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Syllabus of Math 658, Sections 600, 3 hrs

Applied Harmonic Analysis, Credit 3, Spring 2021

Instructor Peter Kuchment

Office Rm. Blocker 614A

E-mail: kuchment@math.tamu.edu, Home Page: http://www.math.tamu.edu/~kuchment


Catalog description

Fourier series and Fourier Transform; discrete(fast) Fourier transform; discrete cosine transform; local cosine transform; Radon transform; filters; harmonic analysis on the sphere; radial, periodic, and spherical basis functions; compressed sensing . Prerequisites: MA 304, MA 308 or equivalent.

Introduction

Harmonic analysis (also called Fourier analysis) is one of the major pillars of the whole mathematics and its applications in physics, engineering, and practically any other area. Thus, the word “applied” in the title neither adds, nor subtracts anything from the content. In the most general terms, the idea of harmonic analysis is: if your problem has symmetries – use them! I will be happy to present it to you.

Prerequisite

MATH 304 and MATH 308 or equivalent. It is highly recommended to overview some linear algebra before the class: Eigenvalues and eigenvectors, self-adjoit (Hermitian) operators, Unitary operators. A wonderful book by Gilbert Strang Linear algebra and applications is available electronically through libcat.tamu.edu.

Recommended textbook

Gerald Folland, Fourier Analysis and its Applications. Various editions are available from Wadsworth & Brooks/Cole and more recently from the AMS.

A very good by Brad Osgood Lectures on the Fourier transform and its applications is also available electronically through the libcat.

Besides these books, I will be distributing some extensive handouts, as well as references to some parts of books available electronically through the A&M library.

The rough plan of the class looks as follows:

Some eCampus details

Every week I will post on eCampus additional materials when needed, as well as bi-weekly homework.
Take home Mid-Term and Final Exams will be given

I will set up a discussion board for the class, where you can exchange opinions, recommendations, questions, etc. (as long as this would not amount to cheating ). I will be monitoring the board and thus will be able to help when needed.

You will be able to communicate with me individually by e-mail, or through ZOOM (in the latter case please arrange the time by e-mail). Questions during the class will be also encouraged.

Please free to contact me
I am looking forward to your participation!

Tentative schedule (might be modified depending on the class situation)

Dates Topic Reference Work
January 20 Introduction
January 25-27 Linear algebra, symmetries, and harmonic analysis A handout HW 1
February 1 shifts and rotations handout
February 3 - 15 Fourier series Ch 2 HW 2
Feb 17 - 22 Discrete and finite Fourier transform Sect. 7.6 HW 3
February 24 - March 3 Distributions Ch 9
March 8 - 22 Fourier transform Ch. 7 Midterm exam. HW 4
March 24 - April 14 Other transforms, miscellanea. Section 7.6, Handouts HW 5
April 19 - 26 A glimpse at non-commutative harmonic analysis Handouts HW 6
May Final Exam

Grading Policy

Bi-weekly home assignments (due the next week) 40%
Midterm exam 30%
Final exam 25%
Class participation 5%

Grading Scale

Class rules

Attendance Policy: University rules related to excused and unexcused absences are located on-line at http://studentrules.tamu.edu/rule07.

Late Work Policy: No late work is accepted for unexcused absences per Section 7.4 of the University Student Rules Policy.

Make-up Policy: Students may be excused from turning in an assignment for the reasons stated in Section 7.1 (http://student-rules.tamu.edu/rule7.htm) or other reason deemed appropriate by the instructor. To be excused the student must notify the instructor in writing (acknowledged e-mail message is acceptable) prior to the date of absence if such notification is feasible. In cases where advance notification is not feasible (e.g. accident, or emergency) the student must provide notification by the end of the second working day after the absence. This notification should include an explanation of why notice could not be sent prior to the class. For approved excuses for missed assignments, an appropriate modified due date will be set by the instructor.

Absences related to Title IX of the Education Amendments of 1972 may necessitate a period of more than 30 days for make-up work, and the timeframe for make-up work should be agreed upon by the student and instructor” (Student Rule 7, Section 7.4.1).

“The instructor is under no obligation to provide an opportunity for the student to make up work missed because of an unexcused absence” (Student Rule 7, Section 7.4.2).

Students who request an excused absence are expected to uphold the Aggie Honor Code and Student Conduct Code. (See Student Rule 24.)

Americans with Disabilities Act (ADA) 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 Disability Services, currently located in the Disability Services building at the Student Services at White Creek complex on west campus or call 979-845-1637. For additional information, visit http://disability.tamu.edu.

