Math 472/572 - Fourier Analysis and Wavelets

### Fall 2021

This class is cross-listed as:

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

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, erratas, 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.

UNM Administrative Mandate on Required Vaccinations: All students, staff, and instructors are required by Administrative Mandate on Required Vaccinations to be fully vaccinated for COVID-19 as soon as possible, but no later than September 30, 2021, and must provide proof of vaccination or of a UNM validated limited exemption or exemption no later than September 30, 2021 to the UNM vaccination verification site. Students seeking medical exemption from the vaccination policy must submit a request to the UNM verification site for review by the UNM Accessibility Resource Center. Students seeking religious exemption from the vaccination policy must submit a request for reasonable accommodation to the UNM verification site for review by the Compliance, Ethics, and Equal Opportunity Office. For further information on the requirement and on limited exemptions and exemptions, see the UNM Administrative Mandate on Required Vaccinations.

UNM Requirement on Masking in Indoor Spaces: All students, staff, and instructors are required to wear face masks in indoor classes, labs, studios and meetings on UNM campuses, see masking requirement. Students who do not wear a mask indoors on UNM campuses can expect to be asked to leave the classroom and to be dropped from a class if failure to wear a mask occurs more than once in that class. Students and employees who do not wear a mask in classrooms and other indoor public spaces on UNM campuses are subject to disciplinary actions. UNM will periodically evaluate and update the mask policy relative to public health conditions.

Communication on change in modality: The President and Provost of UNM may direct that classes move to remote delivery at any time to preserve the health and safety of the students, instructor and community. Please check your email and UNM Learn regularly for updates about our class and please check https://bringbackthepack.unm.edu regularly for general UNM updates about COVID-19 and the health of our community.

Accomodation Statement: In accordance with University Policy 2310 and the Americans with Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. It is imperative that you take the initiative to bring such needs to the instructor's attention, as I am not legally permitted to inquire. Students who may require assistance in emergency evacuations should contact the instructor as to the most appropriate procedures to follow. Contact Accessibility Resource Center at (505) 277-3506 or arcsrvs@unm.edu for additional information.
UNM is committed to providing courses that are inclusive and accessible for all participants. As your instructor, it is my objective to facilitate an accessible classroom setting, in which students have full access and opportunity. If you are experiencing physical or academic barriers, or concerns related to mental health, physical health and/or COVID-19, please consult with me after class, via email/phone or during office hours. You are also encouraged to contact Accessibility Resource Center at arcsrvs@unm.edu or by phone (505) 277-3506.