Fourier Analysis [MA4064] - SS 13
News Timetable Course Material People
Fourier series and the Fourier transform play amazingly many different roles in maths, science, and engineering: they are
- powerful tools in solving mathematical problems which seem unrelated to Fourier analysis (e.g., to solve partial differential equations),
- output data of interest in their own right (experimental X-ray images are essentially optical measurements of Fourier transforms, and the human ear essentially Fourier-decomposes acoustic signals),
- important paradigms of more general concepts (such as diagonalizing transformations of differential operators in functional analysis, or decomposing maps into group homomorphisms in algebra and number theory)
- and even a basic part of fundamental models (they underlie the notion of ''momentum'' in quantum mechanics).
In this course, my aim is to develop the key mathematical results and techniques of Fourier analysis from the beginning, and to convey an understanding of their use in different areas of maths and the sciences. I only assume familiarity with basic analysis, linear algebra, and integration theory.
Some of the material I will present, such as the beautiful Fourier theory underlying X-ray crystallography or digital music technology, is to my knowledge not available in mathematical language in any textbook, but unique to this course. If you have thought of such areas as unrelated to maths, this course will change your mind.
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- 25.09.2013: The repeat exam will take place on Friday, 27.09.2013 from 14:00 to 15:00 in MI 02.08.020. Please be on time.
- Please bring a photo identification, your student identification and black or blue pens (pencils, red and green pens are not allowed).
- The use of any means of help is not permitted. In particular, you may not use any written notes, textbooks or electronic means of help (calculator, mobile phone, notebook,...).
- 07.08.2013: The results of the exam have been published in TUMonline. You can take a look at the graded exams on Wednesday, 14th of August, 2013, in 03.08.011 from 10:00 to 11:00. Please, bring your student ID.
- 31.07.2013: The exam will take place tomorrow, Thursday, 01.08.2013 from 10:00 to 11:00 in Interims Hörsaal 2. Please be on time.
- Please bring a photo identification, your student identification and black or blue pens (pencils, red and green pens are not allowed).
- The use of any means of help is not permitted. In particular, you may not use any written notes, textbooks or electronic means of help (calculator, mobile phone, notebook,...).
- 25.07.2013: Note that a summary of lectures 18 to 20 is now online.
- 17.07.2013: Please learn the proofs of the following three results from the lecture for the exam. One of these (or part of it) will be asked in the exam.
- Lemma 1.1 (Riemann-Lebesgue lemma)
- Theorem 3.1 (Basic properties of the Fourier transform of distributions) (You may use the corresponding results for Schwartz functions)
- Theorem 3.3 (Shannon's sampling theorem) (Without derivation of the Shannon reconstruction formula)
- 17.07.2013: There are no exercises this week (17.07. - 19.07.13).
- 24.06.2013: The exersice group this Thursday, 27.06., will take place at a different time and place: 00.09.022, 16:15 - 17:00.
- 24.06.2013: The exam will be on Thursday, 01.08.2013, 10:00 - 11:00.
- 08.05.2013: Since Thursday, 09.05.2013, is Ascension Day (Christi Himmelfahrt), please try to visit one of the exercise groups on Wednesday and Friday. If this is not possible for you, please write a short eMail to juestel
ma.tum.de.
- 26.04.2013: There is no problem class next week (29.04.13 - 03.05.13).
- 22.04.2013: There will be a fourth exercise group every Thursday, 10:15 - 11:00, 03.08.011.
- 16.04.2013: You can choose your preferred dates for problem class in the following doodle: http://www.doodle.com/u8wdt35wswef6sqr
. This week, problem class will take place on the dates below, plus an additional problem class on Thursday, 18.04.2013, 10:15 - 11:00, 03.08.011.
- 16.04.2013: Please note the information on problem class.
