Discrete Bargmann transform

We define a discrete Bargmann transform for discrete and finite functions by means of the coherent states in the su(2) finite harmonic oscillator model. The transform space is over a corresponding finite square mesh in the complex plane. From there, the inverse discrete Bargmann transform reconstitutes the original function with an average error of 10^-7 for functions of unit norm. Three defining properties of the Bargmann transform also hold in this discrete version: (1) the Bargmann transform of finite su(2) harmonic oscillator functions are complex power functions; (2) the discrete su(1,1) repulsive oscillator functions self-reproduce in the chosen finite interval with little error; and (3) the oscillator dynamics under the fractional Fourier transform in Bargmann's complex space correspond, in this discrete transform, to the dynamics under the su(2) fractional Fourier-Kravchuk transform.