While the periodic sine and cosine curves are used to describe many phenomena in nature and are covered very early in the mathematical education sequence, wavelets are generally ignored. A wavelet can be considered a brief oscillation that starts at zero and then moves in one direction before returning to zero. Wavelets have parameters that can be custom crafted to have specific properties. One of the primary uses for wavelets is in signal processing, specifically for extracting information from a noisy signal. They can also be combined in prescribed ways in a manner similar to Fourier series and Fourier transforms, a topic which is covered in the second half of the book.
When speaking of a mathematics book, the phrase “made easy” must often be interpreted in the broadest possible sense, from being strictly correct to being correct within a narrow context of previous knowledge. That is the case here, although the interpretation is closer than usual to being strictly correct. Students with the full calculus sequence and linear algebra behind them should have no trouble understanding it after a careful reading.
Wavelets are a valuable mathematical tool that is growing more valuable over time as more data is passed between computers as well as between sensors and computers. Engineers and other users of applied mathematics have a growing need for workable knowledge in the area of wavelets and this book is very suitable as a text for an upper division class in wavelets. A large number of exercises, including many based on applications, are included, albeit without solutions. This book could also be used as a resource for self-study by the determined student.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.
Preface.- Outline.- A. Algorithms for Wavelet Transforms.- Haar's Simple Wavelets.- Multidimensional Wavelets and Applications.- Algorithms for Daubechies Wavelets.- B. Basic Fourier Analysis.- Inner Products and Orthogonal Projections.- Discrete and Fast Fourier Transforms.- Fourier Series for Periodic Functions.- C. Computation and Design of Wavelets.- Fourier Transforms on the Line and in Space.- Daubechies Wavelets Design.- Signal Representations with Wavelets. D. Directories.- Acknowledgements.- Collection of Symbols.- Bibliography.- Index.