I read today (ref) that the Continuous Fourier Transform has four eigenvalues: +1, +i, -1, and -i. Associated with each eigenvalue is a space of eigenfunctions: functions which retain their form after undergoing the Fourier transform. Perhaps the best known example is the Gaussian: the Fourier transform of a Gaussian is again Gaussian.
A more general example is the Hermite-Gauss functions (Gaussian multiplied by Hermite polynomial). These are also eigenfunctions of the Fourier transform, which is why TEMxx modes (with Hermite-Gauss transverse profiles) are stable modes of laser beam propagation.
This leads me to wonder a number of things:
- How do I show that the Fourier transform has just these four eigenvalues?
- Given an arbitrary function, how do I find the projections onto the four eigenspaces?