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

摘要

Let D(Omega, Phi) 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 L-2(D(Omega, Phi)), the first component A(0,0)(0) of which is the Bergman space. Moreover, we study the Radon transform on N, and obtain an inversion formula R-1 = (pi)(-2d) LRL which is an extension of that by Strichartz [R. S. Strichartz, LP harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal. 96 (1991), 350-406.]. We devise a subspace of L-2(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.