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Inzell Lectures on Orthogonal Polynomials

W. zu Castell, F. Filbir, B. Forster, Eds.

Advances in the Theory of Special Functions and Orthogonal Polynomials, R. Álvarez-Nodarse, Series Ed., ISBN 1-59454-108-6, Nova Science Publishers, New York NY, USA, 2005, 199 p.



The SIAM Activity Group on Orthogonal Polynomials and Special Functions organized a series of summer schools with the aim of opening the field to young researchers. This volume presents the lecture notes of the second summer school held in Inzell, Germany, and focuses on the interrelation between orthogonal polynomials and harmonic analysis.

Report 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 Lübeck), Brigitte Forster (TU München) and Rupert Lasser (GSF Neuherberg and TU München) 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 University) "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 (University 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 possibilities.

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 Münich and the GSF-National Research Center for Environment and Health, Neuherberg, for having given us all the chance to listen to exciting mathematics.


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