Integration of the intertwining operator

for -harmonic polynomials

associated to reflection groups

Author:
Yuan Xu

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2963-2973

MSC (1991):
Primary 33C50, 33C45, 42C10

DOI:
https://doi.org/10.1090/S0002-9939-97-03986-5

MathSciNet review:
1402890

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the intertwining operator with respect to the reflection invariant measure on the unit sphere in Dunkl's theory on spherical -harmonics associated with reflection groups. Although a closed form of is unknown in general, we prove that

where is the unit ball of and is a constant. The result is used to show that the expansion of a continuous function as Fourier series in -harmonics with respect to is uniformly Cesáro summable on the sphere if , provided that the intertwining operator is positive.

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Additional Information

**Yuan Xu**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Email:
yuan@math.uoregon.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03986-5

Keywords:
Orthogonal polynomials in several variables,
sphere,
$h$-harmonics,
reflection groups,
intertwining operator

Received by editor(s):
May 7, 1996

Additional Notes:
Supported by the National Science Foundation under Grant DMS-9500532

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society