Quantized linear harmonic oscillator is considered in a truncated,
finite-dimensional matrix form restricted by the requirement of
consistency of the Heisenberg equations of motion under the discrete
version of  Fourier transformation. In this approach the authors are
able to display the explicit numerical matrix form of position,
momentum and number operators in dimensions n=3, n=4 and n=5.

MR2515603 Santhanam, Thalanayar S.; Santhanam, Balu The discrete
Fourier transform and the quantum-mechanical oscillator in a
finite-dimensional Hilbert space. J. Phys. A 42 (2009), no. 20,
205303, 7 pp. 81Qxx (42C15 81S05)