In mathematics, a Paley–Wiener theorem is a theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform.
It is named after Raymond Paley (1907–1933) and Norbert Wiener (1894–1964) who, in 1934, introduced various versions of the theorem.
[1] 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).
The original work by Paley and Wiener is also used as a namesake in the fields of control theory and harmonic analysis; introducing the Paley–Wiener condition for spectral factorization and the Paley–Wiener criterion for non-harmonic Fourier series respectively.
[2] These are related mathematical concepts that place the decay properties of a function in context of stability problems.
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
One may then expect to differentiate under the integral in order to verify that the Cauchy–Riemann equations hold, and thus that
; so differentiation under the integral sign is out of the question.
The Paley–Wiener theorem now asserts the following:[3] The holomorphic Fourier transform of
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
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.
Then the holomorphic Fourier transform is an entire function of exponential type
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
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution of compact support on
is an infinitely differentiable function, the expression is well defined.
is a function (as opposed to a general tempered distribution) given at the value
Schwartz's theorem — An entire function
in fact will be supported in the closed ball of center
Additional growth conditions on the entire function
impose regularity properties on the distribution
is an infinitely differentiable function, and vice versa.
Sharper results giving good control over the singular support of