Iterative Variable-Blaschke Factorization
Provides a theoretical extension to Blaschke factorization for mathematicians working in complex analysis and function spaces, but the results are incremental and specialized.
The paper introduces a generalized Blaschke factorization and proves exponential convergence in the Dirichlet space for polynomials under certain conditions, extending the theoretical foundations of Blaschke decompositions.
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.