Paley–Wiener theorem

In mathematics, the Paley–Wiener theorem relates growth properties of entire functions on Cⁿ and Fourier transformation of Schwartz distributions 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

:$v(f)\; =\; v\_x\; left(f(x)\; ight)$

is well defined. In the above expression the variable "x" in "v_x" 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 "C_N" 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) 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 $H^2(mathbf\{R\})$ 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