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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Wavelet transform
and orthogonal decomposition of $L^{2}$ space
on the Cartan domain $BDI(q=2)$

Author: Qingtang Jiang
Journal: Trans. Amer. Math. Soc. 349 (1997), 2049-2068
MSC (1991): Primary 22D10, 81R30; Secondary 42C99
MathSciNet review: 1373641
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Abstract: Let $G=\left ({\mathbb {R}}^{*}_{+}\times SO_{0}(1, n)\right ) \ltimes {\mathbb {R}}^{n+1}$ be the Weyl-Poincaré group and $KAN$ be the Iwasawa decomposition of $SO_{0}(1, n)$ with $K=SO(n)$. Then the ``affine Weyl-Poincaré group'' $G_{a}=\left ({\mathbb {R}}^{*}_{+}\times AN\right ) \ltimes {\mathbb {R}}^{n+1}$ can be realized as the complex tube domain $\Pi ={\mathbb {R}}^{n+1}+iC$ or the classical Cartan domain $BDI(q=2)$. The square-integrable representations of $G$ and $G_{a}$ give the admissible wavelets and wavelet transforms. An orthogonal basis $\{ \psi _{k}\}$ of the set of admissible wavelets associated to $G_{a}$ is constructed, and it gives an orthogonal decomposition of $L^{2}$ space on $\Pi $ (or the Cartan domain
$BDI(q=2)$) with every component $A_{k}$ being the range of wavelet transforms of functions in $H^{2}$ with $\psi _{k}$.

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Additional Information

Qingtang Jiang
Affiliation: Department of Mathematics, Peking University, Beijing 100871, P. R. China
Address at time of publication: Department of Mathematics, The National University of Singapore, Lower Kent Ridge Road, Singapore 119260

Keywords: Weyl-Poincaré group, square-integrable representation, wavelet transform, orthogonal decomposition
Received by editor(s): November 20, 1994
Received by editor(s) in revised form: December 2, 1995
Article copyright: © Copyright 1997 American Mathematical Society