### 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.

Language | English |
---|---|

Pages | 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 |

### Fingerprint

### Keywords

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

### Cite this

*Complex Analysis and Operator Theory*,

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

}

*Complex Analysis and Operator Theory*, vol. 8, no. 6, pp. 1183-1224. https://doi.org/10.1007/s11785-013-0315-5

**Stability of the derivative of a canonical product.** / Langer, Matthias; Woracek, Harald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Stability of the derivative of a canonical product

AU - Langer, Matthias

AU - Woracek, Harald

PY - 2014/8

Y1 - 2014/8

N2 - With each sequence α=(αn)n∈N of pairwise distinct and non-zero points which are such that the canonical product Pα(z):=limr→∞∏∣αn∣≤r(1−z/αn) converges, the sequence α′:=(Pα'(αn))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.

AB - With each sequence α=(αn)n∈N of pairwise distinct and non-zero points which are such that the canonical product Pα(z):=limr→∞∏∣αn∣≤r(1−z/αn) converges, the sequence α′:=(Pα'(αn))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.

KW - canonical product

KW - perturbation of zeros

KW - inverse spectral problem

KW - Krein class

KW - positive definite function

UR - http://www.scopus.com/inward/record.url?scp=84879854747&partnerID=8YFLogxK

UR - http://www.springer.com/birkhauser/mathematics/journal/11785

U2 - 10.1007/s11785-013-0315-5

DO - 10.1007/s11785-013-0315-5

M3 - Article

VL - 8

SP - 1183

EP - 1224

JO - Complex Analysis and Operator Theory

T2 - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

SN - 1661-8254

IS - 6

ER -