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Wavelet transform and orthogonal decomposition of

Authors:	Qingtang Jiang

Institution:	Department of Mathematics, Peking University, Beijing 100871, P. R. China

Abstract:	Let $G=\left ({\mathbb {R}}^{}_{+}\times SO_{0}(1, n)\right ) \ltimes {\mathbb {R}}^{n+1}$ be the Weyl-Poincaré group and be the Iwasawa decomposition of $SO_{0}(1, n)$ with . 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 . The square-integrable representations of 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 ) with every component $A_{k}$ being the range of wavelet transforms of functions in $H^{2}$ with $\psi _{k}$ .

Keywords:	Weyl-Poincaré group square-integrable representation wavelet transform orthogonal decomposition

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