Wavelet Transform and Orthogonal Decomposition of L 2 Space on the Cartan Domain Bdi

Let G = ( R∗+ × SO0(1, n) ) n Rn+1 be the Weyl-Poincaré group and KAN be the Iwasawa decomposition of SO0(1, n) with K = SO(n). Then the “affine Weyl-Poincaré group” Ga = ( R∗+ × AN ) n Rn+1 can be realized as the complex tube domain Π = Rn+1 + iC or the classical Cartan domain BDI(q = 2). The square-integrable representations of G and Ga give the admissible wavelets and wavelet transforms. An orthogonal basis {ψk} of the set of admissible wavelets associated to Ga is constructed, and it gives an orthogonal decomposition of L2 space on Π (or the Cartan domain BDI(q = 2)) with every component Ak being the range of wavelet transforms of functions in H2 with ψk .