Iterative Variable-Blaschke Factorization

TL;DR

This paper introduces a generalized iterative Blaschke factorization method, proving exponential convergence for polynomials in the Dirichlet space and establishing new properties of the factorization process.

Contribution

It develops a novel variation of Blaschke products and analyzes their convergence, extending classical results and providing explicit conditions for convergence.

Findings

Exponential convergence of the series for polynomials in the Dirichlet space.

Explicit conditions for convergence based on parameter choices.

New properties of Blaschke factorization derived from the variable framework.

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.