords:} Computer Aided Geometric Design, flatness test,
control net, B\'ezier net, subdivision, knot insertion, error estimate,
rational element.


\bigskip
\underline{File: chamo1.dvi.Z}

Wavelets, Images, and Surface Fitting, 1994, 459--466

\begin{center}
 Reproduction of Polynomials by Radial Basis Functions \\
R. Schaback
\end{center} 
{\bf Abstract:} For radial basis function interpolation of
scattered data in $\R^d$, the approximative reproduction of high-degree
polynomials is studied.
Results include uniform error bounds and convergence orders
on compact sets.

\newpage
\underline{File: cam.dvi.Z}
\begin{center}
 Lower Bounds for Norms of Inverses of  Matrices for Radial Basis 
Functions \\ 
 Robert Schaback
\end{center}
{\bf Abstract:} Interpolation of scattered data
at  distinct
points  $x_1,\ldots,x_n\in \R^d$ by linear combinations of translates
$\Phi(\|x - x_j\|_2)$ of a radial basis function
$\Phi : \R_{\geq 0} \to \R$
requires the solution of a linear system with the
$n$ by $n$ distance matrix
$ A:=(\Phi(\|x_i-x_j\|_2)$. Recent results of Ball, Narcowich and
Ward, using Laplace transform methods, provide
upper bounds for $\| A^{-1}\|_2$,
while Ball, Sivakumar, and Ward constructed examples
with regularly spaced points to get special lower bounds.
This paper proves general lower bounds by application of results
of classical approximation theory. The bounds
increase with the smoothness
of $\Phi$.
In most cases, they leave no more than a factor of $n^{-2}$ to be gained by
optimization of data placement, starting from regularly distributed data.
This follows from comparison with results of  Ball, Baxter, Sivakumar,
and Ward for points on scaled integer lattices and
supports the hypothesis that
regularly spaced data are near-optimal, as far as the condition of the
matrix $A$ is concerned.

\smallskip
{\bf Keywords:} Radial basis functions, multiquadrics, scattered data
interpolation, lower bounds, condition numbers.

\smallskip
{\bf Classification:} 65D05, 41A05, 41A63


\bigskip
\underline{File: wu2.dvi.Z}
\begin{center}
Shape preserving properties and convergence of \\ 
univariate multiquadric quasi--interpolation \\
Zong--min Wu  and Robert Schaback 
\end{center}
{\bf Abstract:} With a suitable modification at the endpoints of the 
range, quasi--interpolation with univariate multiquadrics $\phi (x)
=\sqrt{c^2+x^2}$ is shown to preserve convexity and monotonicity. If $h$ is
the maximum distance of centres, 
convergence of the quasi--interpolant is of order $\calo (h^2|\log h|)$ if 
$c=\calo (h)$. The log term can not be removed by introducing different 
boundary conditions or special placements of the centres.

\smallskip
{\bf Keywords:} multiquadric, quasi--interpolation, shape preservation,
approximation rate.

\smallskip
{\bf Classification:} 65D05, 65D07, 41A05, 41A63
\end{document}
