### Abstract

With each sequence α=(α_{n})_{n∈N} of pairwise distinct and non-zero points which are such that the canonical product

P_{α}(z):=lim_{r→∞}∏_{∣αn∣≤r}(1−z/α_{n})

converges, the sequence

α′:=(P_{α'}(α_{n}))_{n∈N}

is associated. We give conditions on the difference β−α of two sequences which ensure that β' and α' are comparable in the sense that

∃c,C>0: c|α'_{n}|≤|β'_{n}|≤C|α'_{n}|, n∈N.

The values α'_{n} play an important role in various contexts. As a selection of applications we present: an inverse spectral problem, a class of entire functions and a continuation problem.

Original language | English |
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Pages (from-to) | 1183-1224 |

Number of pages | 42 |

Journal | Complex Analysis and Operator Theory |

Volume | 8 |

Issue number | 6 |

Early online date | 11 Jul 2013 |

DOIs | |

Publication status | Published - Aug 2014 |

### Keywords

- canonical product
- perturbation of zeros
- inverse spectral problem
- Krein class
- positive definite function

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## Cite this

Langer, M., & Woracek, H. (2014). Stability of the derivative of a canonical product.

*Complex Analysis and Operator Theory*,*8*(6), 1183-1224. https://doi.org/10.1007/s11785-013-0315-5