# inner function

If $f:\mathbb{D}\to \u2102$ is an analytic function^{} on the unit disc, we denote by
${f}^{*}({e}^{i\theta})$ the radial limit of $f$ where it exists, that is

$$ |

A bounded analytic function on the disc will have radial limits almost everywhere (with respect to the Lebesgue measure^{} on the $\partial \mathbb{D}$).

###### Definition.

A bounded analytic function $f$ is called an *inner function* if $|{f}^{*}({e}^{i\theta})|=1$ almost everywhere. If $f$ has no zeros on the unit disc, then $f$ is called a *singular inner function*.

###### Theorem.

Every inner function can be written as

$$f(z):=\alpha B(z)\mathrm{exp}\left(-\int \frac{{e}^{i\theta}+z}{{e}^{i\theta}-z}\mathit{d}\mu ({e}^{i\theta})\right),$$ |

where $\mu $ is a positive singular measure^{} on $\mathrm{\partial}\mathit{}\mathrm{D}$, $B\mathit{}\mathrm{(}z\mathrm{)}$
is a Blaschke product^{} and $\mathrm{|}\alpha \mathrm{|}\mathrm{=}\mathrm{1}$ is a constant.

Note that all the zeros of the function come from the Blaschke product.

###### Definition.

Let

$$f(z):=\mathrm{exp}\left(\int \frac{{e}^{i\theta}+z}{{e}^{i\theta}-z}h({e}^{i\theta})\mathit{d}m({e}^{i\theta})\right),$$ |

where $h$ is a real valued Lebesgue integrable^{} function on the unit circle and $m$ is the Lebesgue measure. Then $f$ is called
an *outer function*.

The significance of these definitions is that every bounded holomorphic function^{} can be written as an inner function times an outer function. See the factorization theorem for ${H}^{\mathrm{\infty}}$ functions (http://planetmath.org/FactorizationTheoremForHinftyFunctions).

## References

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

Title | inner function |
---|---|

Canonical name | InnerFunction |

Date of creation | 2013-03-22 15:36:20 |

Last modified on | 2013-03-22 15:36:20 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 30H05 |

Related topic | FactorizationTheoremForHinftyFunctions |

Defines | singular inner function |

Defines | outer function |