Iterative Variable-Blaschke Factorization

Maxime Lukianchikov; Vladyslav Nazarchuk; Christopher Xue

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arXiv · 1810.01458

Iterative Variable-Blaschke Factorization

Maxime Lukianchikov·Vladyslav Nazarchuk·Christopher Xue

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Abstract

Blaschke factorization allows us to write any holomorphic function $F$ as a formal series $$ F = a_0 B_0 + a_1 B_0 B_1 + a_2 B_0 B_1 B_2 + \cdots$$ 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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BibTeX RIS

Maxime Lukianchikov, Vladyslav Nazarchuk, Christopher Xue. 2018-10-02. Iterative Variable-Blaschke Factorization. https://arxiv.org/abs/1810.01458

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