# Fourier analysis, fall 2015

**Teacher:** Kari Astala

**Scope:** 10 cr

**Type:** Advanced studies

**Teaching:** Lectures on weeks 36-42 and 44-50, in room C123, on Tuesdays 10-12 and on Wednesdays 12-14.

First lecture on Tuesday, September 1.

Exercise classes are part of the course; these are on Mondays 12-14, in room C122. Exercise* assignments* are delivered on Wednesdays of the previous week.

First exercise class on Monday, September 7.

**Topics: ** The course is an introduction to modern Fourier Analysis. Fourier analysis is a basic tool both in pure and applied mathematics, from PDE's and harmonic analysis to stochastics, signal processing, physics,...

The first part of the course, roughly weeks 36-42, covers discrete Fourier analysis (Fourier series and also Fast Fourier transform), while the second period, weeks 44-50, studies the continuous Fourier analysis, i.e. Fourier transforms of functions and (Schwartz) distributions in *R** ^{n}*. Throughout the course, and time allowing, many of the applications of Fourier analysis are discussed.

**Prerequisites: **Basic knowledge on measure theory and Lebesgue integration (e.g. the course "Mitta ja integraali") is required. Knowledge of basics of *L ^{p}*

*-*spaces

*will be advantageous (e.g. as discussed in the course Reaalianalyysi I; see below). In addition some facts from functional analysis will be used - if necessary, these can be briefly discussed during the course.*

**Contents:**

- Representing a periodic function as a Fourier series
- Basic properties of Fourier series
- Dirichlet kernel; Fejer kernel and good convolution kernels
- Pointwise convergence, Convergence in
*L*,^{p}*C[0,1]*, etc. -norm - Continuous functions with diverging Fourier series
- Discrete Fourier transform, Fast Fourier transform
- Continuous Fourier transform in
*L*^{1}(R^{n}) - Fourier transform in
*L*; Plancherel's theorem^{2}(R^{n}) - Interpolation and Fourier transform in
*L*^{p}(R^{n}) - Tempered distributions
- Applications of Fourier Analysis

**Course material: **

The course will basically follow the lecture notes: 'Kari Astala, Fourier Analyysi' (course in 2012; notes are in Finnish)

Minor modifications will be made, and new Lecture notes (in Finnish) will be provided on this page as the course proceeds.

**Here are **the complete Lecture notes with Appendix.

**NOTE: **

**Section VII.1.**was left for a self study.**Proof**of Riesz-Thorin Theorem 11.2 (pages 100-106) was left for a self study.

(Riesz-Thorin theorem is proved also e.g. in "Rudin: Real and Complex Analysis")

Except for the distribution theory, the material covered in this course can be found in the books:

- Stein-Shakarchi: Fourier Analysis, An Introduction.
- Rudin: Real and Complex Analysis.

Other books related to the course:

- Grafakos: Classical and Modern Fourier Analysis.
- Rudin: Functional Analysis.
- Duoandikoetxea: Fourier Analysis.

Additional information on real analysis, in particular on basics of *L ^{p}*

*-*spaces, can be found from lecture notes by Ilkka Holopainen attached here: Reaalianalyysi.I (2011).pdf

The information on *L ^{p}*

*-*spaces needed can also be found e.g. in Rudin's book 'Real and Complex analysis' mentioned above; a brief sketch of

*L*

^{p}*-*properties needed is also given in the Appendix of the course notes above.

## News

## Exams

The course will have two exams, one at the end of first period, week 43, and another in the end of second period, week 51.

The second course exam is on Thursday December 17, at 10.00-12.30, in the room C123.

**Topic of Exam/Koealue **for **second exam**: Section VII.2 and Chapters IX - XIII.2 (i.e. Last topic: Fundamental solution)

The first exam is on Thursday Oct. 22., at 10.00-12.30, in the room C 123.

**Topic of Exam/Koealue**: Chapters I - VI from Lecture notes (i.e. Beginning - L^{2} theory of Fourier series)

**Note**: There is also the possibility to pass the course by a **general examination** (Loppukoe/erilliskoe), where topic of exam is the whole course.

If you wish to do so, please contact Kari Astala; an agreement in advance is necessary. We plan to arrange a general examination on *Fourier Analysis* on Dec 21 at 10-14, if there are students who wish to take this.

** Here** you find the corresponding course exams from 2011-2013

**Here ** you find *voluntary review problems*; Aleksis Koski will be on call in his office A415 on Monday 19th October at 12-14 to give help/answers/advice on these problems.

The results of the course and its exams can be found on the department web-page "Kurssitulokset/Exam results"

## Registration

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## Exercises

### Assignments

- Exercises 1
- Exercises 2
- Exercises 3
- Exercises 4
- Exercises 5
- Exercises 6
- Exercises 7
- Exercises 8
- Exercises 9
- Exercises 10
- Exercises 11
- Exercises 12 (voluntary/includes some review problems)

### Exercise classes

Note: You get extra points from solved assignments as follows:

25% = 1p, 35% = 2p, 45% = 3p, 55% = 4p, 65% = 5p and 75% = 6p.

First exercise class on Monday, September 7.

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

1. | Monday | 12-14 | C122 | Aleksis Koski |

## Course feedback

Course feedback can be given at any point during the course. Click here.