On hyperbolic characteristic functions from an analytic and a free-probability point of view

TL;DR

This paper derives integral representations for free-probability Voiculescu transforms, akin to hyperbolic characteristic functions, using only their values on the imaginary axis, simplifying the analysis compared to traditional complex analysis methods.

Contribution

It provides a novel integral representation for Voiculescu transforms based solely on their imaginary axis values, bridging free probability and complex analysis.

Findings

Integral form for Voiculescu transforms obtained

Representation depends only on imaginary axis data

Simplifies analysis of free-probability transforms

Abstract

For free-probability Voiculescu transforms, analogous to hyperbolic characteristic functions, we show how to get their representing measures in an integral form. For that purpose, it is enough to know those transforms only on the imaginary axis. This is in contrast to a complex analysis where one needs to know them in some domains in the complex plane.