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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 1402890
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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
Email: yuan@math.uoregon.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03986-5
PII: S 0002-9939(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