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Iterative Variable-Blaschke Factorization

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Abstract

Blaschke factorization allows us to write any holomorphic function F as a formal series

$$\begin{aligned} F = a_0 B_0 + a_1 B_0 B_1 + a_2 B_0 B_1 B_2 + \cdots \end{aligned}$$

where \(a_i \in \mathbb {C}\) and \(B_i\) is a Blaschke product. We introduce a more general variation of the canonical Blaschke product and study the resulting formal series. We prove that the series converges exponentially in the Dirichlet space given a suitable choice of parameters if F is a polynomial and we provide explicit conditions under which this convergence can occur. Finally, we derive analogous properties of Blaschke factorization using our new variable framework.

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Acknowledgements

The authors were supported by the 2018 Summer Math Research at Yale (SUMRY) program. The authors would like to thank Stefan Steinerberger and Hau-Tieng Wu for their endless mentorship and Lihui Tan for helpful feedback on a preliminary draft of this paper.

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  1. Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT, 06511, USA

    Maxime Lukianchikov, Vladyslav Nazarchuk & Christopher Xue

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  1. Maxime Lukianchikov
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  2. Vladyslav Nazarchuk
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  3. Christopher Xue
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Correspondence to Maxime Lukianchikov.

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Communicated by Daniel Aron Alpay.

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Lukianchikov, M., Nazarchuk, V. & Xue, C. Iterative Variable-Blaschke Factorization. Complex Anal. Oper. Theory 13, 3795–3824 (2019). https://doi.org/10.1007/s11785-019-00931-0

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