Math 472/572 - Fourier analysis and wavelets

### Fall 2009

This class is cross-listed as:

• Math 472 - Call # 33944 - Fourier analysis and wavelets
• Math 572 - Call # 33945 - Fourier analysis and wavelets

Here are quick links to the homework, and to the textbook.

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 applications.
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, mean-square convergence.
• 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 distributions.
• Time/frequency analysis, windowed Fourier Transform, Gabor basis, Wavelets.
• 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 Matlab/Octave 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.

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.

Recommended Texts:

• An Introduction to Wavelets Through Linear Algebra by 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; ISBN 0-534-37660-6
• A Wavelet Tour of Signal Processing. The sparse way by S. Mallat, 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; ISBN o-486-63331-4
• Fourier Analysis: An Introduction by E. M. Stein and R. Shakarchi, Princeton lectures in Analysis I, Princeton University Press, 2003; ISBN 0-691-11384-X.
• Fourier Analysis by T. Korner. Cambridge University Press, 1989; ISBN 0-521-38991-7

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:

More Mathematical
• 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. Weiss, 1996.
• Wavelets and operators, by Yves Meyer, 1992.
More applied/friendlier
• 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.
For a wider audience or emphasis on applications
• 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.
There is a wealth of information available at wavelet digest

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.

Last updated: August 24, 2009