A Density Result for Parametric Representations in Several Complex Variables

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• 作者：Mihai Iancu
• 关键词：Control ; Loewner differential equation ; Parametric representation ; Reachable family ; Univalent mapping ; Primary 32H02 ; Secondary 30C45 ; 93B03
• 刊名：Computational Methods and Function Theory
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：15
• 期：2
• 页码：247-262
• 全文大小：472 KB
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• 作者单位：Mihai Iancu (1)
  
  1. Faculty of Mathematics and Computer Science, Babe?-Bolyai University, 1 M. Kog?lniceanu Str., 400084, Cluj-Napoca, Romania
  
• 刊物主题：Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable;
• 出版者：Springer Berlin Heidelberg
• ISSN：2195-3724

文摘

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 \(S\) 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 \(A\)-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 \(A\)-parametric representation, where \(A\) is a linear operator from \({\mathbb {C}}^n\) to \({\mathbb {C}}^n\) such that \(k_+(A)<2m(A)\), \(k_+(A)\) is the Lyapunov index of \(A\) and \(m(A)=\min \{\mathfrak {R}\langle Az,z\rangle | z\in \mathbb {C}^n,\Vert z\Vert =1\}\).