De Branges canonical systems with finite logarithmic integral

Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on real line with finite Poisson integral. We further develop this area by giving description whose have logarithmic integral converging over line. This result can be viewed as version classical Szego theorem in polynomials orthogonal unit circle. It extends Krein-Wiener completeness theorem, key fact prediction stationary Gaussian processes.