Title: Szego theorem for measures on the real line: optimal results and
applications.
Abstract: Measures on the unit circle for which the logarithmic integral
converges can be characterized in many different ways: e.g., through
their Schur parameters or through the predictability of the future from
the past in Gaussian stationary stochastic process. In this talk, we
consider measures on the real line for which logarithmic integral exists
and give their complete characterization in terms of the Hamiltonian in
De Branges canonical system. This provides a generalization of the
classical Szego theorem for polynomials orthogonal on the unit circle
and complements the celebrated Krein-Wiener theorem in complex function
theory. The applications to Krein strings and Gaussian processes with
continuous time will be discussed (this talk is based on the joint paper
with R. Bessonov).