PlanetMath: Paley-Wiener theorem

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Paley-Wiener theorem

(Theorem)

Let be an entire function such that $\vert f(z)\vert \leq K e^{\gamma \vert z\vert}$ for some $K \geq 0$ and $\gamma > 0$ . If the restriction of to the real line is in $L^2(\mathbb{R})$ , then there exists a function $g(t)\in L^2(-\gamma, \gamma)$ such that

$\displaystyle f(z) = \frac{1}{\sqrt{2\pi}}\int_{-\gamma}^{\gamma}g(t)e^{izt}dt$

for all

"Paley-Wiener theorem" is owned by Gorkem.

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Cross-references: function, line, real, restriction, entire function

This is version 11 of Paley-Wiener theorem, born on 2005-07-27, modified 2007-05-25.
Object id is 7275, canonical name is PaleyWienerTheorem.
Accessed 2860 times total.

Classification:

AMS MSC:

30E99 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Miscellaneous)

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