Generalized shift-invariant systems and frames for subspaces
نویسندگان
چکیده
Given a real and invertible d×d matrix C, we define for k ∈ Zd a generalized translation operator TCk acting on f ∈ L 2(Rd) by (TCkf)(x) = f(x − Ck), x ∈ R . A generalized shift-invariant system is a system of the type {TCjkφj}j∈J,k∈Zd , where {Cj}j∈J is a countable collection of real invertible d×d matrices, and {φj}j∈J ⊂ L 2(Rd). Generalized shift-invariant systems contain the classical wavelet systems and Gabor systems as special cases. Given the matrices {Cj}j∈J , we are interested in functions {φj}j∈J , {φ̃j}j∈J ⊂ L 2(Rd) for which
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