Semester

Fall

Date of Graduation

2007

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Sherman Riemenschneider.

Abstract

This dissertation, entitled Nonlinear Approximation Using Blaschke Polynomials, is motivated by questions arising from Empirical Mode Decomposition (EMD). EMD is a signal processing method which decomposes input signals into components called intrinsic mode functions (IMFs). These IMFs often have the desirable property that the instantaneous frequency of their analytic signals is positive. However, this is not always the case.;The first two chapters are introductions to approximation in general, and Empirical Mode Decomposition, respectively.;The third chapter presents a characterization of which analytic signals have the property of non-negative instantaneous frequency. These 'analytic signals with non-negative instantaneous frequency' (ASNIFs) are described using Hardy spaces on the unit disc. Analogous results are also found for analytic signals which are boundary values fof elements of Hardy spaces on the half-plane.;The fourth chapter describes in general terms how one might construct an approximation method using ASNIFs, or possibly some restricted subclass of ASNIFs, as approximants.;The fifth chapter introduces a special set of ASNIFs: Blaschke polynomials. These are functions which are linear combinations of Blaschke products, which are a special kind of ASNIF. Blaschke polynomials are a natural extension of other classical approximating sets, and have some interesting properties of their own. Some of these properties are explored.;The sixth chapter contains an implementation of a signal decomposition method using Blaschke polynomials, and a discussion of the results.;The seventh chapter further explores the conclusions of this work, and lays out future research questions.

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