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\begin{document}

\begin{center}
{\huge Charles F. Dunkl}

\medskip

{\Large Department of Mathematics, University of Virginia, U.S.A}{\LARGE .}

\bigskip\medskip

\bigskip\textbf{Abstract}

\medskip
\end{center}

{\Large Lecture 1: Orthogonal polynomials of classical type in several variables}

\medskip

This is an overview of methods to deal with orthogonal polynomials on various
domains in $\mathbf{R}^{N}$, such as the sphere, the simplex, and the ball,
equipped with weight functions having certain symmetries. These methods
include differential-difference (\textit{Dunkl}) operators, symmetry-group
invariant differential operators, fractional integral transforms and analogues
of the exponential function. The weight functions mostly occur in the
following three types:

\begin{enumerate}
\item $\prod_{i=1}^{N}\left|  x_{i}\right|  ^{2\kappa_{i}}$, with parameters
$\kappa_{1},\kappa_{2},\ldots,\kappa_{N}>-\frac{1}{2}$;

\item $\prod_{1\leq i<j\leq N}\left|  x_{i}-x_{j}\right|  ^{2\kappa}$, with
parameter $\kappa>-\frac{1}{N};$

\item $\prod_{i=1}^{N}\left|  x_{i}\right|  ^{2\kappa_{1}}\prod_{1\leq i<j\leq
N}\left|  x_{i}^{2}-x_{j}^{2}\right|  ^{2\kappa_{0}}$, with parameters
$\kappa_{0},\kappa_{1}\geq0.$
\end{enumerate}
Polynomials of types (2) and (3) have applications as wave functions in the
solution of Calogero-Sutherland systems. The symmetry groups that arise here
are all finite reflection (\textit{Coxeter}) groups.

\bigskip{\Large Lecture 2: Boundedness and summability problems for
expansions}\medskip

This concerns expansions in orthogonal polynomials on the bounded domains
described in the first lecture. The main methods are Cesaro summability and
Poisson series. Using the general results of M. R\"{o}sler on the positivity
of the intertwining operator, Y. Xu found sufficient conditions on $\delta$
for the boundedness of Cesaro $\left(  C,\delta\right)  $ sums. These results
will be described.

There will be discussion of open problems; for example, explicit forms for the
intertwining operator (intertwining refers to the algebras of partial
derivatives and of Dunkl operators), construction of orthogonal bases for
polynomials on the sphere or the ball in the situation where the symmetry
group is nonabelian.
\end{document}