Nevanlinna class, Dirichlet series and Szeg\"o's problem

TL;DR

This paper explores the connections between Nevanlinna and Smirnov classes, Dirichlet series, and Szeg"o's problem in infinitely many variables, extending classical theories and analyzing their interrelations in an infinite-dimensional context.

Contribution

It generalizes classical Nevanlinna and Smirnov class theories to infinitely many variables and applies these results to Szeg"o's problem in Hardy spaces.

Findings

Extended Nevanlinna and Smirnov classes to infinite variables

Connected Nevanlinna functions with Dirichlet series

Applied theory to Szeg"o's problem in infinite-dimensional Hardy spaces

Abstract

This paper is associated with Nevanlinna class, Dirichlet series and Szeg\"o's problem in infinitely many variables. As we will see, there is a natural connection between these topics. The paper first introduces the Nevanlinna class and the Smirnov class in this context, and generalizes the classical theory in finitely many variables to the infinite-variable setting. These results applied to Szeg\"o's problem on Hardy spaces in infinitely many variables. Moreover, this paper is also devoted to the study of the correspondence between the Nevanlinna functions and Dirichlet series.