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Categories: Theorems | Mathematical analysis

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

$v(f) = v_x \left(f(x)\right)$

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

$\hat{v}(s) = (2 \pi)^{-n/2} v_x\left(e^{-i \langle x, s\rangle}\right)$

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 ∈ Cⁿ,

$|F(z)| \leq C (1 + |z|)^N e^{B| \mathfrak{Im} z|}$

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 ∈ Cⁿ,

$|F(z)| \leq C_N (1 + |z|)^{-N} e^{B| \mathfrak{Im} z|}$

then v is infinitely differentiable and conversely.

The theorem is named for Raymond Paley (1907 - 1933) and Norbert Wiener. 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 $H^2(\mathbf{R})$ 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

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