Paley-Wiener theorem

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

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

for all $z$.

Title	Paley-Wiener theorem
Canonical name	PaleyWienerTheorem
Date of creation	2013-03-22 15:25:42
Last modified on	2013-03-22 15:25:42
Owner	Gorkem (3644)
Last modified by	Gorkem (3644)
Numerical id	14
Author	Gorkem (3644)
Entry type	Theorem
Classification	msc 30E99