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:[3] 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.[4] 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 .