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.
Basic wavelet theory is a natural topic for such a course. By name, 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 sufficiently 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 first two sections of chapter 1. In the remainder of chapter 1 we review linear algebra. Students should be familiar with the basic definitions in sections 1.3 and 1.4. From our viewpoint, linear transformations are the primary object of study; 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 benefit from seeing it again. In section 1.5, we ask how to pick a basis to simplify the matrix representation of a given linear transformation as much as possible. We then focus on the simplest case, when the linear transformation is diagonalizable. In section 1.6, we discuss inner products and orthonormal bases. We end with a statement of the spectral theorem for matrices, whose proof is outlined in the exercises. This is beyond the experience of most undergraduates.
Prologue: Compression of the FBI Fingerprint Files
1. Background: Complex Numbers and Linear Algebra
2. The Discrete Fourier Transform
3. Wavelets on Zn
4. Wavelets on Z
5. Wavelets on R
6. Wavelets and Differential Equations
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