On the uniform stability of recovering sine-type functions with asymptotically separated zeros

TL;DR

This paper establishes the uniform stability of recovering certain entire functions from their zeros, with applications to inverse spectral problems and analysis of differential operators with asymptotically separated eigenvalues.

Contribution

It introduces new stability results for recovering sine-type functions with zeros on a line, including their Lipschitz dependence and zero asymptotics, applicable to spectral theory.

Findings

Lipschitz dependence of functions on zeros within finite radius

Asymptotic zero distribution and infinite product representation

Applicability to inverse spectral problems and differential operators

Abstract

We obtain a uniform stability of recovering entire functions of a special form from their zeros. To this form, one can reduce the characteristic determinants of strongly regular differential operators and pencils of the first and the second orders, including differential systems with asymptotically separated eigenvalues whose characteristic numbers lie on a line containing the origin, and their non-local perturbations. We establish that the dependence of such functions on the sequences of their zeros possesses the Lipschitz property with respect to natural metrics on each ball of a finite radius. Results of this type can be used for studying the uniform stability of inverse spectral problems. In addition, general theorems on the asymptotics of zeros of functions of this class and on their equivalent representation via an infinite product are obtained, which give the corresponding results for many specific operators.