nLab
Paley-Wiener theorem

Theorem

(Paley-Wiener for C 0 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 FF which satisfy the following estimate: there is a positive constant BB such that for every integer n>0n \gt 0 there is a constant C nC_n such that:

F(z)C n(1+z) nexp(BIm(z)) 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)