# factorization theorem for ${H}^{\mathrm{\infty}}$ functions

Let ${H}^{\mathrm{\infty}}$ denote the bounded analytic functions^{} on the unit disc.

###### Theorem.

Every $f\mathrm{\in}{H}^{\mathrm{\infty}}$ can be written as

$$f(z)=\alpha I(z)F(z)$$ |

where $\mathrm{|}\alpha \mathrm{|}\mathrm{=}\mathrm{1}$, $I$ is an inner function and $F$ is a bounded outer function. Conversely any function which can be so written is bounded.

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.

Title | factorization theorem for ${H}^{\mathrm{\infty}}$ functions |
---|---|

Canonical name | FactorizationTheoremForHinftyFunctions |

Date of creation | 2013-03-22 15:36:23 |

Last modified on | 2013-03-22 15:36:23 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 30H05 |

Related topic | InnerFunction |