# Blaschke Decompositions on Weighted Hardy Spaces

## TL;DR

This paper advances the understanding of Blaschke decompositions on weighted Hardy spaces, providing improved bounds and conditions that explain the exponential convergence observed in nonlinear signal approximation methods.

## Contribution

It extends previous work by studying functions in larger Hardy spaces with finitely many roots, improving bounds on weighted norms, and establishing conditions for exponential convergence in signal processing.

## Key findings

- Improved bounds on weighted Hardy norms for Blaschke decompositions.
- New insights into convergence rates on classical function spaces.
- Conditions on weights ensuring exponential convergence for all Hardy space functions.

## Abstract

Recently, several papers have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either nonlinear phase unwinding or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of a complex function $F: \partial\mathbb{D}\to \mathbb{C}$, and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was studied by Coifman and Steinerberger, under the assumption that the complex valued function $F$ has an analytic extension to $\mathbb{D}_{1+\epsilon}$ for some $\epsilon>0$. This provided bounds on weighted Hardy norms involving a function and its Blaschke decomposition. That work also noted that in many examples, the nonlinear unwinding series of a function converges at an exponential rate to the original signal, which when coupled with an efficient algorithm to perform a Blaschke decomposition, has lead to a new and efficient way to approximate signals. In this work, we continue the study of Blaschke decompositions on weighted Hardy Spaces for functions in the larger space $\mathcal{H}^2(\mathbb{D})$ under the assumption that the function has finitely many roots in $\mathbb{D}$. By studying the growth rate of the weights, we improve the bounds provided by Coifman and Steinerberger. This provides us with new insights into Blaschke decompositions on classical function spaces including the Hardy-Sobolev spaces and weighted Bergman spaces. Further, we state a sufficient condition on the weights for our improved bounds to hold for any function in the Hardy space, $\mathcal{H}^2(\mathbb{D})$. These results may help to better explain why the exponential convergence of the unwinding series is seen in many numerical examples.