Self-fourier objects and other self-transform objects

A self-Fourier function (SFF), according to Caola [J. Phys. A 24, L1143 (1991)], is a function that is its own Fourier transform. The Gaussian and Dirac combs are well-known examples. Many more SFF’s have been discovered recently by Caola. This discovery might bear some fruit in optics, since the Fourier transform is perhaps the most important theoretical tool in wave optics. We show that Caola discovered all SFF’s. Furthermore, we study other self-transform functions, which are also tied to some transformations that play a role in coherent optics.