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\begin{document}
\underline{Abstracts of preprints:}

\bigskip
\underline{efficiency1.dvi.Z}
\begin{center}
 On the Efficiency of Interpolation by  Radial Basis Functions \\
Robert Schaback 
\end{center} 
We study the computational complexity, the error behavior, and the 
numerical stability of interpolation by radial basis functions. It turns out
that these issues are intimately connected. For the case of compactly supported
radial basis functions we consider the possibility of getting
reasonably good reconstructions of $d$-variate functions
from $N$ data at ${\cal O}(Nd)$ computational cost and give some 
supporting theoretical results and numerical examples.

\bigskip
\underline{File: dhs.dvi.Z}
\begin{center}
Remarks on High Accuracy Geometric Hermite Interpolation \\
Robert Schaback 
\end{center} 
{\bf Abstract.} 
In a recent paper deBoor, H\"ollig, and Sabin proposed a 
method for local sixth--order
interpolation of convex planar curves by geometrically
$C^2$ piecewise cubic curves, using positional, directional, and
curvature data at breakpoints.
This note extends their results to curves with simple zeros of curvature
and gives some hints to stabilize the numerical construction. 

\smallskip
{\bf Keywords.}  Curves, Interpolation, Geometric Continuity,
Convergence Order.

\bigskip
\underline{File: bs1.dvi.Z}
\begin{center}
Special Cases of Compactly Supported Radial Basis Functions \\
Robert Schaback, Holger Wendland
\end{center}
{\bf  Abstract:} Two instances of positive definite radial basis functions
with compact support are presented. The first is ``Euclid's hat'', obtained
by convolution of the characteristic function of the Euclidean unit
ball in $\R^d$. If $d$ is odd, the result is a piecewise polynomial
function of the Euclidean distance to the origin. In even-dimensional spaces things are
somewhat more complicated, but still easy to calculate.
The second instance is the radialization of tensor products of 
univariate B-splines of even order.
For this we give a recursion via a positive univariate convolution integral that
holds for arbitrary orders and dimensions, and we explicitly calculate the result
for the two-dimensional case. Since the cubic bivariate
tensor-product B-spline on uniform knots does not differ too much from its radialization,
we finally consider the radial function obtained by direct evaluation of the
cubic univariate B-spline as a function of Euclidean distance in $\R^d$.
Calculating its Fourier transform via a power series reprensentation, it turns out
that at least for dimensions $d=2, 3$ and orders $2$ and $4$
of univariate B-splines this approach does not
yield positive definite functions.

\newpage
\underline{File: wu3.dvi.Z}
\begin{center}
 Multivariate Compactly Supported  Positive Definite Radial Functions \\
   Zongmin  Wu
\end{center}
{\bf Abstract: } We provide
criteria for  positive definiteness of  radial functions with
compact support. Based on these criteria we will produce
a series of positive definite 
and compactly supported radial functions, which will be very useful
in applications. The simplest ones are   
{\em cut-off polynomials}, which consist of a single   polynomial 
piece on $[0,1]$ and vanish  on $[1,\infty)$. More precisely, for any
given dimension $n$ and 
prescibed $C^k$ smoothness, there is a  function in $C^k(\R^n)$, which
is a  positive definite radial function with compact support and is a
cut-off polynomial as a function of Euclidean distance. Another example
is derived from  odd-degree
$B$-splines.

\smallskip
{\bf AMS classifications}: 41A15, 41A25, 41A30, 41A63, 65D10, 65D17 .

\smallskip
{\bf Keywords}: Compact support, Positive definite, Radial  function,
 Scattered data interpolation,
Approximation.

\bigskip
\underline{File: wu4.dvi.Z}
\begin{center}
Construction Techniques for Highly Accurate  Quasi -Interpolation Operators \\
---An Extension to Strang -Fix theory--- \\
 R. Schaback and  Zongmin  Wu.
\end{center}
{\bf Abstract:}  
If $\psi \in C^{\ell+1}$,
we can construct quasi -interpolants in the form   
$$
%\bql{QInth}
f_h(x)=\sum^{+\infty}_{j=-\infty} f(hj)\varphi_h\left( \bruch xh -j\right),
\quad x\in \R, \; h>0
$$
%\eq
with  approximation order  $\ell$,
where $\varphi_h(x)$ is a linear combination of  translates 
$\psi(x-jh)$ of the function $\psi$ but depends on  $h$. The order of 
convergence of such operators can be pushed up to a limit that only
depends on the smoothness of the function $\psi$. This approach can be
generalized to the multivariate setting by using  discrete convolutions
with tensor products of odd-degree $B$ -splines.

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

\smallskip
{\bf Keywords}: Radial Basis Function, Quasi -Interpolation, Strang
-Fix conditions, Smoothness, Order of Convergence.

\bigskip

\newpage

\bigskip
\underline{File: texas1.dvi.Z}
\begin{center}
Multivariate Interpolation  and Approximation by  \\
Translates of a Basis Function  \\
R. Schaback
\end{center}
{\bf Abstract:} This contribution  will touch the following topics:
\begin{itemize}
\item{$\bullet$} Short introduction into the theory of multivariate 
interpolation and
approximation by finitely many (irregular) translates of a (not necessarily
radial) {\it basis function}, motivated by optimal recovery of functions
from discrete samples.
\item{$\bullet$} {\it Native}  spaces of functions associated to conditionally 
positive definite functions, and relations between such spaces.
\item{$\bullet$} {\it Error bounds} and {\it condition numbers} for 
interpolation of functions from native spaces.
\item{$\bullet$} {\it Uncertainty Relation}:  Why are good error bounds always 
tied to bad
condition numbers?
\item{$\bullet$} {\it Shift and Scale}: How to cope with the Uncertainty 
Relation?
\end{itemize}
\end{document}
