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.