Paley-Wiener theorem

From Academic Kids

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

<math> v(f) = v_x \left(f(x)\right) <math>

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

<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 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 zCn,

<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 CN such that for all zCn,

<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. Their formulations were not in terms of distributions, a concept not at the time available. The formulation presented here is attributed to Lars Hormander.

In another version, the Paley-Wiener theorem explicitly describes the Hardy space <math>H^2(\mathbf{R})<math> using the unitary 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. Hormander, Linear Partial Differential Operators, Springer Verlag, 1976
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