Perturbation of chains of de Branges spaces

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L²(μ). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures μ₁ and μ₂ 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 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.

Among other situations, we apply—and refine—this general theorem in two important particular situations

1. (1)
   the measures μ₁ and μ₂ differ in essence only on a compact set; then stability of whole chains rather than sections can be shown
    
2. (2)
   the linear space of all polynomials is dense in L²(μ₂); then conditions for density of polynomials in the space L²(μ₂) 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.