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Bohr proved that a uniformly almost periodic function $f$ has a boundedspectrum if and only if it extends to an entire function $F$ of exponentialtype $\tau(F) < \infty$. 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 upperhalf plane. The spectral factor $s$ is unique up to a multiplicative factorhaving modulus $1.$ Krein and Levin constructed $f$ such that $s$ is notuniformly almost periodic and proved that if $f \geq m > 0$ has absolutelyconverging Fourier series then $s$ is uniformly almost periodic and hasabsolutely converging Fourier series. We derive neccesary and sufficientconditions on $f \geq m > 0$ for $s$ to be uniformly almost periodic, weconstruct an $f \geq m > 0$ with non absolutely converging Fourier series suchthat $s$ is uniformly almost periodic, and we suggest research questions.

Note on Spectral Factorization Results of Krein and Levin