M. Langer and H. Woracek,
Stability of the derivative of a canonical product,
Complex Anal. Oper. Theory 8 (2014), 1183–1224
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.