University of Jyväskylä   Department of Mathematics and Statistics
Faculty of Mathematics and Science
  

Mikko Salo

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Fourier analysis (MATS315), period 1, fall 2024

This course gives an introduction to Fourier analysis, which has its origins in the work of Fourier on heat flow in the early 19th century. The basic question in this theory is to represent general functions in terms of elementary pieces, in particular sine and cosine functions. It is a striking fact that very general functions and distributions always have such representations. These representations turn out be instrumental in many areas across mathematics, including partial differential equations, signal processing and number theory.

Topics include multidimensional Fourier series and Fourier transform, related function spaces, elements of distribution theory including tempered distributions, and selected applications of Fourier analysis.

Instructors

The lectures are given by Mikko Salo (office MaD359), and exercise sessions by Hjørdis Schlüter (office MaD303). You are very welcome to contact the instructors at their offices or by email.

Schedule

The lectures are Thursdays and Fridays at 10.15-12.00 in room MaD380 (5 Sep-18 Oct). Exercise sessions are Thursdays at 8.30-10.00 in MaD355 (12 Sep-17 Oct). Written answers to exercises need to be returned on the given date, either in person or by email.

EXCEPTION: lectures on Fri 20.09., Fri 27.09., Fri 04.10., Thu 10.10. and Fri 11.10. will be given online at this Zoom room (meeting ID: 676 0733 4906).

We have covered the following topics in the lecture notes:

  • Week 1 (02.-09.9.): Chapter 1, Section 2.1
  • Week 2 (10.-16.9.): Sections 2.2-2.3
  • Week 3 (17.-22.9.): Section 2.4
  • Week 4 (23.-29.9.): Sections 3.1-3.2
  • Week 5 (30.9.-6.10.): Sections 3.3-3.4
  • Week 6 (07.-13.10.): Section 3.5
  • Week 7 (14.-20.10.): Sections 3.8-3.10

Material

We will follow these lecture notes (26.09.).

Exercises 1 (06.09., return by 20.09., NOTE: exercise 12 was moved to next exercise sheet)
Exercises 2 (20.09., return by 04.10.)
Exercises 3 (04.10., return by 24.10.)

Possible presentation topics (choose by 27.09., due on 24.10.)

Earlier lecture notes may be found here (a longer version of the same course from 2013) or here (by Pu-Zhao Kow from 2022). As additional reading, one could use the following textbooks.

  • Stein-Shakarchi: Fourier analysis: an introduction. Princeton University Press, 2003.
  • Strichartz: A guide to distribution theory and Fourier transforms. CRC Press, 1994.
  • Rudin: Functional analysis. 2nd edition, McGraw-Hill, 1991.
  • Duoandikoetxea: Fourier analysis. AMS, 2001.

Prerequisites

Measure and integration 1. Measure and integration 2 is also recommended (the course employs L^p spaces). Additionally students are expected to be familiar with basic computations for complex numbers and with the notation e^{it}.

Completion

The course can be taken for credit (4 cr, grades 0-5) by attending the lectures, returning written answers to exercises, and by giving a short oral/written presentation. The minimum to pass is to get half of the exercise points and deliver a presentation. The grade will be determined as follows (this may be adjusted slightly later):

  • 50 % of exercises → grade 1
  • 60 % of exercises → grade 2
  • 70 % of exercises → grade 3
  • 80 % of exercises → grade 4
  • 90 % of exercises → grade 5

An excellent presentation can increase the grade.

Each exercise will be graded 0/1/2 points. Everyone must return their own written solutions, but students are encouraged to work together on the exercises. This can be done at the exercise sessions where the assistant will be available to help. As a bonus, 2 additional exercise points will be given for each attendance at the exercise session.

Please register via Sisu.