# Fourier Series & Wavelets

## Math 414-501 — Spring 2022

Instructor: Dr. Francis J. Narcowich
Office: 611D Blocker
E-mail: fnarc@math.tamu.edu
Phone: 979-845-7554 (Messages only.)
URL: http://www.math.tamu.edu/~francis.narcowich/
Office Hours: 12:30-1:30 MTWR, and by appointment.

Catalogue Description: MATH 414. Fourier Series & Wavelets. Fourier series and wavelets with applications to data compression and signal processing. Prerequisite: MATH 323 or MATH 304 or MATH 311

Goals: This is a mathematics course. One of the goals is for you to learn to be able to prove theorems concerning Fourier series, properties of Fourier transforms, filters, discrete Fourier transforms and wavelets. Another is for you to learn how to find and work with these tools. Finally, by means of a project, you will learn how wavelets are applied. To achieve these goals, you must diligently do the homework assignments and read the appropriate sections of the book, as well as additional notes.

Required Text: A First Course in Wavelets and Fourier Analysis, 2nd Edition, by Boggess & Narcowich

Time & Place: MWF 11:30-12:20, BLOC 117

Programming language: Experience with MATLAB would be very helpful.

Grading System & Tests: Your grade will be based on a project, homework, and three in-class tests (February 16, March 23 and April 27).The project will count for 20% of your grade, homework for 20%, and each in-class test for 20%. Your letter grade will be assigned this way: 90-100%, A; 80-89%, B; 70-79%, C; 60-69%, D; 59% or less, F.

Make-up Policy: I will give make-ups (or satisfactory equivalents) only in cases authorized under TAMU Regulations. In borderline cases, I will decide whether or not the excuse is authorized. Also, if you miss a test, contact me as soon as possible.

Homework and Projects: You may consult with each other on homework problem sets, BUT only submit work which is in your own words AND be sure to cite any sources of help (either texts or people). Be aware that usually only some of the problems from an assignment will be graded. Late homework will not be accepted. Information concerning projects may be found on at this webpage: Project Information.

Copying Course Materials:   "All printed hand-outs and web-materials are protected by US Copyright Laws. No multiple copies can be made without written permission by the instructor."

Aggie Honor Code:   "An Aggie does not lie, cheat, or steal or tolerate those who do."
Americans with Disabilities Act Policy Statement: "The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact the Department of Disability Services.

Covid information The latest information about covid policies may be found at the Covid-19 informaiuon link on the TAMU homepage.

Schedule

Week Section Topic
1
1.1.1-1.1.2, 1.2.1-1.2.2 Fourier series (FS): motivation, calculation, examples
2
1.2.1-1.2.3 FS examples, function extensions, symmetry, Fourier cosine/sine series (FCS/FSS), examples
3
1.2.4-1.2.5 Convergence of FS; Complex form of FS; examples
4
1.3.1-1.3.3 Partial sums, Dirichlet (Fourier) kernel, Riemann-Lebesgue Lemma; proof of pointwise convergence (See the Notes on Pointwise Convergence) for a simplified version); uniform convergence
5
Test 1 (2/16/22)
1.3.4
Review, Test 1 (covers 1.2.1-1.2.5, 1.3.1-1.3.3, (Notes on Pointwise Convergence), uniform convergence
6
1.3.4, 0.2-0.5, 1.3,5 inner products, signal spaces ($L^2,\ell^2$), types of convergence, orthogonal bases, Parseval's equation, FS examples
7
2.1, 2.2 Fourier Transform & properties; convolution theorem; Plancherel (Parseval) Theorem
8
2.3, 2.4 Time-invariant filters, causal filters, sampling theorem
3/14-18 N/A Spring Break
9
Test 2 (3/23/22),
3.1
Review, catch up, Test 2 (covers 1.3.2-1.3.5, 2.1-2.4), discrete Fourier transform
10
3.1.1-3.1.4, 3.2.1 Discrete Fourier transform, fast Fourier transform (FFT), applications, discrete signals & filters
11
4.1, 4.2, 4.3 Haar wavelets, decomposition and reconstruction algorithms, filter representation
12
5.1.1, 5.1.2