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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Integration of the intertwining operator for $h$-harmonic polynomials associated to reflection groups
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by Yuan Xu
Proc. Amer. Math. Soc. 125 (1997), 2963-2973
DOI: https://doi.org/10.1090/S0002-9939-97-03986-5

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.
References
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Bibliographic Information
  • Yuan Xu
  • Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
  • MR Author ID: 227532
  • Email: yuan@math.uoregon.edu
  • 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
  • © Copyright 1997 American Mathematical Society
  • 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