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