Self-Fourier functions and self-Fourier operators

The concept of self-Fourier functions, i.e., functions that equal their Fourier transform, is almost always associated with specific functions, the most well known being the Gaussian and the Dirac delta comb. We show that there exists an infinite number of distinct families of these functions, and we provide an algorithm for both generating and characterizing their distinct classes. This formalism allows us to show the existence of these families of functions without actually evaluating any Fourier or other transform-type integrals, a task often challenging and frequently not even possible.