Date of Award

December 2019

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Loredana Lanzani

Second Advisor

Lixin Shen

Keywords

Blaschke Factorization, Weighted Hardy Spaces

Subject Categories

Physical Sciences and Mathematics

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 : ∂D → 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 D_1+ε for some ε > 0. This provided bounds on weighted Hardy norms involving a single zero, α ∈ 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 accomplish three things. Firstly, we continue the study of Blaschke decompositions on weighted Hardy Spaces for functions in the larger space H^2(D) under the assumption that the function has finitely many roots in D. This is meaningful, since there are many functions that meet this criterion but do not extend analytically to D_1+ε for any ε > 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 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, H^2(D), in particular functions with an infinite number of roots in D. Second, we compare the Fourier series and the unwinding series: we show that there are many examples of functions whose unwinding series converges much faster than the Fourier series, but there are also functions for which the Fourier and unwinding series are term wise equal. From the latter, we show the existence of functions that have unwinding series that do not converge exponentially. Lastly, we discuss an efficient algorithm for computing Blaschke decompositions, and apply this algorithm to verify our theoretical results and to gain a better understanding of the underlying mechanics of the unwinding series.

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Open Access

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