Abstract
We study the class of discrete measures in the complex plane with the following property: up to a finite number, all zeros of any Cauchy transform of the measure (with ℓ 2-data) are localized near the support of the measure. We find several equivalent forms of this property and prove that the parts of the support attracting zeros of Cauchy transforms are ordered by inclusion modulo finite sets.
Dedicated to the memory of Serguei Shimorin, a brilliant mathematician and a wonderful person.
The work was supported by the joint grant of Russian Foundation for Basic Research (project 17-51-150005-NCNI-a) and CNRS, France (project PRC CNRS/RFBR 2017-2019 “Noyaux reproduisants dans des espaces de Hilbert de fonctions analytiques”).
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Abakumov, E., Baranov, A., Belov, Y. (2019). Localization of Zeros in Cauchy–de Branges Spaces. In: Aleman, A., Hedenmalm, H., Khavinson, D., Putinar, M. (eds) Analysis of Operators on Function Spaces. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-14640-5_2
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DOI: https://doi.org/10.1007/978-3-030-14640-5_2
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