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 Xray images are essentially optical measurements of Fourier transforms, and the human ear essentially Fourierdecomposes 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 Xray 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 (RiemannLebesgue 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 juestelma.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 2 
Wednesday 
16:00  16:45 (16:00  17:30) 
MW 2050 
Dominik Jüstel 

Group 1 
Wednesday 
15:00  15:45 (14:15  15:45) 
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: 
RiemannLebesgue lemma. 
Lectures12 
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. 
Lectures34 
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. 
Lectures56 
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 1112 
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^pfunctions as distributions. 
Sheet7 solution 
Week 9 

Lecture 15: 
Periodic arrays of delta functions and the Poisson summation formula. 
Lectures 1517 
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: 
Xray 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: 
Xray crystallography, 2: FT of a Bravais lattice of delta functions. 
Lecture 20: 
Comparison theory  experiment: Von Laue images. 
Summary_lectures1820 
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, SelfAdjointness 
Academic Press 
1975 


People
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