Iterative Variable-Blaschke Factorization

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.