Journal of Fourier Analysis and Applications

, Volume 23, Issue 4, pp 778–809 | Cite as

Nonlinear Phase Unwinding of Functions

  • Ronald R. Coifman
  • Stefan Steinerberger


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
$$\begin{aligned} F = a_1 B_1 + a_2 B_1 B_2 + a_3 B_1 B_2 B_3 + \cdots \end{aligned}$$
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.


Blaschke factorization Phase unwinding Dirichlet space Carleson formula 

Mathematics Subject Classification

Primary 30B50 Secondary 30A10 42C40 65T99 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Program in Applied MathematicsYale UniversityNew HavenUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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