Can fourier transform of a function with not empty support be zero on whole range?

Question

We consider $f : [a, b] \subset \mathbb{R} \rightarrow D \subseteq \mathbb{C}$ and support may consist of only one point.

Answer 1

Yes, it can, any function which has a null set as support has zero Fourier transform. This is easy to see from the fact that the Fourier transform is a unitary operator from $L^2$ to $L^2$.