Notes on a special order on $\mathbb{Z}^\infty$

TL;DR

This paper extends the theory of analytic functions on groups with ordered duals to infinite-dimensional settings, characterizing maximal ideals, outer functions, and providing a new proof of the infinite-dimensional Szeg"o's theorem.

Contribution

It introduces a special order on Z^ and analyzes the associated analytic functions, generalizing classical results to infinite dimensions.

Findings

Characterization of maximal ideals in the infinite-dimensional setting

Existence of outer functions corresponding to integrable weights

A new proof of the infinite-dimensional Szegf3's theorem

Abstract

In 1958, Helson and Lowdenslager extended the theory of analytic functions to a general class of groups with ordered duals. In this context, analytic functions on such a group $G$ are defined as the integrable functions whose Fourier coefficients lie in the positive semigroup of the dual of $G$. In this paper, we found some applications of their theory to infinite-dimensional complex analysis. Specifically, we considered a special order on $\mathbb{Z}^\infty$ and corresponding analytic continuous functions on $\mathbb{T}^\omega$, which serves as the counterpart of the disk algebra in infinitely many variables setting. By characterizing its maximal ideals, we have generalized the following theorem to the infinite-dimensional case: For a positive function $w$ that is integrable and log-integrable on $\mathbb{T}^d$, there exists an outer function $g$ such that $w=|g|^2$ if and only if the support of $\hat{\log w}$ is a subset of $\mathbb{N}^d\cap (-\mathbb{N})^d$. Furthermore, we have found the counterpart of the above function algebra in the closed right half-plane, and the representing measures of each point in the right half-plane for this algebra. As an application of the order, we provided a new proof of the infinite-dimensional Szeg\"{o}'s theorem.