I have seen multiple posts about calculating fixed points of the Fourier transform $\mathcal{F}$. Some examples include the Gaussian function, and hyperbolic secant. Moreover, this post gives an easy procedure to take any suitable function $f$ and produce a function $g$ that is a fixed point. Namely, $g = f + \mathcal{F}[f] +\mathcal{F}^2[f]+\mathcal{F}^3[f]$. This post also list a lot of other more perhaps esoteric examples of fixed points.
My question is there anything special about these functions being the fixed point of the Fourier transform? Is being a fixed point useful for answering some questions elsewhere in mathematics? Obviously, the Gaussian function is important, but that isn't because its a fixed point of the Fourier transform. Is there some usefulness to finding and knowing about these Fourier fixed points? Perhaps an application? Or are they just interesting pieces of mathematics for mathematics sake?