Editing Paley–Wiener theorem (section)

==Schwartz's Paley–Wiener theorem==
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a [[distribution (mathematics)|distribution]] of [[compact support]] on <math>\mathbb{R}^n</math> is an [[entire function]] on <math>\mathbb{C}^n</math> and gives estimates on its growth at infinity.  It was proven by [[Laurent Schwartz]] ([[#CITEREFSchwartz1952|1952]]). The formulation presented here is from {{harvtxt|Hörmander|1976}}.

Generally, the Fourier transform can be defined for any [[distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]]; moreover, any distribution of compact support <math>v</math> is a tempered distribution. If <math>v</math> is a distribution of compact support and <math>f</math> is an infinitely differentiable function, the expression

:<math> v(f) = v(x\mapsto f(x)) </math>

is well defined.

It can be shown that the Fourier transform of <math>v</math> is a function (as opposed to a general tempered distribution) given at the value <math>s</math> by

:<math> \hat{v}(s) = (2 \pi)^{-\frac{n}{2}} v\left(x\mapsto e^{-i \langle x, s\rangle}\right)</math>

and that this function can be extended to values of <math>s</math> in the complex space <math>\mathbb{C}^n</math>.  This extension of the Fourier transform to the complex domain is called the [[Fourier–Laplace transform]].

{{math theorem|name=Schwartz's theorem| math_statement=An entire function <math>F</math> on <math>\mathbb{C}^n</math> is the Fourier–Laplace transform of a distribution <math>v</math> of compact support if and only if for all <math>z\in\mathbb{C}^n</math>,

:<math> |F(z)| \leq C (1 + |z|)^N e^{B|\text{Im}(z)|} </math>

for some constants <math>C</math>, <math>N</math>, <math>B</math>.  The distribution <math>v</math> in fact will be supported in the closed ball of center <math>0</math> 
and radius <math>B</math>.}}

Additional growth conditions on the entire function <math>F</math> impose regularity properties on the distribution <math>v</math>. 
For instance:<ref>{{harvnb|Strichartz|1994|loc=Theorem 7.2.2}}; {{harvnb|Hörmander|1990|loc=Theorem 7.3.1}}</ref>

{{math theorem|math_statement= If for every positive <math>N</math> there is a constant <math>C_N</math> such that for all <math>z\in\mathbb{C}^n</math>,

:<math> |F(z)| \leq C_N (1 + |z|)^{-N} e^{B|\mathrm{Im}(z)|} </math>

then <math>v</math> is an infinitely differentiable function, and vice versa.}}

Sharper results giving good control over the [[singular support]] of <math>v</math> have been formulated by {{harvtxt|Hörmander|1990}}. In particular,<ref>{{harvnb|Hörmander|1990|loc=Theorem 7.3.8}}</ref> let <math>K</math> be a convex [[Compact space|compact set]] in <math>\mathbb{R}^n</math> with supporting function <math>H</math>, defined by

:<math>H(x) = \sup_{y\in K} \langle x,y\rangle.</math>  

Then the singular support of <math>v</math> is contained in <math>K</math> [[if and only if]] there is a constant <math>N</math> and sequence of constants <math>C_m</math> such that

:<math>|\hat{v}(\zeta)| \le C_m(1+|\zeta|)^Ne^{H(\mathrm{Im}(\zeta))}</math>

for <math>|\mathrm{Im}(\zeta)| \le m \log(| \zeta |+1).</math>