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

Abstract: 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 ${\theta_k=2\pi k/N}_{k=0}<sup>{N-1}$ for a given mean number $p$ of particles. The resulting Coherent State (CS) subsystem ${\cal S}={|z_k=p<sup>{1/2}e<sup>{i\theta_k}&gt;}$ 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 $p&lt;N$ and/or $N\to\infty$.