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The mathematical description of the quantum harmonic oscillator isessentially based on the Gaussian function. In the case of a quantum oscillatorwith finite-dimensional Hilbert space, the position space consists in a finitenumber of points, and the ground state can be regarded as a finite version ofthe Gaussian function. Conversely, each finite Gaussian can be used in order todefine some finite oscillators. We investigate certain finite Gaussians asconcern their Fourier transform, Wigner function and associated oscillators.

Gaussian type functions defined on a finite set and finite oscillators