An analytic characterization of the symmetric extension of a Herglotz-Nevanlinna function in several variables

TL;DR

This paper provides an analytic characterization of the symmetric extension of multivariable Herglotz-Nevanlinna functions, utilizing variable non-dependence and symmetry properties, and extends the Stieltjes inversion formula for Cauchy-type functions.

Contribution

It introduces a novel analytic characterization of symmetric extensions for multivariable Herglotz-Nevanlinna functions and extends the Stieltjes inversion formula for Cauchy-type functions.

Findings

Characterization of symmetric extension using variable non-dependence and symmetry formula

Extension of Stieltjes inversion formula for Cauchy-type functions

Framework applicable to multivariable Herglotz-Nevanlinna functions

Abstract

In this paper, we derive an analytic characterization of the symmetric extension of a Herglotz-Nevanlinna function in several variables. Here, the main tools used are the so-called variable non-dependence property and the symmetry formula satisfied by Herglotz-Nevanlinna function and Cauchy-type functions. We also provide an extension of the Stieltjes inversion formula for Cauchy-type functions.