Paley–Wiener theorem
In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform.  The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964).  The original theorems did not use the language of distributions, and instead applied to square-integrable functions.  The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration).

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

and allow  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  defines an analytic function. However, this integral may not be well-defined, even for  in ; indeed, since  is in the upper half plane, the modulus of  grows exponentially as ; so differentiation under the integral sign is out of the question. One must impose further restrictions on  in order to ensure that this integral is well-defined.

The first such restriction is that  be supported on : that is, .  The Paley–Wiener theorem now asserts the following: The holomorphic Fourier transform of , defined by

for  in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has

and by dominated convergence, 

Conversely, if  is a holomorphic function in the upper half-plane satisfying

then there exists  such that  is the holomorphic Fourier transform of .

In abstract terms, this version of the theorem explicitly describes the Hardy space . The theorem states that

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 
 of square-integrable functions supported on the positive axis.

By imposing the alternative restriction that  be compactly supported, one obtains another Paley–Wiener theorem. Suppose that  is supported in , so that .  Then the holomorphic Fourier transform

is an entire function of exponential type , meaning that there is a constant  such that

and moreover,  is square-integrable over horizontal lines:

Conversely, any entire function of exponential type  which is square-integrable over horizontal lines is the holomorphic Fourier transform of an 
 function supported in .

Schwartz's Paley–Wiener theorem
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution of compact support on  is an entire function on  and gives estimates on its growth at infinity.  It was proven by Laurent Schwartz (1952). The formulation presented here is from .

Generally, the Fourier transform can be defined for any tempered distribution; moreover, any distribution of compact support  is a tempered distribution. If  is a distribution of compact support and  is an infinitely differentiable function, the expression

is well defined.

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

and that this function can be extended to values of  in the complex space .  This extension of the Fourier transform to the complex domain is called the Fourier–Laplace transform.

Additional growth conditions on the entire function  impose regularity properties on the distribution . 
For instance:

Sharper results giving good control over the singular support of  have been formulated by . In particular, let  be a convex compact set in  with supporting function , defined by

  

Then the singular support of  is contained in  if and only if there is a constant  and sequence of constants  such that

for

Notes

References
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Theorems in Fourier analysis
Generalized functions
Theorems in complex analysis
Hardy spaces