Carathéodory functions on Riemann surfaces and reproducing kernel spaces

Daniel Alpay, Ariel Pinhas, Victor Vinnikov

Research output: Working paperPreprint


Carath\'eodory functions, i.e. functions analytic in the open upper half-plane and with a positive real part there, play an important role in operator theory, $1D$ system theory and in the study of de Branges-Rovnyak spaces. The Herglotz integral representation theorem associates to each Carath\'eodory function a positive measure on the real line and hence allows to further examine these subjects. In this paper, we study these relations when the Riemann sphere is replaced by a real compact Riemann surface. The generalization of Herglotz's theorem to the compact real Riemann surface setting is presented. Furthermore, we study de Branges-Rovnyak spaces associated with functions with positive real-part defined on compact Riemann surfaces. Their elements are not anymore functions, but sections of a related line bundle.
Original languageAmerican English
StatePublished - 7 Dec 2019


  • 46E22, 30F15
  • math.CV


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