Math 472/572 - Fourier analysis and wavelets

### Fall 2013

This class is cross-listed as:

• Math 472 - Call # 44645 - Fourier analysis and wavelets
• Math 572 - Call # 44646 - Fourier analysis and wavelets
(Graduate students please register in Math 572.)

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, 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 based software designed by a team at Stanford and available for free on the Internet. MATLAB 7.12.0 is available in the Mathematics and Statistics Department Computer Laboratory.

Textbook: We will use a book that I wrote with my colleague Lesley Ward from University of South Australia published last year. The book is called Harmonic Analysis: From Fourier to Wavelets , Student Mathematical Library Series, Volume 63, American Mathematical Society 2012. I appreciate all the feedback I can get from you in terms of typos, erratas, and possible improvements for the second edition!

Grades: Grades will be based on homeworks, projects and/or take-home exams.

Prerequisites: Linear algebra and advanced calculus, or permission from the instructor.

Recommended Texts: The literature for Fourier Analysis and Wavelets is large, here you will find a commented list of texts
There is a wealth of information available at wavelet.org

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 19, 2013