SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #11  -------------   OP-SF NET 8.6  -------------  November 15, 2001
From: OP-SF NET Editor (
Subjects: Reports on Inzell Summer School

The Activity Group's Summer School on Orthogonal Polynomials, Harmonic
Analysis, Approximation and Applications was held in Inzell (Germany) during
September 17 - 21, 2001. The following reports from Frank Filbir and
Christian Berg appeared in our October printed Newsletter:

>From Frank Filbir (

After the first SIAG summer school 2000 in Laredo (Spain)  the second one in
a series of four took place from September 17th to 21th in Inzell,
(Germany), a small town located south-east of Munich close to Salzburg.
About 40 participants, mostly PhD students and young researchers from 7
different countries attended the meeting.

The summer school was mainly focused on orthogonal polynomials and their
various applications. We had main lectures (4 hours each) by Holger Dette
(Bochum, Germany), Ryszard Szwarc (Wroclaw, Poland)  and Yuan Xu (Eugene,
Oregon, U.S.A.) and additionally 18 contributed talks.  Unfortunately
Kristian Seip (Trondheim, Norway) who was announced as a invited speaker too
had to cancel his participation suddenly. Instead Rupert Lasser acted for
him. Due to the serious problems with the transatlantic flights during that
period William Connett (St. Louis, U.S.A.) was also not able to participate.

Holger Dette presented a very stimulating lecture on canonical moments and
their relations to the design problems in statistics. Ryszard Szwarc gave a
very interesting and detailed lecture on the problem of non-negative
linearization and the connections to commutative Banach algebras.  Yuan Xu
introduced us to the very challenging field of orthogonal polynomials of
several variables. Finally, Rupert Lasser lectured on polynomial hypergroups
and their applications to stochastics.

Let me mention here only a few of the contributed talks (this is of course a
very personal view). Els Coussement reported on her recent achievements
about multiple orthogonal polynomials and the relations to Bessel functions
of the second kind. In her talk Noemi Lain Fernandez gave us an impression
about the difficulties of the construction of polynomial bases on the sphere
with good localization properties. Finally, I would like to mention here the
very nice presentation of Andreas Ruffing on raising and lowering operators
for Charlier polynomials.

As we all know the success of a meeting depends not only on the scientific
program but also on the social program.  On that score we were lucky to have
gotten some support from the weather.  Starting on Monday with a cloudy sky
and rain we ended up on Wednesday (excursion day) with a nearly perfect
sunny and warm day. Almost all participants went out for excursions in the
nice surroundings of Inzell. However Ryszard Szwarc had to give up his plan
to go skiing since all skiing areas within a reasonable distance were
closed. We will pay more attention on those restrictions next time!

The organizers would like to thank the sponsors of the summer school the GSF
National Research Center and the graduate program "Applied Algorithmic
Mathematics" of the Munich University of Technology. Due to their
sponsorship we were also able to offer financial support to several

Last but not least the organizers wish to express their thanks to all
participants for their various contributions!

>From Christian Berg (

The second of four Summer Schools planned for the years 2000-2003 took place
September 17-21, 2001 at Inzell in Bavaria, Germany close to Salzburg. The
local Organizing Committee consisting of Frank Filbir (MU Lubeck), Brigitte
Forster (TU Munchen) and Rupert Lasser (GSF Neuherberg and TU Munchen) had
selected the very pleasant Hotel Chiemgauer Hof to provide some 40
participants with nice and abundant food and a good lecture hall.
Approximately half of the participants were Ph.D. students or post docs.

The program contained four series of lectures, each a total of 4 full hours
divided into three sessions. In addition approximately 20 contributed talks
were given.

Holger Dette (RU Bochum) told us about "Canonical moments, orthogonal
polynomials with applications to statistics", a subject with a rather new
development as may be seen from the recent monograph by the speaker and W.J.
Studden: "The theory of canonical moments with applications in statistics,
probability and analysis", Wiley 1997. The canonical moments are defined for
a probability measure with compact support, and since they are invariant
under affine transformations of the probability, the definition is usually
considered for measures on [0,1]. Dette presented the formulas for measures
on [-1,1] because of applications to Jacobi polynomials. In many examples
the canonical moments are much simpler than the ordinary moments, and they
have many statistical applications, in particular to optimal design.

Rupert Lasser had accepted with short notice to replace K. Seip, and he told
us about "Applied Harmonic Analysis". He presented the main ideas of
hypergroups--in the discrete setting to avoid technicalities--and
constructed the Banach *-algebra with respect to a left invariant Haar
function. He then applied the Gelfand theory in the commutative case and
went as far as giving the analogues of the theorems of Bochner and
Plancherel. An important example arizes in connection with orthogonal
polynomials having non-negative linearization coefficients like the
Gegenbauer polynomials. Under still further assumptions such a system
defines a commutative hypergroup structure on {0,1,...}.

     The lectures by Lasser prepared us for those of Ryszard Szwarc (Wroclaw
Univ.) "Orthogonal polynomials with applications to Banach algebras". After
a general introduction to the theory of orthogonal polynomials of one
variable, he focused on conditions leading to non-negative linearization
coefficients (called property (P)). He presented Askey's sufficient
condition and his own contributions based on a maximum principle for a
discrete hyperbolic partial difference equation. The exact range of
parameters for the generalized Chebyshev polynomials such that property (P)
holds is still not known.

The fourth series of lectures was given by Yuan Xu (Univ. of Oregon, Eugene)
"Orthogonal polynomials of several variables". Earlier this year appeared
the monograph by Charles Dunkl and the speaker with the same title, and a
few years ago Xu presented many important general results for the several
variable case of orthogonal polynomials in his Pitman Research Notes, vol.
312. Xu gave his version of the three-term recurrence relation with
applications to a Christoffel-Darboux formula and to common zeros of the
orthonormal polynomials of total degree N. There is no agreement as to which
systems of orthogonal polynomials of several variables should be called
classical. Easy cases occur by just taking products of classical weights
from one variable, but also various radially symmetric weights are
important. Xu showed us important systems for balls and the simplex and
relations between orthogonal polynomials on a ball and on the unit sphere,
generalizing the classical spherical harmonics. He also introduced us to the
theory of h-harmonic polynomials due to Dunkl, where h belongs to a finite
group of reflections. Xu stressed several times in his lectures that the
theory of orthogonal polynomials in several variables is still in its very
beginning. There are certainly already many fascinating results but large
unexplored areas are waiting for examination.

    The participants were lucky that Xu had been already one week in Europe,
when the Inzell meeting began. Otherwise he would most probably not have
been able to leave the US. The disaster of September 11 prevented Bill
Connett from participating.

     The meeting started with some rainy days, but on Wednesday afternoon
(set aside for relaxing) the weather was sunny and warm, and the
participants spread out in different groups, hiking on nearby mountains,
going to Salzburg or visiting Konigsee, just to mention some of the many

      On behalf of the participants I wish to thank the local organizers,
the team of initiaters (Branquinho, Koelink, Lasser, Marcellan and Van
Assche) as well as the Sponsors: The Graduate Program "Applied Algorithmic
Mathematics", Technical University of Munich and the GSF-National Research
Center for Environment and Health, Neuherberg for having given us all the
chance to listen to exciting mathematics.

Back to Home Page of
SIAM AG on Orthogonal Polynomials and Special Functions
Page maintained by Bonita Saunders