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Analytic Features of Reproducing Groups for the Metaplectic Representation

Authors:	Elena Cordero Filippo De Mari Krzysztof Nowak Anita Tabacco

Institution:	(1) Dipartimento di Matematica, Universita di Torino, Via Carlo Alberto, 10 10123 Torino, Italy;(2) Dipartimento di Matematica, Universita di Genova, Via Dodecaneso, 35 16146 Genova, Italy;(3) Department of Computer Science, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA;(4) Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24 10129 Torino, Italy

Abstract:	We introduce the notion of admissible subgroup of $G={\mathbb H}^d\rtimes Sp(d,{\mathbb R})$ relative to the (extended) metaplectic representation $\mu_e$ via the Wigner distribution. Under mild additional assumptions, it is shown to be equivalent to the fact that the identity $f=\int_{H}\langle f,\mu_e(h)\phi\rangle\mu_e(h)\phidh$ holds (weakly) for all $f\in L^2({\mathbb R}^d).$ We use this equivalence to exhibit classes of admissible subgroups of $Sp(2,{\mathbb R}).$ We also establish some connections with wavelet theory, i.e., with curvelet and contourlet frames.

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