1 Math 472/572 - Fourier analysis and wavelets

### Fall 2007

This class is cross-listed as:

• Math 472 - Call # 25868 - Fourier analysis and wavelets
• Math 572 - Call # 25875 - Fourier analysis and wavelets

This course is an introduction to Fourier Analysis and Wavelets. It has been specifically designed for engineers, scientists, and mathematicians interested in the basic 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).
• Fourier transform on the line. Time-frequency diccionary. Heisenberg's Uncertainty Principle, Sampling theorems and other applications.
• 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. The use of some Wavelet Toolbox will be encouraged. There exists a WAVELAB package which is Matlab/Octave based software designed by a team at Stanford and available for free on the Internet. MATLAB 7.2 is available in the Mathematics and Statistics Department Computer Laboratory (I think).

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.
• A Wavelet Tour of Signal Processing by S. Mallat, Second Edition, Academic Press, 1999; SBN: 12466606X
• Fourier series and integrals by H. Dym and H.P. McKean. Academic Press, 1986; ISBN: 0122264517

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). In the last ten 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. Hern\'{a}ndez 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

Last updated: August 15, 2007