Enhanced descriptions from Syndetics:

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.

Table of contents provided by Syndetics

• Preface
• Acknowledgments
• 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
• Bibliography
• Index