# Spectral Factorization and Entire Functions

## TL;DR

This paper explores spectral factorization of entire functions and nonnegative trigonometric polynomials, extending classical results through the lens of almost periodic functions and ordered dual groups.

## Contribution

It relates spectral factorizations for archimedean orders to the theory of almost periodic functions, expanding the scope of classical spectral factorization results.

## Key findings

- Extended Fejér-Riesz lemma to entire functions
- Connected spectral factorization with almost periodic functions
- Provided new insights into factorizations on ordered dual groups

## Abstract

The Fej\'{e}r-Riesz spectral factorization lemma, which represents a nonnegative trigonometric polynomial as the squared modulus of a trigonometric polynomial, was extended by Ahiezer to factorize certain entire functions and by Helson and Lowdenslager to factorize certain functions on compact connected abelian groups whose Pontryagin duals are equipped with a linear order. This paper relates these factorizations for archimedian orders using the theory of almost periodic functions.