Abstract
We introduce the notion of admissible subgroup \(H\) 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 \(f=\int_{H}\langle f,\mu_e(h)\phi\rangle\mu_e(h)\phi\;dh\) 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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Cordero, E., De Mari, F., Nowak, K. et al. Analytic Features of Reproducing Groups for the Metaplectic Representation. J Fourier Anal Appl 12, 157–180 (2006). https://doi.org/10.1007/s00041-005-5016-7
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DOI: https://doi.org/10.1007/s00041-005-5016-7