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Revista Matemática Iberoamericana

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Volume 35, Issue 4, 2019, pp. 963–1026
DOI: 10.4171/rmi/1076

Published online: 2019-05-16

Stability of order and type under perturbation of the spectral measure

Anton D. Baranov[1] and Harald Woracek[2]

(1) St. Petersburg State University, Russian Federation
(2) Technische Universität Wien, Austria

It is known that the type of a measure is stable under perturbations consisting 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 $L^2$-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 $\lambda$. 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 closure must contain functions with faster growth than $\lambda$. For the growth function $\lambda(r)=r$, 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 $\leq 2$) turn out to be particular.

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

Baranov Anton, Woracek Harald: Stability of order and type under perturbation of the spectral measure. Rev. Mat. Iberoam. 35 (2019), 963-1026. doi: 10.4171/rmi/1076