Blaschke Decompositions on Weighted Hardy Spaces

Abstract

Recently, several authors have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either the unwinding series or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of the boundary value of an analytic function $F: \partial {\mathbb {D}}\rightarrow {\mathbb {C}}$ and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be efficiently approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was initiated 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 single zero, $\alpha \in {\mathbb {D}}$, of F and its Blaschke decomposition. That work also noted that in many specific examples, the unwinding series of F converges at an exponential rate to F, which when coupled with an efficient algorithm to compute a Blaschke decomposition, has led to the hope that this process will provide 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 ${\mathbb {D}}$. This is meaningful, since there are many functions that meet this criterion but do not extend analytically to ${\mathbb {D}}_{1+\epsilon }$ for any $\epsilon >0$, for example $F(z)=\log (1-z)$. By studying the growth rate of the weights, we improve the bounds provided by Coifman and Steinerberger to obtain new estimates containing all roots of F in ${\mathbb {D}}$. This provides us with new insights into Blaschke decompositions on classical function spaces including the Hardy–Sobolev spaces and weighted Bergman spaces, which correspond to making specific choices for the aforementioned weights. Further, we state a sufficient condition on the weights for our improved bounds to hold for any function in the Hardy space, ${\mathcal {H}}^2({\mathbb {D}})$, in particular functions with an infinite number of roots in ${\mathbb {D}}$. These results may help to better explain why the exponential convergence of the unwinding series is seen in many numerical examples.