ADMISSIBLE WAVELETS ASSOCIATED WITH THE HEISENBERG-GROUP

Let NAK be the Iwasawa decomposition of group SU(n + 1, 1). The Iwasaw a subgroup P = NA can be identified with the generalized upper half-pl ane Un+1 and has a natural representation U on the L-2-space of the He isenberg group L-2(H-n). We decompose L-2(H-n) into the direct sum of the irreducible invariant closed subspaces under U. The restrictions o f U on these subspaces are square-integrable. We characterize the admi ssible condition in terms of the Fourier transform and define the wave let transform with respect to admissible wavelets. The wavelet transfo rm leads to isometric operators from the irreducible invariant closed subspaces of L-2(Hn) to L-2,L-v(Un+1), the weighted L-2-spaces on Un+1 . By selecting a set of mutual orthogonal admissible wavelets, we get the direct sum decomposition of L-2,L-v(Un+1) With the first component A(v)(Un+1), the (weighted) Bergman space.