# An Introduction to Wavelets Through Linear Algebra

## Benefits

• This introduction to wavelets thoroughly covers the basics of the theory Shows non-trivial mathematics leading to natural and important applications, such as video compression and numerical solution of differential equations Includes an interesting prologue which explains the use of wavelet compression in storing the FBIs fingerprint files Requires only a basic linear algebra background along with a bit of analysis Wavelets are a hot area of modern mathematical research

Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xvi
2. Pages 101-164
3. Pages 165-263
4. Pages 265-348
5. Pages 349-450
6. Pages 451-483
7. Back Matter
Pages 484-505

### Introduction

Mathematics majors at Michigan State University take a “Capstone” course near the end of their undergraduate careers. The content of this course varies with each offering. Its purpose is to bring together different topics from the undergraduate curriculum and introduce students to a developing area in mathematics. This text was originally written for a Capstone course. Basicwavelettheoryisanaturaltopicforsuchacourse. Byname, wavelets date back only to the 1980s. On the boundary between mathematics and engineering, wavelet theory shows students that mathematics research is still thriving, with important applications in areas such as image compression and the numerical solution of differential equations. The author believes that the essentials of wavelet theory are suf?ciently elementary to be taught successfully to advanced undergraduates. This text is intended for undergraduates, so only a basic background in linear algebra and analysis is assumed. We do not require familiarity with complex numbers and the roots of unity. These are introduced in the ?rst two sections of chapter 1. In the remainder of chapter 1 we review linear algebra. Students should be familiar with the basic de?nitions in sections 1. 3 and 1. 4. From our viewpoint, linear transformations are the primary object of study; v Preface vi a matrix arises as a realization of a linear transformation. Many students may have been exposed to the material on change of basis in section 1. 4, but may bene?t from seeing it again. In section 1.

### Keywords

Analysis Fourier transform Hilbert space Signal Transformation Wavelet algebra convolution discrete Fourier transform fast Fourier transform fast Fourier transform (FFT) linear algebra

#### Authors and affiliations

1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

### Bibliographic information

• Book Title An Introduction to Wavelets Through Linear Algebra
• Authors Michael W. Frazier
• Series Title Undergraduate Texts in Mathematics
• DOI https://doi.org/10.1007/b97841
• Copyright Information Springer-Verlag New York, Inc. 1999
• Publisher Name Springer, New York, NY
• eBook Packages
• Hardcover ISBN 978-0-387-98639-5
• Softcover ISBN 978-1-4757-7299-9
• eBook ISBN 978-0-387-22653-8
• Series ISSN 0172-6056
• Edition Number 1
• Number of Pages XVI, 503
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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