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"Coherent state(s) quantization'' or "frame quantization'' are generic phrases for naming a certain point of view in analyzing a set X equipped with a measure. The approach matches what physicists understand by quantization when the "observed'' measure space X has a phase space or symplectic structure. It matches also well established approaches by signal analysts, like wavelet analysis. The set X can be finite, countably infinite, or uncountably infinite. The approach is generically simple, of Hilbertian essence, and always the same: one starts from the Hilbert space of complex square integrable functions on X. One chooses an orthonormal set O of vectors in it, satisfying some finiteness condition and a ``companion" Hilbert space H (the space of ``quantum states'') with orthonormal basis in one-to-one correspondence with the elements of O. There results a family C of states |x> (the "coherent states'') in H, which are labelled by elements of X and which resolve the unity operator in H. This is the departure point for analysing the original set and functions living on it from the point of view of the frame (in its true sense) C. We end in general with a non-commutative algebra of operators in H, set aside the usual questions of domains in the infinite dimensional case. There is a kind of manifest universality in this approach. Changing the frame family C produces another quantization, another point of view, possibly equivalent to the previous one, possibly not. Various examples of such explorations will be presented. |
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