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. 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. 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

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; so differentiation under the integral sign in order to ensure that this integral is well-defined.

. The Paley–Wiener theorem now asserts the following:, defined by

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

and by dominated convergence,

is a holomorphic function in the upper half-plane satisfying

.

In abstract terms, this version of the theorem explicitly describes the Hardy space H^2(\mathbb{R}). The theorem states that

of square-integrable functions supported on the positive axis.

be compactly supported, one obtains another Paley–Wiener theorem.. Then the holomorphic Fourier transform

is an entire function of exponential type such that

is square-integrable over horizontal lines:

.

Schwartz's Paley–Wiener theorem

Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution of compact support is an entire function 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 is an infinitely differentiable function, the expression

is well defined.

by

. This extension of the Fourier transform to the complex domain is called the Fourier–Laplace transform.

. For instance:

Sharper results giving good control over the singular support have been formulated by . In particular, be a convex compact set, defined by

if and only if such that

Notes

References

• .

• .

• .

• .