Timetable
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| Lecture |
Monday |
12:00 - 14:00 |
MI HS 3 |
Class  |
Weekday |
Time |
Place |
Tutor |
|
| Group 1 |
Wednesday |
15:00 - 15:45 (14:15 - 15:45) |
MW 2050 |
Dominik Jüstel |
|
| Group 2 |
Wednesday |
16:00 - 16:45 (16:00 - 17:30) |
MW 2050 |
Dominik Jüstel |
|
| Group 3 |
Thursday |
10:15 - 11:00 (10:00 - 11:30) |
03.08.011 |
Dominik Jüstel |
|
| Group 4 |
Friday |
09:00 - 09:45 (08:30 - 10:00) |
00.09.022 |
Dominik Jüstel |
|
| Exam |
|
|
| Thursday, 01.08.2013 |
10:00 - 11:00 |
Interims HS 2 |
| Repeat Exam |
|
|
| Friday, 27.09.2013 |
14:00 - 15:00 |
MI 02.08.020 (M11/GKAAM) |
Course Material
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| Week 1 |
|
| Lecture 1: |
Introduction; plucked string example (analysis meets number theory and music). |
| Lecture 2: |
Riemann-Lebesgue lemma. |
Lectures1-2 |
| Problem Class: |
No induced 7ths. |
Sheet1 solution |
| Week 2 |
|
| Lecture 3: |
Proof of convergence of Fourier series for Hoelder continuous functions. |
| Lecture 4: |
Counterexample of a continuous function whose Fourier series diverges at a point. |
Lectures3-4 |
| Problem Class: |
Gibbs phenomenon. |
Sheet2 solution |
| Week 3 |
|
| Lecture 5: |
What is the probability that a number is not divisible by a square? |
| Lecture 6: |
L2 theory of Fourier series: linear algebra aspects. |
Lectures5-6 |
| No class |
|
|
| Week 5 |
|
| Lecture 9: |
Smoothness and decay of Fourier coefficients. |
Lecture 9 |
| Lecture 10: |
The Fourier transform: definition and basic properties. |
Lecture 10 |
| Problem Class: |
More about regularity and decay of Fourier coefficients. Basic properties of the Fourier transform. |
Sheet4 solution |
| Week 6 |
|
| No lectures |
(Bank holiday -- Pentecost) |
|
| Problem Class: |
Heisenberg uncertainty. |
Sheet5 solution |
| Week 7 |
|
| Lecture 11: |
Proof of the Fourier reconstruction formula. |
| Lecture 12: |
Fourier's original motivation for introducing the FT: Solution of the heat equation. |
Lectures 11-12 |
| Problem Class: |
Why e^ikx? Algebraist's viewpoint and the notion of characters. |
Sheet6 |
| Week 8 |
|
| Lecture 13: |
The Fourier transform on L^2. |
Lecture 13 |
| Lecture 14: |
Distributions and their Fourier transform. The Dirac delta function and plane waves as examples of distributions. |
Lecture 14 |
| Problem Class: |
Getting acquainted with distributions. L^p-functions as distributions. |
Sheet7 solution |
| Week 9 |
|
| Lecture 15: |
Periodic arrays of delta functions and the Poisson summation formula. |
Lectures 15-17 |
| Lecture 16: |
Fourier calculus for distributions. |
|
| Problem Class: |
The wave equation on the line in one dimension. |
Sheet8 solution |
| Week 10 |
|
| Lecture 17: |
Fourier calculus for distributions (proofs). |
| Lecture 18: |
X-ray crystallography, 1: Far field approximation of outgoing radiation is given by FT of electron density. |
|
| Problem Class: |
Plane waves on plain weaves. |
Sheet9 solution |
| Week 11 |
|
| Lecture 19: |
X-ray crystallography, 2: FT of a Bravais lattice of delta functions. |
| Lecture 20: |
Comparison theory -- experiment: Von Laue images. |
Summary_lectures18-20 |
| Problem Class: |
Quasicrystals. |
Sheet10 solution |
| Week 13 [NOT EXAM RELEVANT] |
|
| Lecture 23: |
Free Schroedinger equation and a pedestrian look at the celebrated Feynman path integral. |
Lecture 23 |
| Lecture 24: |
Brief summary of course. Preview of seminar 'Fourier analysis and synthesis of optical and acoustic signals', WS 13/14 |
Lecture 24 |
| Problem Class: |
The phase problem. |
Sheet12 solution |
| Week 14 |
|
| Lecture 25: |
Discussion of the exam / answering your questions about the exam. |
|
Text Books
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| Friesecke |
Lectures on Fourier Analysis |
Lecture Notes |
2007 |
|
|
| Strichartz |
A guide to distribution theory and the Fourier transform |
CRC Press |
1994 |
|
|
| Reed, Simon |
Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness |
Academic Press |
1975 |
|
|
People
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