# Blaschke product

###### Definition.

Suppose that $\{{a}_{n}\}$ is a sequence of complex numbers^{} with $$ and $$, then

$$B(z):=\prod _{n=1}^{\mathrm{\infty}}\frac{|{a}_{n}|}{{a}_{n}}\left(\frac{{a}_{n}-z}{1-{\overline{a}}_{n}z}\right)$$ |

is called the Blaschke product^{}.

This product converges uniformly on compact subsets of the unit disc, and thus $B$ is a holomorhic function on the unit disc. Further it is the function on the disc that has zeros exactly at $\{{a}_{n}\}$. And finally for $z$ in the unit disc, $\left|B(z)\right|\le 1$.

###### Definition.

Sometimes ${B}_{a}(z):=\frac{z-a}{1-\overline{a}z}$ is called the Blaschke factor.

With this definition, the Blascke product becomes $B(z)={\prod}_{n=1}^{\mathrm{\infty}}\frac{|{a}_{n}|}{{a}_{n}}{B}_{{a}_{n}}(z)$.

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | Blaschke product |
---|---|

Canonical name | BlaschkeProduct |

Date of creation | 2013-03-22 14:19:35 |

Last modified on | 2013-03-22 14:19:35 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 30C45 |

Defines | Blascke factor |