# De Branges canonical systems with finite logarithmic integral

## TL;DR

This paper characterizes certain canonical Hamiltonian systems with spectral measures having finite logarithmic integrals, extending classical theorems and connecting spectral theory with orthogonal polynomials and Gaussian process prediction.

## Contribution

It provides a spectral characterization of Hamiltonian systems with measures having convergent logarithmic integrals, extending classical spectral and polynomial orthogonality results.

## Key findings

- Describes canonical Hamiltonian systems with finite logarithmic spectral measures
- Establishes a spectral analogue of Szego's theorem for these systems
- Extends Krein-Wiener completeness theorem in spectral theory

## Abstract

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