TMA4170 Fourier Analysis - Spring 2022

Lecturer

Messages

  • 06/01 The lectures will be in Zoom (at least the first weeks). These classes will be recorded.
  • 06/01 Join Zoom Meeting: Carlos Andrés Chirre Chávez is inviting you to a scheduled Zoom meeting:
    https://NTNU.zoom.us/j/92139246758?pwd=YnQxcDMxQVJCRHNYVjZwUG1tM0d3dz09
    Meeting ID: 921 3924 6758
    Passcode: 392329.
  • 07/01 The first lecture will be on Monday 10 at 12:15pm.
  • 10/01 Schedule of the course:
    1. Monday: 12:15pm - 14:00pm
    2. Tuesday: 8:15am - 10:00am
  • 10/01 Reference group
  • 11/01 Office hours: Wednesday 16:00pm-18:00pm
  • 18/01 Jamboard link: https://jamboard.google.com/d/1o8yFOdVGWtMIn6smYesJGHIQePLv4g8pNBJPmGuhoYM/edit?usp=sharing
  • 18/01 The next class we will start the physical lectures.
    1. Monday 24 January: VE1
    2. Tuesday 25 January: Smia
    3. Monday 31 January: R5
    4. Tuesday 1 February: S6
  • 31/01 Office hours (additional): Friday 17:00pm-18:00pm
  • 15/02 Reference group:
    - Trygve Johan Tegnander: tjtegnan@stud.ntnu.no
    - Robin Lien: robinol@stud.ntnu.no
    Send a message to one (or both) of the members of the reference group talking about the lectures.
  • 25/04 Final session of exercises and Office hours (see below, no recorded).

Lecture notes and problem lists

a) 06/01 Read Section 1 of the Appendix: Integration, in the textbook (pp. 281-289).
b) 06/01 To justify the change of variable in Riemann integration, see: https://www.jstor.org/stable/3614765?seq=1#metadata_info_tab_contents
c) 06/01 Exercises in Chapter 1: 1, 2, 3, 4, 5, 6, 7, 8.
d) 11/01 Problem_set_1
e) 24/01 Problem_set_2
f) 15/02 Problem_set_3
g) 18/03 Problem_set_4
h) 10/04 Problem_set_5

Extra-solutions

a) Solution_35
b) Solution_36
c) Solution_43_44
d) Solution_52
e) Solution_67_68_69a
f) Solution_73
g) Solution_81

Final exam 2021

Final Exam
Solution Final Exam

Textbook

  • Fourier Analysis: An Introduction, Elias M. Stein and Rami Shakarchi. Princeton Lectures in Analysis I. Princeton University Press, 2003.
  • Complex Analysis, Elias M. Stein and Rami Shakarchi. Princeton Lectures in Analysis II. Princeton University Press, 2003.

Syllabus

  • Fourier series
  • Convergence of Fourier series
  • Applications of Fourier series
  • The Fourier transform on R
  • The Riemann zeta-function
  • The Heisenberg Uncertainty principle
  • Paley-Wiener Theorem
  • Interpolation formulas

Classroom

W2: Introduction to Fourier series and examples

W3: Uniqueness of the Fourier series

W4: Convolution and good kernels

W5: Vector spaces and best approximation

W6: Convergence in norm and Parseval's identity

W7: Application: equidistribution of real numbers

W8: Fourier transform I-II

W9: Fourier transform III-IV

W10: Fourier transform: Fourier inversion formula and Plancherel formula

W11: Poisson summation formula and the Gamma-function

W12: The Riemann zeta-function

W13: Uncertainty principle and the Paley-Wiener Theorem

W14: Paley-Wiener theorem and Shannon interpolation formula

W15: Easter

W16: Vaaler interpolation formula

W17: Vaaler interpolation formula

W18: Session of exercises

  • 03/05 Time: 8:15am-10:00 am, Room: S6

W19: Office hours

  • 16/05 Time: 14:00pm-16:00pm
2022-05-15, carlosch