Integration of the intertwining operator for $h$-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 $V$ be the intertwining operator with respect to the reflection invariant measure $h_{\alpha }^{2} d\omega$ on the unit sphere $S^{d-1}$ in Dunkl’s theory on spherical $h$-harmonics associated with reflection groups. Although a closed form of $V$ is unknown in general, we prove that \begin{equation*} \int _{S^{d-1}} Vf(\mathbf {y}) h_{\alpha }^{2}(\mathbf {y}) d\omega = A_{\alpha }\int _{B^{d}} f(\mathbf {x})(1-|\mathbf {x}|^{2})^{|\alpha |_{1} -1} d\mathbf {x}, \end{equation*} where $B^{d}$ is the unit ball of $\mathbb {R}^{d}$ and $A_{\alpha }$ is a constant. The result is used to show that the expansion of a continuous function as Fourier series in $h$-harmonics with respect to $h_{\alpha }^{2} d\omega$ is uniformly Cesáro $(C, \delta )$ summable on the sphere if $\delta > |\alpha |_{1} + (d-2)/2$, 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

MR Author ID:
227532

Email:
yuan@math.uoregon.edu

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