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A Density Result for Parametric Representations in Several Complex Variables

A Density Result for Parametric Representations in Several Complex Variables C. Loewner proved that the class of functions which have a parametric representation obtained by solving the Loewner differential equation with a driving term is dense in the class $$S$$ S of normalized univalent functions on the unit disc, because it contains all the single-slit mappings. I. Graham, H. Hamada, G. Kohr and M. Kohr suggested a generalization of this result to several complex variables, using control theory. We confirm this, namely we prove that the class of mappings which have an $$A$$ A -parametric representation obtained by solving the Loewner differential equation with infinitesimal generators which take values in the set of extreme points of the Carathéodory family in several complex variables is dense in the class of mappings with $$A$$ A -parametric representation, where $$A$$ A is a linear operator from $${\mathbb {C}}^n$$ C n to $${\mathbb {C}}^n$$ C n such that $$k_+(A)<2m(A)$$ k + ( A ) < 2 m ( A ) , $$k_+(A)$$ k + ( A ) is the Lyapunov index of $$A$$ A and $$m(A)=\min \{\mathfrak {R}\langle Az,z\rangle | z\in \mathbb {C}^n,\Vert z\Vert =1\}$$ m ( A ) = min { R ⟨ A z , z ⟩ | z ∈ C n , ‖ z ‖ = 1 } . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

A Density Result for Parametric Representations in Several Complex Variables

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Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-014-0102-y
Publisher site
See Article on Publisher Site

Abstract

C. Loewner proved that the class of functions which have a parametric representation obtained by solving the Loewner differential equation with a driving term is dense in the class $$S$$ S of normalized univalent functions on the unit disc, because it contains all the single-slit mappings. I. Graham, H. Hamada, G. Kohr and M. Kohr suggested a generalization of this result to several complex variables, using control theory. We confirm this, namely we prove that the class of mappings which have an $$A$$ A -parametric representation obtained by solving the Loewner differential equation with infinitesimal generators which take values in the set of extreme points of the Carathéodory family in several complex variables is dense in the class of mappings with $$A$$ A -parametric representation, where $$A$$ A is a linear operator from $${\mathbb {C}}^n$$ C n to $${\mathbb {C}}^n$$ C n such that $$k_+(A)<2m(A)$$ k + ( A ) < 2 m ( A ) , $$k_+(A)$$ k + ( A ) is the Lyapunov index of $$A$$ A and $$m(A)=\min \{\mathfrak {R}\langle Az,z\rangle | z\in \mathbb {C}^n,\Vert z\Vert =1\}$$ m ( A ) = min { R ⟨ A z , z ⟩ | z ∈ C n , ‖ z ‖ = 1 } .

Journal

Computational Methods and Function TheorySpringer Journals

Published: Jan 8, 2015

References