Note on Spectral Factorization Results of Krein and Levin

TL;DR

This paper explores spectral factorization of almost periodic functions, establishing conditions for the spectral factor to be almost periodic and constructing examples with specific Fourier series properties.

Contribution

It provides necessary and sufficient conditions for the spectral factor to be almost periodic and constructs examples with non-absolutely convergent Fourier series.

Findings

Conditions for spectral factor to be almost periodic

Existence of functions with non-absolutely convergent Fourier series

Insights into spectral factorization of almost periodic functions

Abstract

Bohr proved that a uniformly almost periodic function $f$ has a bounded spectrum if and only if it extends to an entire function $F$ of exponential type $\tau(F) < \infty$. If $f \geq 0$ then a result of Krein implies that $f$ admits a factorization $f = |s|^2$ where $s$ extends to an entire function $S$ of exponential type $\tau(S) = \tau(F)/2$ having no zeros in the open upper half plane. The spectral factor $s$ is unique up to a multiplicative factor having modulus $1.$ Krein and Levin constructed $f$ such that $s$ is not uniformly almost periodic and proved that if $f \geq m > 0$ has absolutely converging Fourier series then $s$ is uniformly almost periodic and has absolutely converging Fourier series. We derive neccesary and sufficient conditions on $f \geq m > 0$ for $s$ to be uniformly almost periodic, we construct an $f \geq m > 0$ with non absolutely converging Fourier series such that $s$ is uniformly almost periodic, and we suggest research questions.