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A. Baranov, H. Woracek:
"Stability of order and type under perturbation of the spectral measure";
in: "ASC Report 27/2016", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2016, ISBN: 978-3-902627-09-4, 1 - 71.

It is known that the type of a measure is stable under perturbations con-sisting of exponentially small redistribution of mass and exponentially small additive summands. This fact can be seen as stability of de Branges chains in the corresponding L^2-spaces.
We investigate stability of de Branges chains in L2-spaces under perturbations having the same form, but allow other magnitudes for the error. The admissible size of a perturbation is connected with the maximal growth of functions in the chain and is measured by means of a growth function λ. The main result is a Fast Growth Theorem. It states that an alternative takes place when passing to a perturbed measure: either the original de Branges chain remains dense or its closures must contain functions with faster growth than λ. For the growth function λ(r) = r, i.e. exponentially small perturbations, the afore mentioned known fact is reobtained.
We propose a notion of order of a measure and show stability and monotonicity properties of this notion. The cases of exponential type (order 1) and very slow growth (logarithmic order ≤ 2) turn out to be particular.

de Branges space, order and type of a measure, perturbation of measures, growth function, weighted approximation

http://www.asc.tuwien.ac.at/preprint/2016/asc27x2016.pdf