In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Paley–Wiener theorem is a theorem that relates decay properties of a function or
distribution at infinity with
analyticity of its
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
. It is named after
Raymond Paley (1907–1933) and
Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American computer scientist, mathematician, and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology ( MIT). A child prodigy, Wiener late ...
(1894–1964) who, in 1934, introduced various versions of the theorem. 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
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of Degeneracy (mathematics)#T ...
(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
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
and
harmonic analysis; introducing the
Paley–Wiener condition for
spectral factorization and the
Paley–Wiener criterion for
non-harmonic Fourier series respectively. These are related mathematical concepts that place the decay properties of a function in context of
stability problems.
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 mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
in the
upper half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
. One may then expect to differentiate under the integral in order to verify that the
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equatio ...
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
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
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