A First Course on Wavelets: 1st Edition (Hardback) book cover

A First Course on Wavelets

1st Edition

By Eugenio Hernandez, Guido Weiss

CRC Press

489 pages

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Hardback: 9780849382741
pub: 1996-09-12
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Description

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.

  • Table of Contents

    Bases for L2(R)

    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

    About the Series

    Studies in Advanced Mathematics

    Learn more…

    Subject Categories

    BISAC Subject Codes/Headings:
    MAT000000
    MATHEMATICS / General
    MAT003000
    MATHEMATICS / Applied
    MAT007000
    MATHEMATICS / Differential Equations
    MAT037000
    MATHEMATICS / Functional Analysis