Reproducing groups on analytic featuresfor the metaplectic representation

We introduce the notation of admissible subgroup H of G=R^((1+ε))⋊Sp(1+ε,R) relative to the (extended) metapletic representation μ_e via the wigner distribution. Under mild additional assumptions, it is shown to be equivalent to the fact that the identity ∑_j▒f_j =∫_H▒∑_j▒〈f_j,μ_e (h_j ) ϕ_j 〉 μ_e (h_j)ϕ_j dh_j holds (weakly) for all f_j∈L^2 (R^((1+ε))). They used this equivalence to exhibit classes of admissible subgroups of Sp(2,R) by E. Cordero, F. Damai, K. Nowak, and A. Tabacco [20]. We also eastablish some connections with wavelet theory, i.e., with curvelet and contourlet frames.