Let $\mathcal{F[\omega]}$ denote the space of functions whose Fourier transforms are supported in $[-\omega,\omega]$.
Is $\mathcal{F[\omega]}$ closed under composition? If not, what is its closure?
What about $\bigcup_{0<\omega<\infty} \mathcal{F[\omega]}$?
Do the answers generalize to maps between arbitrary-dimensional Euclidean spaces?
Edit: I realized that for $\mathcal{F[\omega]}$ to have a chance of being closed under composition, the functions' Fourier transforms should be bounded in amplitude somehow. I guess that bounding the $\mathbf{L}^2$ norm (of the transform) should be enough, but I'm not yet sure...