Math 472/572 - Fourier analysis and wavelets
FOURIER ANALYSIS AND WAVELETS
This class is cross-listed as:
- Math 472 - Call # 33944 - Fourier analysis and wavelets
- Math 572 - Call # 33945 -
Fourier analysis and wavelets
(Graduate students please register
in Math 572.)
Here are quick links to the homework, and to the
This course is an introduction to Fourier Analysis and Wavelets.
It has been specifically
designed for engineers, scientists, statisticians and mathematicians interested
in the basic mathematical ideas underlying Fourier analysis, wavelets and their
This course integrates the
classical Fourier Theory with its latest offspring, the Theory of Wavelets.
Wavelets and Fourier analysis are invaluable tools for researchers in many
areas of mathematics and the applied sciences, to name a few:
signal processing, statistics, physics,
differential equations, numerical analysis, geophysics, medical
imaging, fractals, harmonic analysis, etc. It is their
multidisciplinary nature that makes these theories so appealing.
Topics will include:
- Fourier series: pointwise convergence, summability methods,
- Discrete Fourier Transform (including Fast Fourier Transform),
and Discrete Haar Transform (including Fast Haar Transform)
- Fourier transform on the line. Time-frequency diccionary.
Heisenberg's Uncertainty Principle, Sampling theorems and
other applications. Including excursions into Lp spaces and
- Time/frequency analysis, windowed Fourier Transform, Gabor basis,
- Multiresolution analysis on the line.
Prime example: the Haar basis. Basic wavelets examples:
Shannon's and Daubechies' compactly supported wavelets.
Time permiting we will explore variations over the theme
of wavelets: Biorthogonal
wavelets, wavelet packets, wavelets on the interval, and
two-dimentional wavelets for image processing.
Numerical experiments are necessary to fully understand the scope of
the theory. We will let the students explore this realm according
to their interests.
The use of some Wavelet Toolbox will be encouraged.
There exists a
WAVELAB 850 package which is
based software designed by
a team at Stanford and available for free on the Internet.
MATLAB 7.4.0 is
available in the Mathematics and Statistics Department Computer Laboratory.
Grades: Grades will be based on homeworks,
Prerequisites: Linear algebra and advanced calculus, or permission
from the instructor.
Textbook: We will be using a preliminary version of a book
that I am writing with my colleague Lesley Ward from University
of South Australia. I will be posting chapters on the course webpage as
the semester evolves. The book is called
Harmonic Analysis: From Fourier to Haar.
I appreciate all the feedback I can get from you, because now is
our opportunity to make meaningful changes before the book
goes into print.
- An Introduction to Wavelets Through Linear Algebra
Michael Frazier Springer Verlag, Feb 1999; ISBN: 0387986391.
- Introduction to Fourier Analysis and Wavelets
by Mark A. Pinsky.
The Brooks/Cole Series in Advanced Mathematics, 2002;
A Wavelet Tour of Signal Processing. The sparse way
Third Edition, Academic Press, 2008; ISBN 978-0123743701
There are many excellent books devoted to the classical theory of
Fourier analysis (starting with A. Zygmund's Trigonometric Series ,
and following with a long list).
- Fourier series and integrals
by H. Dym and H.P. McKean.
Academic Press, 1986; ISBN: 0122264517
- An Introduction to Harmonic Analysis
by Y. Katznelson.
Dover Publications Inc. New York, NY 1976;
- Fourier Analysis: An Introduction
by E. M. Stein and R. Shakarchi,
Princeton lectures in Analysis I, Princeton University Press, 2003;
- Fourier Analysis
by T. Korner.
Cambridge University Press, 1989;
In the last 15-20 years there have been
published a number of books on wavelets,
as well as countless articles. Here is a limited guide:
- Ten lectures on wavelets, by Ingrid Daubechies, 1992.
- A mathematical introduction to wavelets, by P. Wojtaszczyk, 1997.
- A first course on wavelets, by E. Hernandez and G.
- Wavelets and operators, by Yves Meyer, 1992.
For a wider audience or emphasis on applications
- Wavelets and Filter Banks, by G. Strang and
T. Nguyen, 1996.
- An introduction to wavelets, by C. K. Chui, 1992.
- A friendly guide to wavelets, by G. Keiser, 1994.
There is a wealth of information available at
- The world according to wavelets, by B. Burke Hubbard,
2nd edition, 1998.
- Wavelets: Tools for science and technology, by
S. Jaffard, Y. Meyer, R. D. Ryan, 2001.
- The illustrated wavelet transform handbook,
by P. S. Addison, 2002.
Americans with Disabilities Act:
Qualified students with disabilities needing appropriate academic
adjustments should contact me as soon as possible to ensure your needs are
met in a timely manner. Handouts are available in alternative accessible
formats upon request.
Department of Mathematics and Statistics,
University of New Mexico
Last updated: August 24, 2009