# nLab Paley-Wiener theorem

###### Theorem

(Paley-Wiener for ${C}_{0}^{\infty }$)

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:

$F\left(z\right)\le {C}_{n}\left(1+\mid z\mid {\right)}^{-n}\mathrm{exp}\left(B\phantom{\rule{thickmathspace}{0ex}}\mid Im\left(z\right)\mid \right)$F(z) \le C_n (1 + |z|)^{-n} \exp{ (B \; |\operatorname{Im}(z)|)}
Created on April 28, 2010 18:04:25 by Urs Schreiber (131.211.233.6)