Localization of zeros in Cauchy-de Branges spaces

Abstract

We study the class of discrete measures in the complex plain with the following property: up to a finite number, all zeros of any Cauchy transform of the measure (with ${\ell }^2$-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.

This paper is a continuation of arXiv:1312.6706. It extends the results obtained in arXiv:1312.6706 for the Cauchy transforms of measures with support on the real line to the case of general discrete measures in the plane. A new elementary proof of the ordering theorem for attraction sets is given which does not use the de Branges theory and the estimates of subharmonic functions due to Heins