Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T17:43:36.405Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Askey, R., Some basic hypergeometric extensions of integrals ofSelberg and Andrews, SIAM J. Math. Anal. 11(1980), 938–951.Google Scholar
2. van der Corput, J. and Schaake, G., Ungleichungenfiir Polynôme und trigonometrische Polynôme, Composite Math. 2(1935), 321–361.Google Scholar
3. Dunkl, C., An addition theorem for Heisenberg harmonics. Conference on Harmonic Analysis in honor of A. Zygmund, Wadsworth International, 1983, 690–707.Google Scholar
4. Dunkl, C., Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197(1988), 33–60.Google Scholar
5. Dunkl, C., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311(1989), 167–183.Google Scholar
6. Dunkl, C., Poisson and Cauchy kernels for orthogonal polynomials with dihedral symmetry, J. Math. Anal. Appl. 143(1989), 459–470.Google Scholar
7. Dunkl, C., Operators commuting with Coxeter group actions on polynomials. Invariant Theory and Tableaux, D. Stanton (éd.), Springer-Verlag, 1990, 107–117.Google Scholar
8. Erdèlyi, A., Axially symmetric potentials and fractional integration, SIAM J. 13(1965), 216–228.Google Scholar
9. Heckman, G.J., A remark on the Dunkl differential-difference operators. Proceedings of Bowdoin conference on Harmonic Analysis of Reductive Groups, August 1989, to appear.Google Scholar
10. Ikeda, M., On spherical functions for the unitary group, I, II, III, Mem. Fac. Engrg, Hiroshima Univ. 3(1967), 17–75.Google Scholar
11. Kanjin, Y., A convolution measure algebra on the unit disc, Tôhoku Math. J. 28(1976), 105–115.Google Scholar
12. Macdonald, I.G., Some conjectures for root systems, SIAM J. Math. Anal. 13(1982), 988–1007.Google Scholar
13. Opdam, E.M., Some applications of hypergeometric shift operators, Inventiones Math. 98(1989), 1–18.Google Scholar
14. Selberg, A., Bemerkninger om et multiplet integral, Norsk. Mat. Tidsskrift 26(1944), 71–78.Google Scholar
15. Szegö, G., Orthogonal polynomials. 3rd éd., AMS Colloquium Publications, Providence, 1967.Google Scholar