Bound Volume Number

4

Document Type

Honors Capstone Project

Date of Submission

Spring 5-1-2015

Capstone Advisor

Professor Leonid Kovalev

Honors Reader

Professor Claudia Miller

Capstone Major

Mathematics

Capstone College

Arts and Science

Audio/Visual Component

no

Keywords

applied linear algebra, finite frame, Fourier series, measure theory

Capstone Prize Winner

no

Won Capstone Funding

no

Honors Categories

Social Sciences

Subject Categories

Algebra | Applied Mathematics

Abstract

In applied linear algebra, the term frame is used to refer to a redundant or linearly dependent coordinate system. The concept was introduced in the study of Fourier series and is pertinent in signal processing, where the reconstruction property for finite frames allows for redundant transmission of data to guard against losses due to noise. We give a brief introduction to the theory of finite frames in Section 1, including the major results that allow for the easy construction and description of frames. The subsequent sections relate to the theoretical importance of frames. As a natural extension of the definition of basis, we are lead naturally to ask the same topological questions for the space of frames as we do for the space of bases, GLn(R). Two particular questions are explored in Sections 2 and 3. The reconstruction property for finite frames leads to a natural generalization in the realm of measure theory. This is the scope of Section 4, culminating in the approximation theorem for “frame measures”.

Creative Commons License

Creative Commons Attribution 3.0 License
This work is licensed under a Creative Commons Attribution 3.0 License.

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