Finite systems,fractional Fourier transforms and their finite phase spaces

Various harmonic oscillator models define — in a sense to be explained here — fractional Fourier transforms (up to a phase). The fractionalization of the Fourier integral transform is well understood; the finite case is less. There are several discrete and finite oscillator models that contract to the continuous, integral model. The Ankara model can be thought as a ring of point masses joined by springs to their equilibrium positions and to each other; the Cuernavaca model uses the su(2) algebra with a distinct physical interpretation. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.