Fourier transform of a function with compact support on an interval

Question

is it possible to have a function $ f(x) $ with compact support $ [a,b] $

whose Fourier transform has a compact support on $ [c,d] $

here $ [a,b] $ and $ [c,d] $ are an interval or a disjoint union of intervals

Answer 1

This issue is an easy case of "Paley-Wiener" -type theorems. In particular, the answer is "no": the Fourier transform of a compactly-supported $L^1$ function (so the Fourier transform exists as a literal, absolutely convergent integral) extends to an entire function on $\mathbb C$. In particular, it cannot be compactly supported unless it's identically $0$.

In fact, the same conclusion holds for compactly-supported distributions (which are provably tempered, so have Fourier transforms, even if not defined by the literal integrals).

So, in a strong sense, Fourier transforms of compactly supported "functions" are never compactly supported (unless identically $0$).