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Fast convergence of the phase unwinding series for most functions in H^2

Determine whether the nonlinear phase unwinding series defined by iterative Blaschke factorization converges "fast" for "most" functions in the Hardy space H^2 on the unit disk.

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Background

The unwinding series is motivated by the goal of improving convergence behavior relative to classical Fourier series by adaptively selecting Blaschke factors to reduce coefficients or exploit structure. Numerical experiments have suggested very fast convergence of this series in practice.

Despite progress on convergence for special MT systems, the paper notes that the question of whether the unwinding series converges rapidly for most functions in H2 remains explicitly open.

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)