Remarks on restricted Nevanlinna transforms

TL;DR

This paper explores the restricted Nevanlinna transform, showing it can be expressed as a Laplace transform of a measure's Fourier transform, and discusses its implications for free and boolean convolutions.

Contribution

It establishes a novel representation of the restricted Nevanlinna transform as a Laplace transform of the Fourier transform of a measure.

Findings

Restricted Nevanlinna transform equals Laplace transform of Fourier transform

Relation between Voiculescu and boolean convolutions is indicated

Provides new analytical tools for complex analysis and free probability

Abstract

The Nevanlinna transform K(z), of a measure and a real constant, plays an important role in the complex analysis and more recently in the free probability theory (boolean convolution). It is shown that its restriction k(it) (the restricted Nevanlinna transform) to the imaginary axis can be expressed as the Laplace transform of the Fourier transform (characteristic function) of the corresponding measure. Finally, a relation between the Voiculescu and the boolean convolution is indicated.