Approximations and almost periodicity

Bohr proved that a uniformly almost periodic function $f : \mathbb R \rightarrow \mathbb C$ extends to a continuous function $\widetilde f : G(f) \rightarrow \mathbb C,$ where $G(f)$ is a compactification of $\mathbb R$ whose Pontryagin dual $\widehat {G(f)}$ is generated by the spectrum $\Omega(f).$ He showed that $\Omega(f)$ is bounded iff $f$ extends to an entire function $F$ of exponential type $\tau(F) < \infty,$ and Krein proved that $f \geq 0$ implies $f = |s|^2$ 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 used properties of zero sets to construct $f$ so $s$ is not uniformly almost periodic. We use approximation methods to

• characterize $f$ so $s$ is uniformly almost periodic,
• relate $s$ to Helson–Lowdenslager's spectral factorization of $\widetilde f$,
• construct uniformly almost periodic $h$ with unbounded Hilbert transform, bounded spectrum, and rank $\widehat {G(h)} = 2,$ and
• construct a Bohr–type compactification for and homotopy classification of a class of uniformly periodic discrete subsets of $\mathbb R^d.$