# 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 ${L}^{2}(\mathbb{R})$, then there exists a function
$g(t)\in {L}^{2}(-\gamma ,\gamma )$ such that

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

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 |