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In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients.

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  • Fonction harmonique positive (fr)
  • Positive harmonic function (en)
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  • En mathématiques, une fonction harmonique positive du disque unité à valeurs complexes est caractérisée comme l'intégrale de Poisson d'une mesure positive finie sur le cercle. Ce résultat, le théorème de représentation de Herglotz, a été prouvé par Gustav Herglotz en 1911. Il peut être utilisé pour donner une caractérisation des fonctions holomorphes du disque unité ayant une partie réelle positive. Ces fonctions ont déjà été caractérisées en 1907 par Constantin Carathéodory en termes de la fonction définie positive de leurs coefficients de Taylor. (fr)
  • In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients. (en)
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  • En mathématiques, une fonction harmonique positive du disque unité à valeurs complexes est caractérisée comme l'intégrale de Poisson d'une mesure positive finie sur le cercle. Ce résultat, le théorème de représentation de Herglotz, a été prouvé par Gustav Herglotz en 1911. Il peut être utilisé pour donner une caractérisation des fonctions holomorphes du disque unité ayant une partie réelle positive. Ces fonctions ont déjà été caractérisées en 1907 par Constantin Carathéodory en termes de la fonction définie positive de leurs coefficients de Taylor. (fr)
  • In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients. (en)
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