Studies in tight frames and polar derivatives

Keywords

Approximation theory; Frames (Combinatorial analysis); Harmonic analysis

Abstract

Since their discovery in the early 1950's, frames have emerged as an important tool in many mathematical applications, especially in the area of signal processing. Frames play a significant role in the study of wavelets, and the recent interest in that branch of mathematics has reawakened interest in the formal study of the mathematical theory of frames. In chapter one, we provide an overview of some of the basic definitions and concepts underlying frame theory. Special attention is given to finite frames in real and complex Hilbert spaces and the use of frame potential as a tool to analyze the structure of these frames. In the next three chapters of the thesis, we investigate finite frames with specific structural constraints. In the second chapter, we study the theory of equiangular and equal-IP tight frames-frames in which the angular structure is of paramount importance. In the third chapter, we develop a theoretical analysis of tight frames on the unit square in IR2 . These frames allow us to consider issues in the construction and analysis of frames that are both nonuniform and tight. In the fourth chapter, we analyze uniform tight frames that locally minimize the total potential of several selected central force functions. In the fifth chapter, we consider two definitions for a polar derivative for rational functions. We outline some of the difficulties involved in the definition process and develop several inequalities based on the proposed definitions.

Notes

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Graduation Date

2003

Advisor

Han, Deguang

Degree

Doctor of Philosophy (Ph.D.)

College

College of Arts and Sciences

Department

Electrical Engineering and Computer Science

Format

PDF

Pages

94 p.

Language

English

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

Identifier

DP0029093

Subjects

Arts and Sciences -- Dissertations, Academic; Dissertations, Academic -- Arts and Sciences

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