TMA4170 Fourier Analysis 2024

• Instructor: Yan He
• Email: yan.he@ntnu.no
• Office: 602
• Textbook: Fourier analysis, an introduction by Stein & Shakarchi.
• The course will be delivered in English.
• Lectures: Tuesday 14:15 - 16:00 and Friday 10:15-12:00 in GL-RFB R D4-132.
• Exercise sessions: Monday 17:15-18:15 in F4.

Johanne Sand: johanne.k.sand@ntnu.no

It is not mandatory to turn in your homework and I will not grade it. The aim to collect homework is get feedback from you and make the exercise sessions more efficient.

There will be a written exam. 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. Yellow paper from math department is not required.

1. The first lecture will be on Tuesday Jan 9th.
2. No exercise session on Monday Jan 15th. The first homework is problem set 1 in the lecture reader https://yanhe.curve.space/fouriernew/exercise-1 The due is Friday Jan 19.
3. Second exercise is problem set 2 https://yanhe.curve.space/fouriernew/exercise-2, due on Sunday Jan 28th by email.
4. The exercise for week n is by default problem set n. Due on Sunday by email.
5. Reference group meeting on Tuesday, Feb 6. The reference group expressed positive feedback on the course materials. Additionally, they suggested that the course should focus less on direct calculations and include more emphasis on discussing ideas. Moreover, the reference group pointed out that the textbook should be written clearly on the course webpage.
6. No exercise session on Monday, Feb 12. Solution to problem set 4 will be uploaded.
7. No lectures on Tuesday Apr 8 and Friday Apr 12.

1. Introducing Fourier series
2. Convergence of Fourier series
3. Application in wave and heat equations
4. Application in isoperimetric inequality and Weyl equidistribution.

1. Schwartz functions
   1. Schwartz functions and their Fourier transforms, Poisson summation formula
   2. Pluncherel identity and applications
2. Distribution theory
   1. Definition and examples of distributions and tempered distributions
   2. Fourier transform, convolution theorem, shifting and scaling theorem
   3. Sampling theorem, operation on sound waves
   4. Pauley-Wiener theorem
   5. Uncertainty principle, central limit theorem

1. n-dim Schwartz functions
   1. Calculus: change of variable formula and spherical integral.
      1. smooth functions, Schwartz functions, distribution
      2. vector fields
      3. differential forms
      4. integration on spheres
   2. Basic properties, Fourier transform of sphere measure
   3. n-dimensional wave equation

1. X-ray and Radon transform

1. Fourier transform on Z/(n) and FFT
2. Fourier transform on Abelian groups
3. n-dimensional lattices

1. Gauss' law of quadratic reciprocity
2. Dirichlet theorem

1. Week 1: Periodic functions, Fourier series, uniqueness of Fourier series for continuous functions. Derivation of heat equation.
2. Week 2: Basic properties of convolutions. Good kernel, Fejer's theorem. Here is the handwritten note week 2-1
3. Week 3: 2-dimensional heat equation, differential forms, Poisson Kernel, harmonic functions on the disk. handwritten note
4. Week 4: Mean square convergence and applications of Fourier series.
5. Week 5: Summery of Fourier series part, definition of Fourier transform, Schwarz functions.
6. Week 6: Introduce Fourier transform, Poisson summation formula
7. Week 7-9: Schwartz functions, distribution, tempered distribution
8. Week 10: Fourier Laplace transform
9. Week 11: Pauley-Weiner theorem, uncertainty principle
10. Week 12: Probability theory
11. Week 14: Proof of central limit theorem
12. Week 15: n-dimensional Fourier transform, wave equation
13. Week 16: Elliptic operators
14. Week 17: Elliptic operators
15. Week 18: Solving mock and previous exam problems.

- The course reader will be hosted on https://yanhe.curve.space/fouriernew , the notes will be updated.

- Video for exercise 1: https://www.filemail.com/d/eqneuxxcofkjuqb (Note that the video is only available to download for 7 days!) Here is the Scratch paper

The scope of exam covers the material Lectures 1-15 and n-dimensional Fourier transforms from the lecture notes. Appendices in lectures, probability theory and the special topic about elliptic operators will be excluded in the exam. One should focus on the key computations in the proofs.

Suggestion to the distribution theory part: Although it will be good to fully understand the theory, my minimal expectation to students is to be able to work on specific examples, especially the Delta measures and SHA measures.

In the mock exam there are two problems about discrete Fourier transform because it was the one of last year. This year discrete Fourier transform and analytic number theory are not included in the lectures so they will not appear in the exam.

Good Luck on the Exam!