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Branched circle packings and discrete Blaschke products


Author: Tomasz Dubejko
Journal: Trans. Amer. Math. Soc. 347 (1995), 4073-4103
MSC: Primary 30D50; Secondary 30G25, 52C15
DOI: https://doi.org/10.1090/S0002-9947-1995-1308008-1
MathSciNet review: 1308008
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Abstract: In this paper we introduce the notion of discrete Blaschke products via circle packing. We first establish necessary and sufficient conditions for the existence of finite branched circle packings. Next, discrete Blaschke products are defined as circle packing maps from univalent circle packings that properly fill $ D = \{ z:\left\vert z \right\vert < 1\} $ to the corresponding branched circle packings that properly cover $ D$. It is verified that such maps have all geometric properties of their classical counterparts. Finally, we show that any classical finite Blaschke product can be approximated uniformly on compacta of $ D$ by discrete ones.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1308008-1
Keywords: Circle packing, Blaschke products, discrete analytic functions
Article copyright: © Copyright 1995 American Mathematical Society

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