Synopses & Reviews
This text was originally written for a "Capstone" course at Michigan State University. A Capstone course is intended for undergraduate mathematics majors, as one of the final courses taken in their undergraduate curriculum. Its purpose is to bring together different topics covered in the undergraduate curriculum and introduce students to current developments in mathematics and their applications. Basic wavelet theory seems to be a perfect topic for such a course. As a subject, it dates back only to 1985. Since then there has been an explosion of wavelet research, both pure and applied. Wavelet theory is on the boundary between mathematics and engineering. In particular it is a good topic for demonstrating to students that mathematics research is thriving in the modern day: Students can see non-trivial mathematics ideas leading to natural and important applications, such as video compression and the numerical solution of differential equations. The only prerequisites assumed are a basic linear algebra background and a bit of analysis background. This text is intended to be as elementary an introduction to wavelet theory as possible. It is not intended as a thorough or authoritative reference on wavelet theory.
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."
The mathematical theory of wavelets is less than 15 years old, yet already wavelets have become a fundamental tool in many areas of applied mathematics and engineering. This introduction to wavelets assumes a basic background in linear algebra (reviewed in Chapter 1) and real analysis at the undergraduate level. Fourier and wavelet analyses are first presented in the finite-dimensional context, using only linear algebra. Then Fourier series are introduced in order to develop wavelets in the infinite-dimensional, but discrete context. Finally, the text discusses Fourier transform and wavelet theory on the real line. The computation of the wavelet transform via filter banks is emphasized, and applications to signal compression and numerical differential equations are given. This text is ideal for a topics course for mathematics majors, because it exhibits and emerging mathematical theory with many applications. It also allows engineering students without graduate mathematics prerequisites to gain a practical knowledge of wavelets.
This book offers a solid, yet basic view of wavelet theory as an active part of modern day mathematical research leading to important applications such as video compression and the numerical solution of differential equations.
Wavelet theory is on the boundary between mathematics and engineering, making it ideal for demonstrating to students that mathematics research is thriving in the modern day. Students can see non-trivial mathematics ideas leading to natural and important applications, such as video compression and the numerical solution of differential equations. The only prerequisites assumed are a basic linear algebra background and a bit of analysis background. Intended to be as elementary an introduction to wavelet theory as possible, the text does not claim to be a thorough or authoritative reference on wavelet theory.
Includes bibliographical references (p. 484-490) and index.
Table of Contents
Prologue: Compression of the FBI Fingerprint Files
1 Background: Complex Numbers and Linear Algebra
1.1 Real Numbers and Complex Numbers
1.2 Complex Series, Euler's Formula, and the Roots of Unity
1.3 Vector Spaces and Bases
1.4 Linear Transformations, Matrices, and Change of Basis
1.5 Diagonalization of Linear Transformations and Matrices
1.6 Inner Products, Orthonormal Bases, and Unitary Matrices
2 The Discrete Fourier Transform
2.1 Basic Properties of the Discrete Fourier Transform
2.2 Translation-Invariant Linear Transformations
2.3 The Fast Fourier Transform
3 Wavelets on $bZ_N$
3.1 Construction of Wavelets on $bZ_N$: The First Stage
3.2 Construction of Wavelets on $bZ_N$: The Iteration Step
3.3 Examples and Applications
4 Wavelets on $bZ$
4.1 $\ell ^2(bZ)$
4.2 Complete Orthonormal Sets in Hilbert Spaces
4.3 $L^2([-\pi ,\pi ))$ and Fourier Series
4.4 The Fourier Transform and Convolution on $\ell ^2(bZ)$
4.5 First-Stage Wavelets on $bZ$
4.6 The Iteration Step for Wavelets on $bZ$
4.7 Implementation and Examples
5 Wavelets on $bR$
5.1 $L^2(bR)$ and Approximate Identities
5.2 The Fourier Transform on $bR$
5.3 Multiresolution Analysis and Wavelets
5.4 Construction of Multiresolution Analyses
5.5 Wavelets with Compact Support and Their Computation
6 Wavelets and Differential Equations
6.1 The Condition Number of a Matrix
6.2 Finite Difference Methods for Differential Equations
6.3 Wavelet-Galerkin Methods for Differential Equations