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Computing Polynomial Conformal Models for Low-Degree Blaschke Products

Computing Polynomial Conformal Models for Low-Degree Blaschke Products For any finite Blaschke product B, there is an injective analytic map $$\varphi :{\mathbb {D}}\rightarrow {\mathbb {C}}$$ φ : D → C and a polynomial p of the same degree as B such that $$B=p\circ \varphi $$ B = p ∘ φ on $${\mathbb {D}}$$ 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 $$\varphi $$ φ . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Computing Polynomial Conformal Models for Low-Degree Blaschke Products

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-018-0259-x
Publisher site
See Article on Publisher Site

Abstract

For any finite Blaschke product B, there is an injective analytic map $$\varphi :{\mathbb {D}}\rightarrow {\mathbb {C}}$$ φ : D → C and a polynomial p of the same degree as B such that $$B=p\circ \varphi $$ B = p ∘ φ on $${\mathbb {D}}$$ 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 $$\varphi $$ φ . 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.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Jan 18, 2019

References