nLab
Paley-Wiener theorem

Theorem

(Paley-Wiener for $C_0^{\mathrm{\infty }}$)

The vector space of smooth compactly supported functions is (algebraically and topologically) isomorphic, via the Fourier transform, to the space of entire functions $F$ which satisfy the following estimate: there is a positive constant $B$ such that for every integer $n>0$ there is a constant $C_n$ such that:

$$F(z)\le C_n(1+\mid z\mid {)}^{-n}\exp (B\mid Im(z)\mid )$$F(z) \le C_n (1 + |z|)^{-n} \exp{ (B \; |\operatorname{Im}(z)|)}