# Paley-Wiener theorem

(Redirected from Paley–Wiener theorem)

A function $f \in L _ {2} ( - \infty , + \infty )$ vanishes almost everywhere outside an interval $[ - A , A ]$ if and only if its Fourier transform

$$F ( y) = \ \int\limits _ {- \infty } ^ { {+ } \infty } f ( x) e ^ {ixy} d x ,\ y \in \mathbf R ,$$

satisfies

$$\int\limits _ {- \infty } ^ { {+ } \infty } | F ( y) | ^ {2} d y < \infty$$

and is the restriction to the real line of a certain entire analytic function $F ( z)$ of a complex variable $z$ satisfying $| F ( z) | \leq e ^ {A | z | }$ for all $z \in \mathbf C$( see ). A description of the image of a certain space of functions or generalized functions on a locally compact group under the Fourier transform or under some other injective integral transform is called an analogue of the Paley–Wiener theorem; the most frequently encountered analogues of the Paley–Wiener theorem are a description of the image of the space $C _ {0} ^ \infty ( G)$ of infinitely-differentiable functions of compact support and a description of the image of the space $S ( G)$ of rapidly-decreasing infinitely-differentiable functions on a locally compact group $G$ under the Fourier transform on $G$. Such analogues are known, in particular, for Abelian locally compact groups, for certain connected Lie groups, for certain subalgebras of the algebra $C _ {0} ^ \infty ( G)$ on real semi-simple Lie groups, and also for certain other integral transforms.

How to Cite This Entry:
Paley–Wiener theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Paley%E2%80%93Wiener_theorem&oldid=22884