Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T01:05:34.010Z Has data issue: false hasContentIssue false
  • English
  • Français

Intertwining Operator and h-Harmonics Associated With Reflection Groups

Published online by Cambridge University Press:  20 November 2018

Yuan Xu
Yuan Xu*
Affiliation:
Department of Mathematics University of Oregon Eugene, Oregon 97403-1222 U.S.A. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

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.

We study the intertwining operator and $h$-harmonics in Dunkl's theory on $h$–harmonics associated with reflection groups. Based on a biorthogonality between the ordinary harmonics and the action of the intertwining operator $V$ on the harmonics, the main result provides a method to compute the action of the intertwining operator $V$ on polynomials and to construct an orthonormal basis for the space of $h$-harmonics.

Keywords

33C5033C45h-harmonicsintertwining operatorreflection group
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 1 , 01 February 1998 , pp. 193 - 209
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Askey, R., Orthogonal polynomials and special functions.SIAM, Philadelphia, 1975.Google Scholar
2. Dunkl, C., Reflection groups and orthogonal polynomials on the sphere. Math. Z. 197 (1988), 33–60.Google Scholar
3. Dunkl, C., Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1989), 167–183.Google Scholar
4. Dunkl, C., Possion and Cauthy kernels for orthogonal polynomials with dihedra symmetry. J. Math. Anal. Appl. 143 (1989), 459–470.Google Scholar
5. Dunkl, C., Integral kernels with reflection group invariance. Canad. J. Math. 43 (1991), 1213–1227.Google Scholar
6. Dunkl, C., Intertwining operators associated to the group S3. Trans.Amer.Math. Soc. 347 (1995), 3347–3374.Google Scholar
7. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Higher transcendental functions.McGraw-Hill 2 , New York, 1953.Google Scholar
8. Stein, E.M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces.Princeton Univ. Press, Princeton, New Jersey, 1971.Google Scholar
9. Szegő, H., Orthogonal polynomials. 4th ed., Amer.Math. Soc., Colloq. Publ. 23 , Providence, Rhode Island, 1975.Google Scholar
10. Vilenkin, N.J., Special functions and the theory of group representations.Amer. Math. Soc., Trans. Math. Monographs 22 , Providence, Rhode Island, 1968.Google Scholar
11. Xu, Y., On multivariate orthogonal polynomials. SIAM J. Math. Anal. 24 (1993), 783–794.Google Scholar
12. Xu, Y., On orthogonal polynomials in several variables. In: Special functions, q-series, and related topics, The Fields Institute for Research in Math. Sci., Communications Series 14 (1997), 247–270.Google Scholar
13. Xu, Y., Orthogonal polynomials for a family of product weight functions on the spheres. Canad. J. Math. 49 (1997), 175–192.Google Scholar
14. Xu, Y., Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups. Proc. Amer. Math. Soc. 125 (1997), 2963–2973.Google Scholar
15. Xu, Y. , Orthogonal polynomials and cubature formulae on spheres and on balls. SIAM. J. Math. Anal., to appear.Google Scholar