1st Edition

A First Course on Wavelets

By Eugenio Hernandez, Guido Weiss Copyright 1996
    512 Pages
    by CRC Press

    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.
  • Bases for L2(R)
    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
    The LemariƩ-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


    Eugenio Hernandez