Wavelet Theory (PDF)

A self-contained, elementary introduction to wavelet theory and
applications

Exploring the growing relevance of wavelets in the field of
mathematics, Wavelet Theory: An Elementary Approach with
Applications provides an introduction to the topic, detailing the
fundamental concepts and presenting its major impacts in the world
beyond academia. Drawing on concepts from calculus and linear
algebra, this book helps readers sharpen their mathematical proof
writing and reading skills through interesting, real-world
applications.

The book begins with a brief introduction to the fundamentals of
complex numbers and the space of square-integrable functions. Next,
Fourier series and the Fourier transform are presented as tools for
understanding wavelet analysis and the study of wavelets in the
transform domain. Subsequent chapters provide a comprehensive
treatment of various types of wavelets and their related concepts,
such as Haar spaces, multiresolution analysis, Daubechies wavelets,
and biorthogonal wavelets. In addition, the authors include two
chapters that carefully detail the transition from wavelet theory
to the discrete wavelet transformations. To illustrate the
relevance of wavelet theory in the digital age, the book includes
two in-depth sections on current applications: the FBI Wavelet
Scalar Quantization Standard and image segmentation.

In order to facilitate mastery of the content, the book features
more than 400 exercises that range from theoretical to
computational in nature and are structured in a multi-part format
in order to assist readers with the correct proof or solution.
These problems provide an opportunity for readers to further
investigate various applications of wavelets. All problems are
compatible with software packages and computer labs that are
available on the book's related Web site, allowing readers to
perform various imaging/audio tasks, explore computer wavelet
transformations and their inverses, and visualize the applications
discussed throughout the book.

Requiring only a prerequisite knowledge of linear algebra and
calculus, Wavelet Theory is an excellent book for courses in
mathematics, engineering, and physics at the upper-undergraduate
level. It is also a valuable resource for mathematicians,
engineers, and scientists who wish to learn about wavelet theory on
an elementary level.