文档介绍：An Introduction
to Wavelets Through
Linear Algebra
Michael W. Frazier
Springer
Preface
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 pression and the numericalsolution
of differential equations. The author believes that the essentials of
wavelet theory are sufficiently elementary to be taught essfully
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 plex 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 and . From our
viewpoint, linear transformations are the primary object of study;
v
vi Preface
a matrix arises as a realization of a linear transformation. Many
students may have been exposed to the materialon change of basis
in section , but may benefit from seeing it again. In section ,
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 , we discuss inner products and orthonormalbases. We
end with a statement of the spectraltheorem for matrices, whose
proof is outlined in th