Editing Paley–Wiener theorem (section)

==Holomorphic Fourier transforms==
The classical Paley–Wiener theorems make use of the [[Holomorphic function|holomorphic]] Fourier transform on classes of [[square-integrable function]]s supported on the real line.  Formally, the idea is to take the integral defining the (inverse) Fourier transform

:<math>f(\zeta) = \int_{-\infty}^\infty F(x)e^{i x \zeta}\,dx</math>

and allow <math>\zeta</math> to be a [[complex number]] in the [[upper half-plane]].  One may then expect to differentiate under the integral in order to verify that the [[Cauchy–Riemann equations]] hold, and thus that <math>f</math> defines an analytic function. However, this integral may not be well-defined, even for <math>F</math> in <math>L^2(\mathbb{R})</math>; indeed, since <math>\zeta</math> is in the upper half plane, the modulus of <math>e^{ix\zeta}</math> grows exponentially as <math>x \to -\infty</math>; so [[Leibniz integral rule|differentiation under the integral sign]] is out of the question. One must impose further restrictions on <math>F</math> in order to ensure that this integral is well-defined.

The first such restriction is that <math>F</math> be supported on <math>\mathbb{R}_+</math>: that is, <math>F\in L^2(\mathbb{R}_+)</math>.  The Paley–Wiener theorem now asserts the following:<ref>{{harvnb|Rudin|1987|loc=Theorem 19.2}}; {{harvnb|Strichartz|1994|loc=Theorem 7.2.4}}; {{harvnb|Yosida|1968|loc=§VI.4}}</ref> The holomorphic Fourier transform of <math>F</math>, defined by

:<math>f(\zeta) = \int_0^\infty F(x) e^{i x\zeta}\, dx</math>

for <math>\zeta</math> in the [[upper half-plane]] is a holomorphic function. Moreover, by [[Plancherel's theorem]], one has

:<math>\int_{-\infty}^\infty \left |f(\xi+i\eta) \right|^2\, d\xi \le \int_0^\infty |F(x)|^2\, dx</math>

and by [[dominated convergence]], 

:<math>\lim_{\eta\to 0^+}\int_{-\infty}^\infty \left|f(\xi+i\eta)-f(\xi) \right|^2\,d\xi = 0.</math>

Conversely, if <math>f</math> is a holomorphic function in the upper half-plane satisfying

:<math>\sup_{\eta>0} \int_{-\infty}^\infty \left |f(\xi+i\eta) \right|^2\,d\xi = C < \infty</math>

then there exists <math>F\in L^2(\mathbb{R}_+)</math> such that <math>f</math> is the holomorphic Fourier transform of <math>F</math>.

In abstract terms, this version of the theorem explicitly describes the [[Hardy space]] [[H square|<math>H^2(\mathbb{R})</math>]]. The theorem states that

:<math> \mathcal{F}H^2(\mathbb{R})=L^2(\mathbb{R_+}).</math>

This is a very useful result as it enables one to pass to the Fourier transform of a function in the Hardy space and perform calculations in the easily understood space 
<math>L^2(\mathbb{R}_+)</math> of square-integrable functions supported on the positive axis.

By imposing the alternative restriction that <math>F</math> be [[compact support|compactly supported]], one obtains another Paley–Wiener theorem.<ref>{{harvnb|Rudin|1987|loc=Theorem 19.3}}; {{harvnb|Strichartz|1994|loc=Theorem 7.2.1}}</ref> Suppose that <math>F</math> is supported in <math>[-A,A]</math>, so that <math>F\in L^2(-A,A)</math>.  Then the holomorphic Fourier transform

:<math>f(\zeta) = \int_{-A}^A F(x)e^{i x\zeta}\,dx</math>

is an [[entire function]] of [[exponential type]] <math>A</math>, meaning that there is a constant <math>C</math> such that

:<math>|f(\zeta)|\le Ce^{A|\zeta|},</math>

and moreover, <math>f</math> is square-integrable over horizontal lines:

:<math>\int_{-\infty}^{\infty} |f(\xi+i\eta)|^2\,d\xi < \infty.</math>

Conversely, any entire function of exponential type <math>A</math> which is square-integrable over horizontal lines is the holomorphic Fourier transform of an 
<math>L^2</math> function supported in <math>[-A,A]</math>.