Characterizations of Herglotz–Nevanlinna Functions Using Positive Semi-Definite Functions and the Nevanlinna Kernel in Several Variables

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 Ne...

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Bibliographic Details
Published in:Complex analysis and operator theory Vol. 15; no. 7
Main Author: Nedic, Mitja
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.10.2021
Springer Nature B.V
Subjects:
Analysis
Analytic functions
Higher Dimensional Geometric Function Theory and Hypercomplex Analysis
Kernels
Mathematical analysis
Mathematics
Mathematics and Statistics
Operator Theory
Positive semi-definite functions
Poisson-type functions
32A36
Nevanlinna kernel
32A99
Herglotz–Nevanlinna functions
ISSN:1661-8254, 1661-8262
Online Access:Get full text
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Summary: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.
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ISSN:1661-8254
1661-8262
DOI:10.1007/s11785-021-01155-x