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:
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.
Preliminaries
Orthonormal Bases Generated by a Single Function: The Balian-Low Theorem
Smooth Projections on L2(R)
Local Sine and Cosine Bases and the Construction of Some Wavelets
The Unitary Folding Operators and the Smooth Projections
Notes and References
Multiresolution Analysis and the Construction of Wavelets
Multiresolution Analysis
Construction of Wavelets from a Multiresolution Analysis
The Construction of Compactly Supported Wavelets
Better Estimates for the Smoothness of Compactly Supported Wavelets
Notes and References
Band-Limited Wavelets
Orthonormality
Completeness
The Lemarié-Meyer Wavelets Revisited
Characterization of Some Band-Limited Wavelets
Notes and References
Other Constructions of Wavelets
Franklin Wavelets on the Real Line
Spline Wavelets on the Real Line
Orthonormal Bases of Piecewise Linear Continuous Functions for L2(T)
Orthonormal Bases of Periodic Splines
Periodization of Wavelets Defined on the Real Line
Notes and References
Representation of Functions by Wavelets
Bases for Banach Spaces
Unconditional Bases for Banach Spaces
Convergence of Wavelet Expansions in LP(R)
Pointwise Convergence of Wavelets Expansions
H1 and BMO on R
Wavelets as Unconditional Bases for H1(R) and LP(R) with 1
Biography
Eugenio Hernandez
