# nLab Paley-Wiener theorem

###### Theorem

(Paley-Wiener for $C^{\infty}_0$)

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 $n \gt 0$ there is a constant $C_n$ such that:

$F(z) \le C_n (1 + |z|)^{-n} \exp{ (B \; |\operatorname{Im}(z)|)}$
Created on April 28, 2010 18:04:25 by Urs Schreiber (131.211.233.6)