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Nonlinear Phase Unwinding of Functions

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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

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

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.

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Authors and Affiliations

  1. Department of Mathematics, Program in Applied Mathematics, Yale University, New Haven, CT, 06510, USA

    Ronald R. Coifman

  2. Department of Mathematics, Yale University, New Haven, CT, 06510, USA

    Stefan Steinerberger

Authors
  1. Ronald R. Coifman
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  2. Stefan Steinerberger
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Correspondence to Stefan Steinerberger.

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Communicated by Thomas Strohmer.

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Coifman, R.R., Steinerberger, S. Nonlinear Phase Unwinding of Functions. J Fourier Anal Appl 23, 778–809 (2017). https://doi.org/10.1007/s00041-016-9489-3

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  • DOI: https://doi.org/10.1007/s00041-016-9489-3

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