Discretization of the phase space for a q-deformed harmonic oscillator   with q a root of unity
  The ``position'' and ``momentum'' operators for the q-deformed oscillator with q being a root of unity are proved to have discrete eigenvalues which are roots of deformed Hermite polynomials. The Fourier transform connecting the ``position'' and ``momentum'' representations is also found   The phase space of this oscillator has a lattice structure, which is a non-uniformly distributed grid. Non-equidistant lattice structures also occur in the cases of the truncated harmonic oscillator and of the q-deformed parafermionic oscillator, while the parafermionic oscillator corresponds to a uniformly distributed grid. 
D. Bonatsos, C. Daskaloyannis, D. Ellinas and A. Faessler