Math 472/572 - Fourier Analysis and Wavelets

### Spring 2024

• Instructor: Cristina Pereyra
• E-mail: crisp @ math . unm . edu
• Office: SMLC 320
• E-mail: crisp(at)math(dot)unm(dot)edu
• Schedule: TuTh 12:30-1:45pm, SMLC 352
• Office Hours: TTh 1:45-2:30pm (after class), Wed 4:30-6:00pm (in zoom), and by appointment

This class is cross-listed as:

• Math 472 - Call # 60743 - Fourier analysis and wavelets
• Math 572 - Call # 62189 - Fourier analysis and wavelets
(Graduate students please register in Math 572.)

Here is a quick link to the homework.

Here is a nice 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. Matlab is now available for free for all UNM students. It is installed in all pod computers. You can download it onto your laptop or home computer. Follow the instructions on it.unm.edu/download. You are allowed to install it on two devices. Once installed on your computer, you run it by clicking on it (Mac,Windows) or typing matlab (Linux).

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, errata, and possible improvements for the second edition! Here is a list of errata so far compiled.

Recommended Texts: The literature for Fourier Analysis and Wavelets is large, here you will find a commented list of texts

Grades: Grades will be based on homework assignments and a final group project.

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

Accomodation Statement: UNM is committed to providing equitable access to learning opportunities for students with documented disabilities. As your instructor, it is my objective to facilitate an inclusive classroom setting, in which students have full access and opportunity to participate. To engage in a confidential conversation about the process for requesting reasonable accommodations for this class and/or program, please contact Accessibility Resource Center ( (https://arc.unm.edu/) at arcsrvs@unm.edu or by phone (505) 277-3506.

Credit-hour statement: This is a three credit-hour course. Class meets for two 75-minute sessions of direct instruction for fifteen weeks during the Spring 2024 semester. Please plan for a minimum of six hours of out-of-class work (or homework, study, assignment completion, and class preparation) each week.