- Paley–Wiener theorem
In
mathematics , the Paley–Wiener theorem relates growth properties ofentire function s on Cn andFourier transform ation ofSchwartz distribution s ofcompact 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:v(f) = v_x left(f(x) ight)
is well defined. In the above expression the variable "x" in "vx" 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
:hat{v}(s) = (2 pi)^{-n/2} v_xleft(e^{-i langle x, s angle} ight)
and that this function can be extended to values of "s" in the complex space Cn. This extension of the Fourier transform to the complex domain is called the
Fourier-Laplace transform .Theorem. An entire function "F" on Cn is the Fourier-Laplace transform of distribution "v" of compact support if and only if for all "z" ∈ C"n",
:F(z)| leq C (1 + |z|)^N e^{B| mathfrak{Im} z
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 "CN" such that for all "z" ∈ C"n",
:F(z)| leq C_N (1 + |z|)^{-N} e^{B| mathfrak{Im} z
then "v" is infinitely differentiable and conversely.
The theorem is named for
Raymond Paley (1907 - 1933) andNorbert Wiener (1894 - 1964). Their formulations were not in terms of distributions, a concept not at the time available. The formulation presented here is attributed toLars Hörmander .In another version, the Paley–Wiener theorem explicitly describes the
Hardy space H^2(mathbf{R}) using the unitary Fourier transform mathcal{F}. The theorem states that:mathcal{F}H^2(mathbf{R})=L^2(mathbf{R_+}).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 L^2(mathbf{R_+}) 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
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