Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel.
type: Item
instance of: scholarly article
title: Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel.
zbMATH Open document ID: 1052.33014
author: Fethi Soltani
published in: Pacific Journal of Mathematics
publication date: 2004-09-17T00:00:00Z
review text: Dunkl operators on \(\mathbb{R}^d\) generalize the classical partial derivatives; they generate a commutative algebra of differential-difference operators. Due to the work of Dunkl, Opdam, de Jeu, Rösler and others, it is well-known that there exist associated Dunkl kernels and Dunkl transforms (generalizing the classical exponentials and Fourier tranform) as well as heat semigroups and Hermite polynomials. In the paper under review, Fock spaces and the chaos transform associated with Dunkl operators are introduced. In particular, relations to Hermite functions and commutation relations between Dunkl and multiplication operators are studied (while these relations are partially known). These commutation relations in particular lead to Weyl commutation relations.
Mathematics Subject Classification ID: 33C99
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 46E99
Mathematics Subject Classification ID: 47B99
zbMATH DE Number: 2101159
zbMATH Keywords: Dunkl operators
zbMATH Keywords: Fock spaces
zbMATH Keywords: commutation relations
zbMATH Keywords: Hermite polynomials
reviewed by: Michael Voit
MaRDI profile type: Publication
full work available at URL: pjm.2004.214.379
OpenAlex ID: W2093141326
DOI: 10.2140/PJM.2004.214.379
description: scientific article; zbMATH DE number 2101159