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H. Woracek:
"Existance of zerofree functions $N$-associated to a de Branges Pontryagin space";
in: "ASC Report 23/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

In the theory of de Branges Hilbert spaces of entire functions, so-called
`functions associated to a space´ play an important role. In the present
paper we deal with a generalization of this notion in two directions, namely
with functions N-associated (N 2 Z) to a de Branges Pontryagin space.
Let a de Branges Pontryagin space P and N 2 Z be given. Our aim
is to characterize whether there exists a real and zerofree function Nassociated
to P in terms of Kre˘ın´s Q-function associated with the multiplication
operator in P. The conditions which appear in this characterization
involve the asymptotic distribution of the poles of the Q-function
plus a summability condition.
Although this question may seem rather abstract, its answer has a variety
of nontrivial consequences. We use it to answer two questions arising
in the theory of general (indefinite) canonical systems. Namely, to characterize
whether a given generalized Nevanlinna function is the intermediate
Weyl-coefficient of some system in terms of its poles and residues, and to
characterize whether a given general Hamiltonian ends with a specified
number of indivisible intervals in terms of the Weyl-coefficient associated
to the system. In addition, we present some applications, e.g. dealing with
admissible majorants in de Branges spaces or the continuation problem
for hermitian indefinite functions.

de Branges space, Pontryagin space, associated function, canonical

http://dx.doi.org/10.1007/s00605-010-0203-2

http://www.asc.tuwien.ac.at/preprint/2009/asc23x2009.pdf