Almost Complete Coherent State Subsystems and Partial Reconstruction of Wave Functions in the Fock-Bargmann Phase-Number Representation

We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number $N$ of phase samples $\{\theta_k=2\pi k/N\}_{k=0}^{N-1}$ for a given mean number $p$ of particles. The resulting Coherent State (CS) subsystem ${\cal S}=\{|z_k=p^{1/2}e^{i\theta_k}>\}$ is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, ${\cal S}$ is "almost complete" (a \textit{pseudo-frame}) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when $p<N$ and/or $N\to\infty$.