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Fourier Analysis [MA4064] - SS 13

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Fourier series and the Fourier transform play amazingly many different roles in maths, science, and engineering: they are

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.

News

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Timetable

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Lecture   Monday    12:00 - 14:00   MI HS 3

Class    Weekday    Time    Place   Sorted ascending Tutor  
Group 4   Friday    09:00 - 09:45 (08:30 - 10:00)    00.09.022    Dominik Jüstel     
Group 3   Thursday    10:15 - 11:00 (10:00 - 11:30)    03.08.011    Dominik Jüstel     
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     

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 4
Lecture 7: L2 theory of Fourier series: analysis aspects. Lecture7
Lecture 8: Some basic mathematics of string instruments. Lecture8    Movie of waveshape
Problem Class: The ideal plucked damped string: Fourier analysis; experiments with synthesized sound as well as real instruments. Sheet3       solution
  Effect of changing the frequency   440-1-1/7     (2/3)*440-1-1/7  
  Effect of changing the plucking point     sintone     440-1-1/2     440-1-1/7     440-1-1/20  

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 12
Lecture 21: Proof of Lemma 3.1: Schwartz space is invariant under the FT. Lecture 21
Lecture 22: Digital music technology, I: Shannon sampling. Lecture 22
Problem Class: Digital music technology, II: Undersampling. Sheet11       solution
  original     undersample15    undersample35  

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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  Name eMail Room Consultation
Lecturer   Gero Friesecke  gfma.tum.de   MI 03.08.054   lecture break and after the lecture
Tutor   Dominik Jüstel  juestelma.tum.de   MI 03.06.021   Wednesday, 14:15 - 15:00, MW2050