Lawton Wayne
Note on Results of Krein and Levin

Аннотация:
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 τ (F) < ∞. If f ≥ 0 then a result of Krein implies that f admits a factorization f = |h|^2 where h extends to an entire function H of exponential type τ (H) = τ (F)/2 having no zeros in the open upper half plane. The spectral factor h is unique up to a multiplicative factor having modulus 1. Krein and Levin constructed f such that h is not uniformly almost periodic and proved that if f ≥ m > 0 has absolutely converging Fourier series then h is uniformly almost periodic and has absolutely converging Fourier series. We derive necessary and sufficient conditions on f ≥ m > 0 for h to be uniformly almost periodic, we construct an ≥ m > 0 with non absolutely converging Fourier series such that h is uniformly almost periodic, and we suggest research questions.