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In this paper, we give several characterizations of Herglotz-Nevanlinnafunctions in terms of a specific type of positive semi-definite functionscalled Poisson-type functions. This allows us to propose a multidimensionalanalogue of the classical Nevanlinna kernel and a definition of generalizedNevanlinna functions in several variables. Furthermore, a characterization ofthe symmetric extension of a Herglotz-Nevanlinna function is also given. Thesubclass of Loewner functions is also discussed, as well as an interpretationof the main result in terms of holomorphic functions on the unit polydisk withnon-negative real part.

Characterizations of Herglotz-Nevanlinna functions using positive semi-definite functions and the Nevanlinna kernel in several variables