Fourier signal and image analysis in phase space: The Fourier-Kravchuk and Fourier-Hahn transforms

We give an overview of current research of the Optica Matematica project where we have studied a set of finite Fourier-like transforms based on the finite oscillator model, where the position and momentum spaces are discrete and finite ab initio. These include the fractional Fourier-Kravchuk, Fourier-q-Kravchuk, and Hankel-Hahn transforms. These are based on well-known group-theoretical results from the three- and four-dimensional rotation groups, and lead to unitary evolution, equally-spaced energy levels (bounded from below and from above), a Schrodinger difference equation, and admit a Wigner function which is covariant with the dynamical group of the system.