• 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
  (Graduate students please register in Math 572.)

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.

Grades: Grades will be based on homeworks, projects and/or 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

Accomodation Statement: Accessibility Services (Mesa Vista Hall 20121, 277-3506) provides academic support to students who have dissabilities. If you think you need alternative accesible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner. If you need local assistance in contacting Accessibility Services, see the Bachelor and Graduate Programs office.

Academic Integrity: The University of New Mexico believes that academic honesty is a foundation principle for personal and academic development. All university policies regarding academic honesty apply to this course. Academic dishonesty includes, but is not limited to, cheating or copying, plagiarism (claiming credit for the words or works of another from any type of source such as print, Internet or electronic database, or failing to cite the source), fabricating information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. The University's full statement on academic honesty and the consequences for failure to comply is available in the college catalog and in the Pathfinder.

Gender Discrimination: In an effort to meet obligations under Title IX, UNM faculty, Teaching Assistants, and Graduate Assistants are considered "responsible employee" by the Department of Education (see pg 15 - http://www2.ed.gov/about/offices/list/ocr/docs/qa-201404-title-ix.pdf ). This designation requires that any report of gender discrimination which includes sexual harassment, sexual misconduct and sexual violence made to a faculty member, TA, or GA must be reported to the Title IX Coordinator at the Office of Equal Opportunity (oeo.unm.edu). For more information on the campus policy regarding sexual misconduct, see: https://policy.unm.edu/university-policies/2000/2740.html

Return to: Department of Mathematics and Statistics, University of New Mexico

Last updated: August 20, 2019