2021/04/18 by Wayne Lawton, Lawton, Wayne M.
Computer Science · Mathematics · #30D15 #42A75 #47A68 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2104.08917
openalex publication_date 2021/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 = |s|2 where s extends to an entire function S of exponential type τ(S) = τ(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 ≥ 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 ≥ m > 0 for s to be uniformly almost periodic, we construct an f ≥ m > 0 with non absolutely converging Fourier series such that s is uniformly almost periodic, and we suggest research questions.