Note on Spectral Factorization Results of Krein and Levin

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. 2010 Mathematics Subject Classification:47A68;42A75;30D15 1 1 Notation := means ‘is defined to equal’ and iff means ‘if and only if’. N = {1, 2, 3, ...}, Z, Q, R, C are the natural, integer, rational, real, and complex numbers. For z ∈ C, x := Rz; y := Iz are its real; imaginary coordinates. D := {z ∈ C : |z| ≤ 1} is the closed unit disk, D := {z ∈ C : |z| ≤ 1} is the open unit disk, and T := {z ∈ C : |z| = 1} is the circle. U := { z ∈ C : Iz ≥ 0 } is the closed upper half–plane and U := { z ∈ C : Iz ≥ 0 } is the open upper half plane. For z ∈ C, χz(x) := e. For closed K ⊂ C, Cb(K) is the C–algebra of bounded continuous complex–valued functions on K with norm ||f || := supz∈K |f(z)|. For ρ > 0, Dρ : Cb(R) → Cb(R) is the dilation operator (Dρf)(x) := f(ρ x). Zeros of nonzero entire functions are denoted by sequences zn, n ≥ 1 of finite (possibly zero) or infinite length. This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of Regional Centers for Mathematics Research and Education (Agreement No. 075-02-2020-1534/1).