See coherent state.
Bargmann-Segal transform is the integral transform whose kernel is the overlap between the projective measure corresponding to the coherent states and a measure corresponding to an orthonormal basis comeing from some polarization for -sections. The kernel is a special case of a Bergman kernel? in complex analysis.
Classical case of Heisenberg group:
Vectors in the Hilbert space can be represented in the coherent state representation: ; if is in then is a holomorphic function and this passage is called the Bargmann-Segal transform (referring to Irving Segal); this way certain Hilbert space of holomorphic function appears, the Bargmann-Fock space.
Further generalization is to Perelomov coherent states.
More recent generalized Segal-Bargmann transform of Hall: