H. Woracek:

"Perturbation of chains of de Branges spaces";

in: "ASC Report 42/2014", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2014, ISBN: 978-3-902627-07-0, 1 - 32.

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L2( ). Thereby, we follow a perturbative approach. The main result is a growth dependent stability theorem: Assume that two measures 1 and 2 are close to each other in a sense quantified relative to a proximate order.

Consider the sections of corresponding chains of de Branges spaces C1 and C2 which consist of those spaces whose elements have finite (possibly zero) type w.r.t. the given proximate order. Then these sections coincide, or one is smaller than the other

but its complement consists only of a (finite or infinite) sequence of spaces.Among others we apply { and refine { this general theorem in two important particular situations. (1) the given measures 1 and 2 differ in essence only on a compact set; then stability of whole chains rather than sections can be shown. (2) the linear space of all polynomials is dense in L2( 1); then conditions for density of

polynomials in the space L2( 2) are obtained.In the proof of the main result we employ a method used by P.Yuditskii in the

context of density of polynomials. Another vital tool is the notion of the index of a chain, which is a generalisation of the index of determinacy of a measure having all power moments. We undertake a systematic study of this index, which is also of interest on its own right.

De Branges space, perturbation of spectral measure, stability of chains,

http://www.asc.tuwien.ac.at/preprint/2014/asc42x2014.pdf

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