• Instructor Yan He
  
• Email: yan.he@ntnu.no
  
• Office: 602
  
• The course will be delivered in English.
• Lectures: Monday 15:15-17:00, Tuesday 10:15-12:00, in GL-RFB R D4-132
• Exercise sessions: Thursday 16:15-18:00, in the same place.
• Office hours: By appointment.

Maximilian Rønseth: maximilian.ronseth@ntnu.no

The exam will be written. You are allowed to take 3 pages (total 6 sides) of handwritten notes with you to the exam. You are not allowed to take book or any printed material to the exam.

Fourier analysis appears in many fields of mathematics and has applications in physics and engineering. This course attempts to give the students an overview of Fourier analysis and its related mathematics, the idea and the math behind its real life applications. Due to its richness, it is not possible to cover the interests from different backgrounds. For this reason, there will be four projects for you to choose based on your background and interest. There will be four projects, two from pure math and two from applied, each student is required to choose two from the four projects. In the exam, there will be four questions that are directly related to corresponding projects and one should choose and only choose two to answer.

In our journey we are going to see Fourier series and its convergence problems and its relation to heat equation and Poisson equations, a detailed and rigorous examine of the Fourier transform and inverse Fourier transform, convolution and central limit theorem, Fourier transform of distributions, sampling of music, the fast Fourier transform, higher dimensional case, image processing, etc. The students are expected to understand the idea of Fourier transform and its applications the right way, to know and to be able to explain the correct math behind its applications, in particular, to comfortably work on specific examples.

• Textbook: E. M. Stein & R. Shakarchi: Fourier Analysis, An Introduction, Princeton University Press, 2003
• The Stanford EE course is a very nice and elegant exposition with a lot of examples, motivations and applications. Our course will give more details on the theoretical part.

Unlike previous years, this year I want to include also some computer things for you to better understand and visualize the math. These will not be required at all in exams.

• Sagemath is a tool to work on mathematics with computers. Unlike matlab, it does symbolic calculations. We'll use sagemath to do some calculation and plot some pictures. You are encouraged to install sagemath on your computers.
• Jupyter notebook is a nice tool to write interactive python and sagemath code. The lecture notes will be delivered in both the jupyter format and pdf format. The advantage of using jupyter is that it's interactive so that you can run and modify my sage codes in the lecture notes.
• The main thing sage can help you do is summerized here.

• The first lecture will take place in GL-RFB R D4-132 on 01.09.2023 at 15:15. The full course calendar is available here.
  
• Please take this survey for me to know better your math background. This will be helpful for me to make decisions about things to present in detail in lectures.
• 01.20.23. For better version control and readability, the notes are now served at curvenote. You can find the whole lecture notes and problem sets here.
• 01.26.23 We'll decide the reference group next week. Please let me know if you want to be the reference group.
• 01.02.23 On Monday Maximilian volunteered to be in the reference group. We might need one more who are interested in applied math or physics or EE. Please let me know if you want to be reference group.
• 02.02.23 We have finished the Fourier series part. Here is an annonymous survery to check how you understand convergence of Fourier series.
• 29.03.23 No exercise session on Mar 30.
• There will be no projects.
• 03.05.23 Mock exam and solution uploaded to curvenote. There is an update on exam policy in the Course Information section.

The lecture notes will be provided in three formats. One is jupyter notebook, in which you can run and modify the sagemath code directly. You need to have jupyter notebook and sagemath installed to successfully open and run it. If you do not have sage and jupyter, the second file is export of jupyter containing all outputs of sage. Note that it comes from an automatic script via LaTex so may have some issue on the numbering. The third is pdf without code output. If a note has no code at all, only pdf will be provided.

• Lecture 1: The first lecture covers chapter 2.1, 2.2 Stein-Shakarchi book.
  • Jupyter notebook: lect1.ipynb, lect1-code-export.pdf
  • PDF: lect1.pdf

• Lecture 2: Chapter 2.3 and 4.4 of Stein-Shakarchi.
  • PDF: lecture_2.pdf

• Lecture 3: Good kernels, Cesaro sums and Fejer theorem. Chapter 2.4 and 2.5 of Stein-Shakarchi.
• Lecture 4: Poisson integrals, chapter 2.6 of Stein-Shakarchi, but with a complex analysis point of view, following chapter 2-3 of Hoffman's book. You can find the lecture notes for lectures 3-4 on curvenote.

All future lecture notes will be on curvenote.

• Lecture 5-6: Mean square convergence of Fourier series, Riemann-Lebesgue lemma, brief introduction to Lebesgue integrals, equidistribution theorem (part). The lecture notes are on curvenote.
• Plan for lecture 6-7: Finish the proof of equidistribution theorem, high-dimensional Fourier series, introduce the Fourier transform.
• Lecture 7-8 Isoperimetric inequality, summery of convergence of Fourier series, example of continuous but nowhere differentiable functions.
• Lecture 9-10 Fourier transform: motivation, definition, properties and Schwartz functions. Poisson summation formula.
• Lecture 11: Test functions and distributions.
• Lecture 12: tempered distributions and its Fourier transform. The material for lecture 11,12,13,14 will not be on Stein's book. You can read https://wiki.math.ntnu.no/_media/tma4170/2019v/fourier_1.pdf section 2.4.
• Lecture 13-14: Properties of Fourier transform of tempered distributions, a brief introduction to sampling theorem, periodic distributions and its Fourier series.
• Lecture 15: Fundamental solution of D+s, some examples of distributions and its Fourier transform: Heviside function and Cauchy principal value.
• Lecture 16: Cauchy integral formula (with partition of unity), fundamental solution of dbar-operator, Fourier-Laplace transform of timelimited signal.
• Lectures 17-18: The Paley-Wiener-Schwartz theorem, distributions with compact support, distribution version of Paley-Wiener theorem. Brief introduction to probability theory.
• Lecture 19: Uncertainty principle and Central limit theorem.
• Lecture 20: High dimensional Fourier transform, wave equation and its solution. (Following Stein-Shakarchi 176-189 but also including some high dimensional distribution theory).
• Lecture 21-22: Wave equations (Stein-Shakarchi page 189-198).
• Lecture 23-24: Radon transform, chapter 6 Stein-Shakarchi finished.
• Lecture 25-26: Discrete Fourier transform and Fast Fourier transform. Fourier transform on abelian groups (Chapter 7 Stein-Shakarchi) finished.
• Lecture 26-27: Zeta function and Dirichlet L function (Page 251-261 of Stein Shakarchi).
• Lecture 27-28: Gauss' law of quadratic reciprocity (page 222-226 of Chapter 4 of Dym-Mckean), theta function (page 155-156 Stein Shakarchi).
• Lecture 29-30: [Plan] More on theta functions.
• Lecture 31-32: [Plan] Review.
• Lecture 33-34: [Plan] Review.

• Problem set 1: exercise_1.pdf
• Problem set 2: On curvenote
• All future problem sets will also be on curvenote.

• Final Exam exam.pdf
• Solution exam_2_5.pdf
• Explanation of grade:
  • A: You understand and have finished all problems. Your solution indicates that you understand all the topics involved very well.
  • B: You have finished all problems very well with one or a few exceptions. Your response indicates that you have understand the topics pretty good.
  • C: You can solve about half of the problems. Your response indicates that you understands the topics in a satisfactory way.
  • D: You can only solve a few problems. Your response indicates that your understanding to the topics is not good but OK.
  • E: You basically have failed the exam but your response is savable by loosening the grading standard.
  • F: Unfortunately, your response is much worse than the passing standard and I have no idea what else to do.