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. Journal of Fourier Analysis and Applications Springer Journals

Nonlinear Phase Unwinding of Functions

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Springer US
Copyright © 2016 by Springer Science+Business Media New York
Mathematics; Fourier Analysis; Signal,Image and Speech Processing; Abstract Harmonic Analysis; Approximations and Expansions; Partial Differential Equations; Mathematical Methods in Physics
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