Abstract
In this paper, we give several characterizations of Herglotz–Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the classical Nevanlinna kernel and a definition of generalized Nevanlinna functions in several variables. Furthermore, a characterization of the symmetric extension of a Herglotz–Nevanlinna function is also given. The subclass of Loewner functions is discussed as well, along with an interpretation of the main result in terms of holomorphic functions on the unit polydisk with non-negative real part.
Highlights
Let us denote by class Nκ (C+n) the poly-upper half-plane in Cn, i.e.C+n := (C+)n = z ∈ Cn ∀ j = 1, 2, . . . , n : Im[z j ] > 0 .Communicated by H
For Herglotz–Nevanlinna functions of one variable, it follows as an immediate consequence of the integral representation theorem mentioned above that a holomorphic function q : C+ → C has non-negative imaginary part if and only if the function (z, w)
Let us begin by shortly recalling how Herglotz–Nevanlinna functions in one variable are characterized via positive semi-definite functions, see e.g. [15]
Summary
Let us denote by C+n the poly-upper half-plane in Cn, i.e. C+n := (C+)n = z ∈ Cn ∀ j = 1, 2, . . . , n : Im[z j ] > 0. Definition 1.1 A function q : C+n → C is called a Herglotz–Nevanlinna function if it is holomorphic and has non-negative imaginary part This is a well-studied class of functions, appearing e.g. in [2,3,25,26,27,28,33,37,38]. For Herglotz–Nevanlinna functions of one variable, it follows as an immediate consequence of the integral representation theorem mentioned above that a holomorphic function q : C+ → C has non-negative imaginary part if and only if the function (z, w). This result allows us to derive an analogous characterization to the one mentioned above using a suitable analogue of the Nevanlinna kernel in several variables This is presented in Theorem 4.4, which establishes that a function q as before is a Herglotz–Nevanlinna function if and only if the function.
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