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Admissible wavelets associated with the Heisenberg group. (English) Zbl 0885.22012
Let $$NAK$$ be the Iwasawa decomposition of the group $$SU (n+1, 1)$$. The Iwasawa subgroup $$P= AN$$ can be identified with the generalized upper half-plane $$U^{n+1}$$ and has a natural representation $$U$$ on the $$L^2$$-space of the Heisenberg 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 of $$U$$ on these subspaces are square-integrable. We characterize the admissible condition in terms of the Fourier transform and define the wavelet transform with respect to admissible wavelets. The wavelet transform leads to isometric operators from the irreducible invariant closed subspaces of $$L^2 (H^n)$$ to $$L^{2, \nu} (U^{n+ 1})$$, the weighted $$L^2$$-spaces on $$U^{n+ 1}$$. By selecting a set of mutual orthogonal admissible wavelets, we get the direct sum decomposition of $$L^{2, \nu} (U^{n+ 1})$$ with the first component $$A^\nu (U^{n+ 1})$$, the (weighted) Bergman space.
Reviewer: H.Liu (Beijing)

##### MSC:
 22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 43A85 Harmonic analysis on homogeneous spaces 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010) 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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