Computing Polynomial Conformal Models for Low-Degree Blaschke Products
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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 FingerprintsMathematics Subject Classification
Primary 30J10 Secondary 30C35 30C10Notes
Acknowledgements
The authors thank Don Marshall and the anonymous referees for several helpful suggestions.
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