CRN: 10703

Time and Place: TTh 9:00–10:15am in SCHM 309

Instructor: Arshak Petrosyan

Office Hours: TTh 10:30–11:45am, or by appointment, in MATH 836

Course Description: Credit Hours: 3.00. Topics include: Fourier series, convolutions, kernels, summation methods, Fourier transforms, and applications to the wave, heat, and Laplace equations.

Learning Outcomes:

  1. Know applications of Fourier series to problems in physics and other areas.
  2. Know convergence theorems and proofs for Fourier series.
  3. Know the relationship between convolutions and Fourier series.
  4. Know applications of Fourier transforms to problems in physics and other areas.

Textbook: [Stein-Shakarchi] E.M. Stein & R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003

Course Content: Syllabus is essentially the first six chapters in [Stein-Shakarchi]:

  1. The Genesis of Fourier Analysis
  2. Basic Properties of Fourier Series
  3. Convergence of Fourier Series
  4. Some Applications of Fourier Series
  5. The Fourier Transform on $\mathbb{R}$
  6. The Fourier Transform on $\mathbb{R}^d$ (excluding the higher dimensional wave equation)

Particlular topics include: Fourier series, uniqueness, convolutions, good kernels, Cesaro and Abel summation, Fejer and Poisson kernels, Parseval’s identity, Fourier transform, Schwarz class, Gaussian kernels, Plancherel’s identity, Poisson summation formula; applications to the wave, heat, and Laplace equations, the isoperimetric inequality, equidistribution theorems.

Homework: There will be weekly homework assignments to be collected through Gradescope, typically due at 11:59pm on Thur. No late homeworks will be accepted, however the lowest homework score will be dropped. For more information, see Homework page.

Exams: There will be two midterm exams and a final, either in open-book in-person or in take-home format. The exact information will be posted on the Exams page at least two weeks in advance.

Grading: You homework and exam scores will be available in Gradescope. Your final score will be calculated as the maximum of scores obtained by the two schemes

	Scheme I = (3/10)ME1 + (3/10)ME2 + (1/5)FE + (1/5)HW,
	Scheme II = (3/8)ME1 + (3/8)ME2 + (1/4)HW,

where FE, MEi, HW are the scores (in %) for Final Exam, Midterm Exam i, Homework, respectively. Thus, you have an option to skip the final exam.

Grade cutoffs: Students who get at least 97% of the total points in this course are guaranteed an A+, 93% guarantees an A, 90% an A-, 87% a B+, 83% a B, 80% a B-, 77% a C+, 73% a C, 70% a C-, 67% a D+, 63% a D, and 60% a D-; for each of these grades, it’s possible that at the end of the semester a lower percentage will be enough to get that grade.

Important dates:

  • Mon, Jan 26: Last day to withdraw from a course without it appearing on record
  • Thur, Apr 16: Last day to withdraw from a course with a W (advisor approval required)

For other important dates see Purdue Registration Calendars.

Academic Integrity: As a reminder, all students must comply with Purdue’s policy for academic integrity: https://www.purdue.edu/odos/osrr/academic-integrity/

Academic integrity is one of the highest values that Purdue University holds. Individuals are encouraged to alert university officials to potential breeches of this value by either emailing integrity@purdue.edu or by calling 765-494-8778. While information may be submitted anonymously, the more information that is submitted provides the greatest opportunity for the university to investigate the concern.

Use of Online Resources, Collaboration, and AI Tools: Collaboration and the use of outside resources can support learning when done responsibly. Homework is an opportunity to develop the independent understanding required for quizzes and exams, where no outside aids are permitted.

All submitted work must be written in the student’s own words. Any assistance from classmates, textbooks, online resources, or AI tools and large language models (such as ChatGPT or similar systems) must be fully disclosed, including the specific sources and the type of help received. Submitting work that is not one’s own without disclosure will result in zero points on the assignment and may lead to an academic misconduct report.

Students with Disabilities: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247, as soon as possible.

If the Disability Resource Center (DRC) has determined reasonable accommodations that you would like to utilize in this class, you must send your Course Accommodation Letter to the instructor. Instructions on sharing your Course Accommodation Letter can be found by visiting:

https://www.purdue.edu/drc/students/course-accommodation-letter.php

Additionally, you are strongly encouraged to contact the instructor as soon as possible to discuss implementation of your accommodations.

Attendance Policy: This course follows Purdue’s academic regulations regarding attendance, which states that students are expected to be present for every meeting of the classes in which they are enrolled. When conflicts or absences can be anticipated, such as for many University-sponsored activities and religious observations, the student should inform the instructor of the situation as far in advance as possible. For unanticipated or emergency absences when advance notification to the instructor is not possible, the student should contact the instructor as soon as possible by email or phone. For cases that fall under the University’s excused absence regulations, the student or the student’s representative should contact or go to the Office of the Dean of Students (ODOS) website to complete appropriate forms for instructor notification. Under academic regulations, excused absences may be granted by ODOS for cases of grief/bereavement, military service, jury duty, parenting leave, or emergent medical care. Absences outside of those covered by the University’s excused class absence regulations are at the instructor’s discretion. Purdue expects each student to be responsible for class-related work missed due to an unavoidable absence. Students should contact their instructors directly to discuss the absence and opportunity to complete missed coursework. This work may be made up at the discretion of the instructor.

Mental Health/Wellness Statement: Purdue University is committed to advancing the mental health and well-being of its students. If you or someone you know is feeling overwhelmed, depressed, and/or in need of mental health support, services are available. For help, such individuals should contact Counseling and Psychological Services (CAPS) at 765-494-6995 during and after hours, on weekends and holidays, or by going to the CAPS office on the second floor of the Purdue University Student Health Center (PUSH) during business hours.

Nondiscrimination Statement: Purdue University is committed to maintaining a community which recognizes and values the inherent worth and dignity of every person; fosters tolerance, sensitivity, understanding, and mutual respect among its members; and encourages each individual to strive to reach his or her own potential. The University believes that intellectual and cultural diversity among its many members strengthens the institution, stimulates creativity, promotes the exchange of ideas, and enriches campus life.

For more information, please see Purdue’s full Nondiscrimination Policy Statement.

Emergencies: In the event of a major campus emergency, course requirements, deadlines and grading percentages are subject to changes that may be necessitated by a revised semester calendar or other circumstances beyond the instructor’s control. Relevant changes to this course will be posted onto the course website or can be obtained by contacting the instructor via email or phone. You are expected to read your @purdue.edu email on a frequent basis.