Nonlinear Phase Unwinding of Functions

Nonlinear Phase Unwinding of Functions 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 $$\begin{aligned} F = a_1 B_1 + a_2 B_1 B_2 + a_3 B_1 B_2 B_3 + \cdots \end{aligned}$$ F = a 1 B 1 + a 2 B 1 B 2 + a 3 B 1 B 2 B 3 + ⋯ where $$a_i \in \mathbb {C}$$ a i ∈ C and $$B_i$$ 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 $$L^2$$ L 2 for functions in the Dirichlet space $$\mathcal {D}$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Fourier Analysis and Applications Springer Journals

Nonlinear Phase Unwinding of Functions

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Fourier Analysis; Signal,Image and Speech Processing; Abstract Harmonic Analysis; Approximations and Expansions; Partial Differential Equations; Mathematical Methods in Physics
ISSN
1069-5869
eISSN
1531-5851
D.O.I.
10.1007/s00041-016-9489-3
Publisher site
See Article on Publisher Site

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 $$\begin{aligned} F = a_1 B_1 + a_2 B_1 B_2 + a_3 B_1 B_2 B_3 + \cdots \end{aligned}$$ F = a 1 B 1 + a 2 B 1 B 2 + a 3 B 1 B 2 B 3 + ⋯ where $$a_i \in \mathbb {C}$$ a i ∈ C and $$B_i$$ 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 $$L^2$$ L 2 for functions in the Dirichlet space $$\mathcal {D}$$ 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.

Journal

Journal of Fourier Analysis and ApplicationsSpringer Journals

Published: Jul 11, 2016

References

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