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Integral Kernels with Reflection Group Invariance

Published online by Cambridge University Press:  20 November 2018

Charles F. Dunkl
Charles F. Dunkl*
Affiliation:
Department of Mathematics, Mathematics-Astronomy Building, Charlottesville, VA 22903-3199, U. S. A.
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Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 6 , 01 December 1991 , pp. 1213 - 1227
Copyright
Copyright © Canadian Mathematical Society 1991

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