Vol. 14, No. 5, 2021

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De Branges canonical systems with finite logarithmic integral

Roman V. Bessonov and Sergey A. Denisov

Vol. 14 (2021), No. 5, 1509–1556

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 Szegő theorem in the theory of polynomials orthogonal on the unit circle. It extends the Krein–Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.

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Szegő class, canonical Hamiltonian systems, inverse problem, entropy
Mathematical Subject Classification 2010
Primary: 42C05, 34L40, 34A55
Received: 14 June 2019
Revised: 9 December 2019
Accepted: 9 February 2020
Published: 22 August 2021
Roman V. Bessonov
St. Petersburg State University
St. Petersburg
St. Petersburg Department of Steklov Mathematical Institute
Russian Academy of Sciences
St. Petersburg
Sergey A. Denisov
Department of Mathematics
University of Wisconsin
Madison, WI
United States
Keldysh Institute of Applied Mathematics
Russian Academy of Sciences