# Paley-Wiener theorem

Let $f(z)$ be an entire function such that $|f(z)|\leq Ke^{\gamma|z|}$ for some $K\geq 0$ and $\gamma>0$. If the restriction of $f$ to the real line is in $L^{2}(\mathbb{R})$, 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 PaleyWienerTheorem 2013-03-22 15:25:42 2013-03-22 15:25:42 Gorkem (3644) Gorkem (3644) 14 Gorkem (3644) Theorem msc 30E99