# 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 [1]). 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.

#### References

 [1] N. Wiener, R.E.A.C. Paley, "Fourier transforms in the complex domain" , Amer. Math. Soc. (1934) [2] V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1977) (Translated from Russian) [3] I.M. Gel'fand, M.I. Graev, N.Ya. Vilenkin, "Generalized functions" , 5. Integral geometry and representation theory , Acad. Press (1966) (Translated from Russian) [4] D.P. Zhelobenko, "Harmonic analysis of functions on semi-simple complex Lie groups" , Moscow (1974) (In Russian) [5] W. Rudin, "Functional analysis" , McGraw-Hill (1973)

Let $\phi \in C _ {0} ^ \infty ( \mathbf R )$ with $\supp \phi \subset [- A, A]$. Then the Fourier transform $\widehat \phi$ of $\phi$ can be extended to an entire analytic function on $\mathbf C$ satisfying: for any integer $m \geq 0$ there is a constant $c _ {m} > 0$ such that for all $w \in \mathbf C$,
$$\tag{* } | \widehat \phi ( w) | \leq c _ {m} ( 1+ | w | ) ^ {-} m e ^ {2 \pi A | \mathop{\rm Im} w | } .$$
Conversely, let $F: \mathbf C \rightarrow \mathbf C$ be an entire function which satisfies (*) (replacing $\widehat \phi$ with $F$), for some $A > 0$. Then there exists a $\phi \in C _ {0} ^ \infty ( \mathbf R )$ with $\supp \phi \subset [- A, A]$ and $\widehat \phi = F$.