TMA4170 Fourier Analysis - Spring 2021

Lecturer

Messages

  • 12/01 The lectures will be in Zoom (at least this first week).
  • 12/01 Date and time:
    Tuesdays, 8:15-10:00,
    Thursdays, 10:15-12:00,
    Thursdays, 16:15-17:00 (By zoom)
  • 12/01 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.
  • 14/01 Reference group: Send me a message if you want to be the reference group in the course.
  • 01/02 This week our class is still by Zoom.
  • 04/02 First lecture in F2: Tuesday 09/02
  • 09/02 First physical lecture in F2, and also broadcast live by zoom.
  • 10/02 Time of exercises for Thursday 11: 16:00-17:00 hour office (948) (CANCELLED), and by zoom 17:00-17:45.
  • 10/02 CANCELLATION of the second physical lecture. We will have our classes by zoom.
  • 16/02: Reference Group:
    Markus Valås Hagen (markus.v.hagen@ntnu.no)
    Emil Gautesen (emilga@stud.ntnu.no)
    First meeting: 22/02
  • 18/02 Jamboard link (17nd class): https://jamboard.google.com/d/1MZYo6Vle9punnIO6NDGvD-4ANnMB9_AiUATiBF2bne4/edit?usp=sharing
  • 23/02: Second physical lecture in F2 in 23/02, and also broadcast live by zoom.
  • 04/03: Exercises time each Thursday 16:15-17:00, by zoom.
  • 23/03: The final topic that we will discuss will be interpolation formulas.
  • 24/03: CANCELLATION of the physical lecture tomorrow. We will have our classes by zoom.
  • 15/04 Jamboard link (38nd class): https://jamboard.google.com/d/1FeWRfL6AOcjJsX_fEWLajnfSawO9YRUJGhOfatSlcl0/edit?usp=sharing
  • 20/04: The final classes (from now) will be physical.
  • 21/05: The final exam and its solution are enabled (see below).
  • 09/08: The oral exam will be 12 Thursday. It will be on zoom and the language will be English. The students that can take this exam have received a message with the necessary information.

Lecture notes and problem lists

a) 12/01 Read Section 1 of the Appendix: Integration, in the textbook (pp. 281-289).
b) 12/01 To justify the change of variable in Riemann integration, see: https://www.jstor.org/stable/3614765?seq=1#metadata_info_tab_contents
c) 12/01 Exercises in Chapter 1: 1, 2, 3, 4, 5, 6, 7, 8.
d) 14/01 Problem_set_1
e) 31/01 Problem_set_2
f) 21/02 Problem_set_3
g) 15/03 Problem_set_4
h) 08/04 Problem_set_5
i) 22/04 Problem_set_6
j) 26/04 Notes on complex analysis

Final exam

*Exam

Textbook

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

Syllabus

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

Classroom

First week: Introduction to Fourier series and examples

Second week: Uniqueness of Fourier series and convolutions

Third week: Convolution, good kernels and Cesáro summability

Fourth week: Vector spaces of functions on the circle and mean-square convergence

Fifth week: Convergence of Fourier series

Sixth week: Applications of Fourier series

Seventh week: Fourier transform I

Eighth week: Fourier transform II

Ninth week: Fourier transform III

Tenth week: Poisson summation formula

Eleventh week: The Gamma-function and the Riemann zeta-function

Twelfth week: Easter

Thirteenth week: The Riemann zeta-function

Fourteenth week: Paley-Wiener theorem and Uncertainty principle

Fifteenth week: Shannon interpolation formula and Vaaler interpolation formula

Sixteenth week: Review

2021-11-22, carlosch