Math 472/572 - Fourier analysis and wavelets

### Fall 2019

• Instructor: Cristina Pereyra
• E-mail: crisp @ math . unm . edu
• Office: SMLC 320
• Phone: 277-4613 (best by e-mail)
• Schedule: TuTh 9:30-10:45, DSH 227
• Office Hours: Tu 4-5pm, Wed 1-2pm, Th 2-3pm or by appointment

This class is cross-listed as:

• Math 472 - Call # 65998 - Fourier analysis and wavelets
• Math 572 - Call # 65999 - Fourier analysis and wavelets

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

Here is a very well made video about Fourier series recommended by my former PhD student David Weirich.

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 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 important 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. 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! Here is a list of errata so far compiled.