Orthogonal polynomials and reflection groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 0997.33005
author: Charles F. Dunkl
publication date: 2002-11-12T00:00:00Z
Mathematics Subject Classification ID: 33C52
Mathematics Subject Classification ID: 33C80
zbMATH DE Number: 1684156
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: reflection groups
zbMATH Keywords: Dunkl operators
zbMATH Keywords: \(h\)-harmonic polynomials
zbMATH Keywords: generalized Fourier transform
MaRDI profile type: Publication
title: Orthogonal polynomials and reflection groups
review text: In this survey article, the author describes his work on orthogonal polynomials of several variables which are invariant under reflection groups. After presenting some motivating classical one-variable and multivariable examples (e.g. Hermite and Laguerre polynomial and their multivariable versions), he introduces a commuting family of first order differential-difference operators \({\mathcal D}_i\) associated with a reduced root system \(R\) and a multiplicity function \(k\) on \(R\). The \({\mathcal D}_i\), commonly known after the author as Dunkl operators, can be considered as a deformation of the usual partial derivatives, which correspond to the limit \(k \rightarrow 0\). Connection with inner product structure are given. In particular, the \(h\)-harmonic polynomials are introduced as analogues of the spherical harmonics corresponding to a certain weight function \(h\) which is invariant under the Weyl group of \(R\). The analogue \(K(x,y)\) of the exponential function \(\exp(<x,y>)\) (with \(x,y \in \mathbb R^d\)) is constructed by means of certain intertwining operators. Integration against \(K(x,-iy)\) yields a generalized Fourier transform which can be considered as the spectral resolution of the operators \({\mathcal D}_i\). The \(h\)-harmonic polynomials can be used to describe an orthogonal basis for \(L^2(\mathbb R^d, h(x)^2 dx)\). NEWLINENEWLINENEWLINEProofs are omitted in this article, but precise references are provided. See e.g. the recent monograph by \textit{C. F. Dunkl} and \textit{Y. Xu} [Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications. 81. Cambridge: Cambridge University Press (2001; Zbl 0964.33001)].NEWLINENEWLINEFor the entire collection see [Zbl 0969.00053].
reviewed by: Angela Pasquale
description: scientific article; zbMATH DE number 1684156