Paley–Wiener theorem
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In [[mathematics]], the '''Paley–Wiener theorem''' relates growth properties of [[entire function]]s on '''C'''<sup>n</sup> and [[Fourier transform]]ation of [[Schwartz distribution]]s of [[compact support]].
Generally, the Fourier transform can be defined for any [[tempered distribution]]; moreover, any distribution of compact support ''v'' is a tempered distribution. If ''v'' is a distribution of compact support and ''f'' is an infinitely differentiable function, the expression
:<math> v(f) = v_x \left(f(x)\right) </math>
is well defined. In the above expression the variable ''x'' in ''v<sub>x</sub>'' is a dummy variable and indicates that the distribution is to be applied with the argument function considered as a function of ''x''.
It can be shown that the Fourier transform of ''v'' is a function (as opposed to a general tempered distribution) given at the value ''s'' by
:<math> \hat{v}(s) = (2 \pi)^{-n/2} v_x\left(e^{-i \langle x, s\rangle}\right)</math>
and that this function can be extended to values of ''s'' in the complex space '''C'''<sup>n</sup>. This extension of the Fourier transform to the complex domain is called the [[Fourier-Laplace transform]].
'''Theorem'''. An entire function ''F'' on '''C'''<sup>n</sup> is the Fourier-Laplace transform of distribution ''v'' of compact support if and only if for all ''z'' ∈ '''C'''<sup>''n''</sup>,
:<math> |F(z)| \leq C (1 + |z|)^N e^{B| \mathfrak{Im} z|} </math>
for some constants ''C'', ''N'', ''B''. The distribution ''v'' in fact will be supported in the closed ball of center 0 and radius ''B''.
Additional growth conditions on the entire function ''F'' impose regularity properties on the distribution ''v'': For instance, if for ''every'' positive ''N'' there is a constant ''C<sub>N</sub>'' such that for all ''z'' ∈ '''C'''<sup>''n''</sup>,
:<math> |F(z)| \leq C_N (1 + |z|)^{-N} e^{B| \mathfrak{Im} z|} </math>
then ''v'' is infinitely differentiable and conversely.
The theorem is named for [[Raymond Paley]] (1907 - 1933) and [[Norbert Wiener]] (1894 - 1964). Their formulations were not in terms of distributions, a concept not at the time available. The formulation presented here is attributed to [[Lars Hörmander]].
In another version, the '''Paley–Wiener theorem''' explicitly describes the [[Hardy space]] <math>H^2(\mathbf{R})</math> using the unitary [[continuous Fourier transform|Fourier transform]] <math>\mathcal{F}</math>. The theorem states that
:<math> \mathcal{F}H^2(\mathbf{R})=L^2(\mathbf{R_+})</math>.
This is a very useful result as it enables one pass to the Fourier transform of a function in the Hardy space and perform calculations in the easily understood space <math>L^2(\mathbf{R_+})</math> of square-integrable functions supported on the positive axis.
==References==
See section 3 Chapter VI of
* K. Yosida, ''Functional Analysis'', Academic Press, 1968
See also Theorem 1.7.7 in
* L. Hörmander, ''Linear Partial Differential Operators'', Springer Verlag, 1976
See Paley–Wiener Theorems (7.22 - 7.23) of:
* W. Rudin, ''Functional Analysis'', McGraw-Hill Book Company, 1973 First Edition
[[Category:Mathematical theorems]]
[[Category:Generalized functions]]
[[Category:Complex analysis]]
[[Category:Fourier analysis]]
[[Category:Hardy spaces]]
[[ru:Теорема Палея–Винера]]