Note on Spectral Factorization Results of Krein and Levin

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.