Factorizations of analytic self-maps of the upper half-plane

TL;DR

This paper generalizes Krein's factorization for analytic self-maps of the upper half-plane, enabling the construction of functions with specified zeros and poles, and providing a deeper understanding of their structure.

Contribution

It extends Krein's factorization to all analytic self-maps of the upper half-plane, allowing for tailored function construction with prescribed zeros and poles.

Findings

Provides a new factorization method for analytic self-maps

Enables construction of functions with specific zeros and poles

Enhances understanding of the structure of such functions

Abstract

We extend a factorization due to Krein to arbitrary analytic functions from the upper half-plane to itself. The factorization represents every such function as a product of fractional linear factors times a function which, generally, has fewer zeros and singularities than the original one. The result is used to construct functions with given zeros and poles on the real line.