Outline of Book on First Course in Wavelets with Fourier Analysis

by Al Boggess and Fran Narcowich

The goal of this book is to present some of the recent advances in Fourier analysis, most notably wavelets, to an advanced undergraduate audience. The book starts with the classical ideas of Fourier series and the Fourier transform and progresses to the construction of Daubechies' orthogonal wavelets. Most current books on Fourier analysis at the undergraduate level develop the tools on Fourier analysis and then apply these tools to the solution of ordinary and partial differential equations. In this book, our motivation is signal analysis and the decomposition of a signal into its frequency components.

The development of Fourier analysis in this book serves two purposes. First, Fourier analysis is important in its own right. Second the construction of wavelets uses the tools from Fourier analysis. The development of wavelets is viewed as an extension of Fourier analysis. Wavelets provide the time localization that is not part of standard Fourier series and this time localization is presented as the motivation for looking at wavelets.

We intend this as a book to be used as a reference for a one semester course with a diverse audience of students of mathematics, science and engineering. As a consequence, we keep the level of explanation at a low key level. The key ideas are explained without excessive mathematical rigor. The technical details of some of the proofs are placed in an appendix for the interested reader. We only assume the reader has a background in advanced calculus and linear algebra (for example, the calculus and linear algebra courses taken by the typical engineering student should suffice). At the same time, any physical concepts from signal analysis will be explained in simple terms and without the technical jargon that typically is used in the field.

Outline of Topics

  1. Introduction and Motivation.
    1. The decomposition of a signal into its components
    2. Filtering of noise and image reconstruction
    3. Data compression
    4. Limitations on Trignometric Series
  2. Inner Product Spaces
    1. Motivation.
    2. Definition of Inner product.
    3. The space L^2
    4. Schwarz and Triangle inequalities
    5. Orthogonality
    6. Linear operators and their adjoints
    7. Least squares and linear predictive coding
  3. Fourier Series.
    1. Motivation - historical perspective, pdes, signal analysis
    2. Computation of Fourier Series
    3. Convergence theorems for Fourier series
  4. The Fourier Transform
    1. Informal Development of the Fourier Transform and its inverse
    2. Properties of the Fourier Transform
    3. Linear filters
    4. The Sampling Theorem
    5. Uncertainty Principle
    6. Discrete Fourier Analysis - Fast Fourier Transform
    7. Z-transform
  5. Wavelet Analysis
    1. Motivation
    2. Haar wavelets
    3. Decomposition and Reconstruction via Haar wavelets
    4. Multiresolution analysis
    5. General Decompostion algorithm
    6. General Reconstruction algorithm
    7. Fourier Transform criteria
    8. Daubechies' wavelets
  6. Other Wavelet Topics
    1. Quadrature Mirror Filters
    2. Wavelet Transform
  7. Appendix
    1. Proof of the Fourier Inversion formula
    2. Rigorous convergence proofs.