# Perturbation of chains of de Branges spaces

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## Abstract

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space *L*^{2}(*μ*). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures *μ*_{1} and *μ*_{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 C_{1} and C_{2} which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate order. Then either these sections coincide or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces.

- (1)
the measures

*μ*_{1}and*μ*_{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

*L*^{2}(*μ*_{2}); then conditions for density of polynomials in the space*L*^{2}(*μ*_{2}) are obtained.

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