The support of a Fourier transform of a function in the positive real axis

Question

I vaguely remember reading a theorem stating that if a function in $L^2(\mathbb{R})$ has support in the positive semiaxis, then the support of the Fourier transform must contain points in the negative semiaxis. Can someone provide a reference for this?

Answer 1

In fact if $f\in L^2$ is supported on $(0,\infty)$ then the Fourier transform extends to be holomorphic in the upper half-plane (hence the boundary values cannot vanish on $(-\infty,0)$ unless $f=0$). See for example the chapter on "Holomorphic Fourier Transforms" in Rudin Real and Complex Analysis.