Localization of zeros in Cauchy-de Branges spaces

TL;DR

This paper investigates the localization of zeros of Cauchy transforms in Cauchy-de Branges spaces, showing that zeros are primarily near the measure's support and establishing their ordered structure.

Contribution

It introduces new characterizations of zero localization in Cauchy-de Branges spaces and proves the ordered inclusion of support parts attracting zeros.

Findings

Zeros of Cauchy transforms are localized near the measure's support.

Parts of the support attracting zeros are ordered by inclusion.

Several equivalent characterizations of the zero localization property.

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.