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. 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.
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
f(\zeta)=
infty | |
\int | |
-infty |
F(x)eidx
and allow
\zeta
f
F
L2(R)
\zeta
eix\zeta
x\to-infty
F
The first such restriction is that
F
R+
F\in
2(R | |
L | |
+) |
F
f(\zeta)=
infty | |
\int | |
0 |
F(x)eidx
for
\zeta
infty | |
\int | |
-infty |
\left|f(\xi+iη)\right|2d\xi\le
infty | |
\int | |
0 |
|F(x)|2dx
and by dominated convergence,
\lim | |
η\to0+ |
infty | |
\int | |
-infty |
\left|f(\xi+iη)-f(\xi)\right|2d\xi=0.
Conversely, if
f
\supη>0
infty | |
\int | |
-infty |
\left|f(\xi+iη)\right|2d\xi=C<infty
then there exists
F\in
2(R | |
L | |
+) |
f
F
In abstract terms, this version of the theorem explicitly describes the Hardy space H2(R)
l{F}H2(R)=L
| ||
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
2(R | |
L | |
+) |
By imposing the alternative restriction that
F
F
[-A,A]
F\inL2(-A,A)
f(\zeta)=
A | |
\int | |
-A |
F(x)eidx
A
C
|f(\zeta)|\leCeA|\zeta|,
and moreover,
f
infty | |
\int | |
-infty |
|f(\xi+iη)|2d\xi<infty.
Conversely, any entire function of exponential type
A
L2
[-A,A]
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution of compact support on
Rn
Cn
Generally, the Fourier transform can be defined for any tempered distribution; moreover, any distribution of compact support
v
v
f
v(f)=v(x\mapstof(x))
is well defined.
It can be shown that the Fourier transform of
v
s
\hat{v}(s)=(2
| ||||
\pi) |
v\left(x\mapstoe-i\right)
and that this function can be extended to values of
s
Cn
Additional growth conditions on the entire function
F
v
Sharper results giving good control over the singular support of
v
K
Rn
H
H(x)=\supy\in\langlex,y\rangle.
Then the singular support of
v
K
N
Cm
|\hat{v}(\zeta)|\le
Ne | |
C | |
m(1+|\zeta|) |
H(Im(\zeta))
for
|Im(\zeta)|\lemlog(|\zeta|+1).