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Computing Polynomial Conformal Models for Low-Degree Blaschke Products

  • Trevor Richards
  • Malik YounsiEmail author
Article
  • 13 Downloads

Abstract

For any finite Blaschke product B, there is an injective analytic map \(\varphi :{\mathbb {D}}\rightarrow {\mathbb {C}}\) and a polynomial p of the same degree as B such that \(B=p\circ \varphi \) on \({\mathbb {D}}\). Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial p or the associated conformal map \(\varphi \). In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke products of arbitrary finite degree whose zeros are equally spaced on a circle centered at the origin.

Keywords

Conformal models Blaschke products Polynomials Fingerprints 

Mathematics Subject Classification

Primary 30J10 Secondary 30C35 30C10 

Notes

Acknowledgements

The authors thank Don Marshall and the anonymous referees for several helpful suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceMonmouth CollegeMonmouthUSA
  2. 2.Department of MathematicsUniversity of Hawai’i at ManoaHonoluluUSA

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