Nonlinear phase unwinding of functions

Abstract

We study a natural nonlinear analogue of Fourier series. Iterative Blaschke factorization allows one to formally write any holomorphic function $F$ as a series which successively unravels or unwinds the oscillation of the function $$ F = a_1 B_1 + a_2 B_1 B_2 + a_3 B_1 B_2 B_3 + \dots$$ where $a_i \in \mathbb{C}$ and $B_i$ is a Blaschke product. Numerical experiments point towards rapid convergence of the formal series but the actual mechanism by which this is happening has yet to be explained. We derive a family of inequalities and use them to prove convergence for a large number of function spaces: for example, we have convergence in $L^2$ for functions in the Dirichlet space $\mathcal{D}$. Furthermore, we present a numerically efficient way to expand a function without explicit calculations of the Blaschke zeroes going back to Guido and Mary Weiss.