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Almost everywhere convergence of the phase unwinding series in H^2

Establish almost everywhere convergence on the unit circle for the nonlinear phase unwinding series associated with a function F in the Hardy space H^2 on the unit disk, where the series is defined by iteratively factoring F−F(0) by the Blaschke product of its zeros and expanding F(z)=F(0)+F1(0)B1(z)+...+Fn(0)B1(z)...Bn(z)+... .

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Background

The paper discusses the Malmquist–Takenaka (MT) system and a related nonlinear phase unwinding decomposition defined via iterative Blaschke factorization. Given F in H2, one constructs B1 as the Blaschke product with the zeros of F−F(0), sets F1=(F−F(0))/B1, and iterates to obtain a formal series F(z)=F(0)+F1(0)B1(z)+... . This unwinding series coincides with an MT series for a particular choice of the points defining the Blaschke products.

While numerical simulations suggest rapid convergence of the unwinding series, general convergence properties remain unresolved. The present paper proves almost everywhere convergence for a specific wavelet-type MT system, but explicitly notes that the almost everywhere convergence of the unwinding series for arbitrary F in H2 remains an open question.

References

Whether the unwinding series unwinding converges almost everywhere for $F\in H2()$ or whether unwinding converges "fast" for "most" functions remain interesting open questions.

unwinding:

F(z)=F(0)+F1(0)B1(z)+⋯+Fn(0)B1(z)…Bn(z)+⋯ .F(z) = F(0)+F_1(0)B_1(z)+\dots + F_n(0)B_1(z)\dots B_n(z)+\cdots \, .

— Almost everywhere convergence of a wavelet-type Malmquist-Takenaka series (2404.13296 - Mnatsakanyan, 20 Apr 2024) in Section 1 (Introduction), after Equation (1.4)