Academic Integrity Statement:

AGGIE HONOR CODE “An Aggie does not lie, cheat, or steal or tolerate those who do”. Upon accepting admission to Texas A&M University, a student immediately assumes a commitment to uphold the Honor Code, to accept responsibility for learning, and to follow the philosophy and rules of the Honor System. Students will be required to state their commitment on examinations, research papers, and other academic work. Ignorance of the rules does not exclude any member of the TAMU community from the requirements or the processes of the Honor System.

“Texas A&M University students are responsible for authenticating all work submitted to an instructor. If asked, students must be able to produce proof that the item submitted is indeed the work of that student. Students must keep appropriate records at all times. The inability to authenticate one’s work, should the instructor request it, may be sufficient grounds to initiate an academic misconduct case” (Section 20.1.2.3, Student Rule 20).

For additional information please visit: http://aggiehonor.tamu.edu/.

Scholastic Dishonesty:

Copying work done by others, either in class or out of class, looking on other students papers during exams or quizzes, having possession of unapproved information in your calculator, and/or having someone else do your work for you are all acts of scholastic dishonesty. These acts, and other acts that can be classified as scholastic dishonesty, will be prosecuted to the full extent allowed by University policy. Punishment can range from a zero on the assignment/quiz/exam to expulsion from the university. In any case of scholastic dishonesty, the student forfeits their right to Qdrop the class. In this class, collaboration on assignments, either in class or out of class, is forbidden unless permission to do so is granted by the instructor.

Title IX and Statement on Limits to Confidentiality:

Texas A&M University is committed to fostering a learning environment that is safe and productive for all. University policies and federal and state laws prohibit gender-based discrimination and sexual harassment, including sexual assault, sexual exploitation, domestic violence, dating violence, and stalking.

With the exception of some medical and mental health providers, all university employees (including full and part-time faculty, staff, paid graduate assistants, student workers, etc.) are Mandatory Reporters and must report to the Title IX Office if the employee experiences, observes, or becomes aware of an incident that meets the following conditions (see University Rule 08.01.01.M1):

> • The incident is reasonably believed to be discrimination or harassment.

• The incident is alleged to have been committed by or against a person who, at the time of the incident, was (1) a student enrolled at the University or (2) an employee of the University.

Mandatory Reporters must file a report regardless of how the information comes to their attention – including but not limited to face-to-face conversations, a written class assignment or paper, class discussion, email, text, or social media post. Although Mandatory Reporters must file a report, in most instances, you will be able to control how the report is handled, including whether or not to pursue a formal investigation. The University’s goal is to make sure you are aware of the range of options available to you and to ensure access to the resources you need.

Students wishing to discuss concerns in a confidential setting are encouraged to make an appointment with Counseling and Psychological Services (CAPS).

Students can learn more about filing a report, accessing supportive resources, and navigating the Title IX investigation and resolution process on the University’s Title IX web page.

COVID-19 Temporary Amendment to Minimum Syllabus Requirements

Campus Safety Measures
To promote public safety and protect students, faculty, and staff during the coronavirus pandemic, Texas A&M University has adopted policies and practices for the Fall 2020 academic term to limit virus transmission. Students must observe the following practices while participating in face-to-face courses and course-related activities (office hours, help sessions, transitioning to and between classes, study spaces, academic services, etc.):

Personal Illness and Quarantine
Students required to quarantine must participate in courses and course-related activities remotely and must not attend face-to-face course activities. Students should notify their instructors of the quarantine requirement. Students under quarantine are expected to participate in courses and complete graded work unless they have symptoms that are too severe to participate in course activities.
Students experiencing personal injury or Illness that is too severe for the student to attend class qualify for an excused absence (See Student Rule 7, Section 7.2.2.) To receive an excused absence, students must comply with the documentation and notification guidelines outlined in Student Rule 7. While Student Rule 7, Section 7.3.2.1, indicates a medical confirmation note from the student's medical provider is preferred, for Fall 2020 only, students may use the Explanatory Statement for Absence from Class form in lieu of a medical confirmation. Students must submit the Explanatory Statement for Absence from Class within two business days after the last date of absence.

Statement on Mental Health and Wellness

Texas A&M University recognizes that mental health and wellness are critical factors that influence a student’s academic success and overall wellbeing. Students are encouraged to engage in healthy self-care by utilizing the resources and services available from Counseling & Psychological Services (CAPS). Students who need someone to talk to can call the TAMU Helpline (979-845-2700) from 4:00 p.m. to 8:00 a.m. weekdays and 24 hours on weekends. 24- hour emergency help is also available through the National Suicide Prevention Hotline (800- 273-8255) or at https://suicidepreventionlifeline.org

Copyright Policy:

All printed materials disseminated in class or on the web are protected by Copyright laws. One copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.
Last revised January 5th, 2021