MMG710/TMA362, Fourier Analysis, Autumn 18

Overview

Welcome to the course! The schedule can be found here.

The exam (with solutions) on October 27, 2018 can be found here and here.

The exam (with solutions) on January 9, 2019, can be found here and here.

The exam (with solutions) on August 22, 2019, can be found here and here.

The course roughly splits into three parts: (FS), (FT), (LT) - the abbreviations stand for

• FS = Fourier series.
• FT = Fourier transform.
• LT = Laplace transform.

Each part is further partitioned into modules. Every module has a motivating theme (an application of the material that will be covered in the module):

• Module I (FS): Benford's Law - will serve as motivation for Fejer's Theorem (which we will use to establish Weyl's famous equidistribution criterion), as well as various elaborations of this theorem; leads to the important notions of Fourier series and Dirac sequences. Handwritten notes: Part 1, Part 2

• Module II (FS): The Isoperimetric Inequality - motivates Parseval's Theorem (via the Wirtinger inequalities) and leads to the "Hilbert space"-perspective on Fourier series. Handwritten notes: Part 1, Part 2

• Module III (FS): The heat equation in the disk - we discuss how to solve various PDE's using Fourier techniques. An overview of Weierstrass famous example of a continuous, but nowhere differentiable function, is given (see notes here)

• Module IV (FT): Central Limit Theorem - motivates the Fourier tranform and its approximation properties. Handwritten notes: Part I, Part II.

• Module V (FT):  Shannon's Sampling Theorem - motivates the "Hilbert space"-perspective on the Fourier transform - and the necessity to find good implementations of its "L^2"-version (leads to Hermite functions). Handwritten notes.

• Module VI (LT): Stability of casual LTI-systems - after a crash course on LTI-systems, we introduce the Laplace transform and show how it can be used to analyze stability of such systems. Short notes can be found here.

Exercises (with solutions) are given at the end of each hand-out. The "ringed" ones are strongly recommended. Additional exercises can be found here.

Instead of giving a detailed time plan, we will go through the modules one by one. I expect that the Module I will take 1 and a half week, Module II, III, IV, V and VI one week each. In particular, (most of) the two last weeks will be devoted to repetition.

Every week, 4 classes (twice 45 min) are given. The last one of each classes will be an exercise session (we will go through some of the recommended exercises, and related tricks), while the second one will be a mixed lecture/exercise session (except the first week).

The exam will have problems of a total worth of 25 points (You will need at least 12 points to pass the course with the grade G, and at least 18 points to pass the course with the grade VG). For Chalmers students: 3 (12-17 points), 4 (18-21 points) and 5 (22-24 points).

A summary of the proofs of the main theorems can be found here.

During the course, 3 assignments will be given, to be finished individually, and will render at most 2 + 2 + 1 bonus points to be used on the exam (and the following two extra exams).

First bonus point round (Discrete Fourier transform) - Deadline, September 20 (at noon).

--- The aim of these exercises is for you to see the notions of Fourier series/transform in the discrete setting where convergence issues disappear, and the whole thing just reduces to linear algebra. ---

Second bonus point round (Fourier transform)- Deadline, October 11 (at noon).

-- The aim of these exercises is to provide an overview of some typical problems in linear system theory which can be approached using the Fourier transform. ---

Third bonus point round (Laplace transform) - Deadline October 26 (at noon). Note: In the first version, the definition of the function a_lambda was missing, it is corrected now.

Teacher

Course literature

• Handwritten course notes (including exercises/solutions) for each module can be found above.

• Additional material will be distributed as the course progresses.

• A recommended book for the PDE-part of the course is Folland's "Fourier Analysis and Its Applications" - available at Cremona bookstore. Although the course notes will be sufficient material to pass the course, it is always good to have other sources to compare the expositions and find the one most suitable for you.

Course requirements

The learning goals of the course can be found in the course plan.

Examination

The first exam will be on Saturday, October 27, 8.30-12.30 at Campus Johanneberg (exact location will be determined later). The last day to register for the exam is October 11.

The exam will deviate slightly from earlier years, so in order to prepare you for the (slight) change, a few "model exams" will be distributed in good time before the exam. Older exams will also be posted here, as they contain a lot of useful material. It is strongly recommended that your work through them as well.

Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal, where you also can read about what rules apply to examination at University of Gothenburg.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and retrieved at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

Model exams and old exams

Old exams, see previous years courses, e.g. here.