An introduction to the basic properties of wavelets, for mathematicians, engineers, physicists, and graduate students familiar with the Fourier transform and its basic properties. Illustrates the construction of these algorithms by means of multiresolution analyses, and shows how they can be used in applications to characterize various stypes of function spaces. Contains chapters on sine and cosine bases, band-limited wavelets, representation of functions and function space, frames, and discrete transforms and algorithms. Annotation c. by Book News, Inc., Portland, Or.
Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory. In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function. This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications. The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets.
The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications. Very little mathematical background is needed to follow this material. A complete treatment of band-limited wavelets follows. These are characterized by some elementary equations, allowing the authors to introduce many new wavelets. Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported wavelets.
Some of the topics treated include:
Several bases generated by a single function via translations and dilations
Multiresolution analysis, compactly supported wavelets, and spline wavelets
Band-limited wavelets
Unconditionality of wavelet bases
Characterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functions
The authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations. Material related to applications is provided, and constructions of splines wavelets are presented.
Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets.
