Admissible wavelets and inverse radon transofrm associated with the affine homogeneous Siegel domains of type II

Abstract

Let D(Ω,Φ) be the affine homogeneous Siegel domain of type II, whose Silov boundary N is a nilpotent Lie group of step two. In this article, we develop the theory of wavelet analysis on N . By selecting a set of mutual orthogonal wavelets we give a direct sum decomposition of L2(D(Ω,Φ)), the first component A0,0 of which is the Bergman space. Moreover, we study the Radon transform on N , and obtain an inversion formula R−1 = (π)−2dLRL which is an extension of that by Strichartz [R. S. Strichartz, L harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal. 96 (1991), 350–406.]. We devise a subspace of L2(N) on which the Radon transform is a bijection. Using wavelet inverse transform, we establish an inversion formula of the Radon transform in the weak sense.