Stability of the derivative of a canonical product

Langer, Matthias and Woracek, Harald (2014) Stability of the derivative of a canonical product. Complex Analysis and Operator Theory, 8 (6). pp. 1183-1224. ISSN 1661-8254

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    Abstract

    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.