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Distributional representations of N κ ( ∞ ) ‐functions
Author(s) -
Langer Matthias,
Woracek Harald
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300280
Subject(s) - mathematics , pure mathematics , generalized function , operator (biology) , space (punctuation) , cauchy distribution , distribution (mathematics) , context (archaeology) , class (philosophy) , multiplication (music) , infinity , function (biology) , pontryagin's minimum principle , measure (data warehouse) , mathematical analysis , combinatorics , optimal control , philosophy , repressor , database , artificial intelligence , linguistics , chemistry , computer science , biology , paleontology , mathematical optimization , biochemistry , evolutionary biology , transcription factor , gene
The subclasses N κ ( ∞ )of the classes N κ of generalized Nevanlinna functions appear in the context of Pontryagin space models, where they correspond to model relations having a particular spectral behaviour. Applications are found, for instance, in the investigation of differential expressions with singular coefficients. We study representations of N κ ( ∞ ) ‐functions as Cauchy‐type integrals in a distributional sense and characterize the class of distributions occurring in such representations. We make explicit how the Pontryagin space model of an N κ ( ∞ ) ‐function is related to the multiplication operator in the L 2 ‐space of the measure which describes the action of the representing distribution away from infinity. Moreover, we determine the distributional representations of a pair of functions associated with a symmetric generalized Nevanlinna function.