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Hermite distributions associated to the group $O(p,q)$

Author(s): Gerald B. Folland
Journal: Proc. Amer. Math. Soc. 126 (1998), 1751-1763.
MSC (1991): Primary 33E30; Secondary 33C15, 35C05
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Abstract: We calculate the tempered $O(p,q)$ -invariant eigendistributions of the -invariant Hermite operator

$\begin{equation*}-{\textstyle {\frac{1}{2}}}(\Delta _{x}- \Delta _{y}) +{\textstyle {\frac{1}{2}}}(|x|^{2}-|y|^{2})\qquad (x\in \mathbb{R}^{p}, y\in \mathbb{R}^{q}).\end{equation*}$

They are singular on the cone $|x|=|y|$ and are given elsewhere in terms of confluent hypergeometric functions.

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6.: A. Tengstrand, Distributions invariant under an orthogonal group of arbitrary signature, Math. Scand. 8 (1960), 201-218. MR 23:A3450


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