# Fourier Analysis [MA4064] - WS 16/17

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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.

## News

• 02.08.2016: Welcome! There will be a 2h-lecture and a 1h-exercise class each week.
• 16.10.2016: The FIRST LECTURE is on Thu 20.10.2016, 10:15-12:00, MA HS3. At the end of the lecture we will present/discuss possible time slots for the problem classes.
• 20.10.2016: The first tutorial is on Tue 25.10.2016, 09:00-10:00, MI 03.06.011.
• 20.10.2016: Please register for Übungen zu Fourieranalysis [MA4064] at TUMOnline even if you are not going to attend the problem classes (just register for any group). There is a moodle forum for Übungen zu Fourieranalysis participants, https://www.moodle.tum.de/mod/forum/view.php?f=42010.
• 10.11.2016 Solutions for the 3rd sheet will appear online with a short delay, on 11.11.2016 before 4 p.m.
• 12.12.2016 Tutorial Group 1 (Tue., 13.12.2016, 09:15--10:00) this week is CANCELLED. Please attend the tutorials on Wednesday instead. Next week all groups will be held as usual.
• 22.01.2017 Office hours of Arseniy Tsipenyuk on Tuesday, January the 24th, are held between 14:00 and 16:30 instead of the usual office hours.
• 27.01.2017 The exam will take place on 09.02.2017 between 9:00 and 10:00 in MW 2050 (Hap - Z) and in MW 1250 (A - Hao). 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,...).
• The following material is
Needed in exam Not needed in exam
All definitions in theoretical sections (e.g. definitions of Fourier coefficients, FT of distributions, Schwartz space etc.) You should be able to check and explain whether a definition applies to an object (e.g. whether a one-dimensional lattice of delta functions is a tempered distribution). Definitions in application sections
All theorems in theoretical sections (e.g. Thm. 4.1 (2.1 in uploaded lecture notes), Thm. 7.1 (3.1 in uploaded lecture notes), Poisson, decay of Fourier coefficients of smooth functions) and Shannon's sampling theorem Theorems in application sections, except Shannon and generalized Poisson summation with arbitrary invertible matrices in the argument (derived in Sheet 11)
Proofs of Thm. 4.1 1)-7) (Thm. 2.1 in lecture notes; proof of the reconstruction formula is not needed); Shannon (only reconstruction of $\hat f$); proof of Thm. 1.7 (smoothness implies decay of Fourier coefficients) All other proofs
Compute Fourier coefficients and Fourier transformations Fourier transformations of explicit functions
Solve partial differential equations applying methods from lectures
If you have further questions regarding the table above, contact Arseniy Tsipenyuk via tsipenyuma.tum.de (comments to the Q&A session held on 2.2.2017 were sent out by e-mail to everybody who registered for the tutorials in TUMonline).

• 21.02.2017: The results of the exam have been published in TUMonline. You can take a look at the graded exams on Thursday, February the 23rd, 2017, from 11:30 to 12:30, in 03.08.022. Please, bring your student ID.
• 24.04.2017 The repeat exam will take place on Wednesday, 26.04.2017 between 15:00 and 16:00 in MI HS 3. 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,...).

## Timetable

 Lecture Thursday 10:15 - 12:00 MI HS 3

Class    Weekday    Time    Place    Tutor
Group 1   Tuesday    09:15 - 10:00    MI 03.06.011    Arseniy Tsipenyuk
Group 2   Wednesday    12:00 - 12:45    MI 02.08.020    Arseniy Tsipenyuk
Group 3   Wednesday    14:00 - 14:45    MI 03.08.022    Arseniy Tsipenyuk

## Course Material

Typed lecture notes will be uploaded here directly after each lecture.

Lecture Appendices

Week 1
Lecture 1 Part 1: Introduction; plucked string example (analysis meets number theory and music).
Lecture 1 Part 2: Riemann-Lebesgue lemma.   Lecture 1
Problem Class: No induced 7ths.   Sheet 1       Solution

Week 2
Lecture 2 Part 1: Proof of convergence of Fourier series for Hoelder continuous functions.
Lecture 2 Part 2: Counterexample of a continuous function whose Fourier series diverges at a point.   Lecture 2
Problem Class: Gibbs phenomenon.   Sheet 2       Solution

Week 3
Lecture 3 Part 1: What is the probability that a number is not divisible by a square?
Lecture 3 Part 2: L2 theory of Fourier series: analysis aspects. Lecture 3
Problem Class: Using Fourier to calculate infinite sums; Fourier transform with even modes.   Sheet 3       Solution

Week 4
Lecture 4 Part 1: L2 theory of Fourier series: linear algebra aspects. Lecture 4 Part 1
Lecture 4 Part 2: Some basic mathematics of string instruments. Lecture 4 Part 2
Problem Class: The ideal plucked damped string: Fourier analysis; experiments with synthesized sound as well as real instruments. Sheet 4       Solution

Week 5
Lecture 5 Part 1: Smoothness and decay of Fourier coefficients. Lecture 5 Part 1
Lecture 5 Part 2: The Fourier transform: definition. Lecture 5 Part 2
Problem Class: More about regularity and decay of Fourier coefficients. Basic properties of the Fourier transform. Sheet 5       Solution

Week 6
Lecture 6 Part 1: The Fourier Transform: basic properties. (See Lecture 5 Part 2)
Lecture 6 Part 2: Proof of the Fourier reconstruction formula. Lecture 6
Problem Class: Heisenberg uncertainty. Sheet 6       Solution

Week 7
Lecture 7 Part 1: Fourier's original motivation for introducing the FT: Solution of the heat equation. Lecture 7 Part 1
Lecture 7 Part 2: The Fourier transform on L^2. Lecture 7 Part 2
Problem Class: Why e^ikx? Introduction to the notion of characters. Sheet 7       Solution

Week 8
Lecture 8 Part 1: Why e^ikx? Algebraist's viewpoint and the notion of characters. (See Sheet 7)
Lecture 8 Part 2: Distributions and their Fourier transform. The Dirac delta function and its Fourier transform. Lecture 8 Part 2
Problem Class: Getting acquainted with distributions. L^p-functions as distributions. Sheet 8       Solution

Week 9
Lecture 9 Part 1: The Fourier transform of plane waves, and of ordinary functions viewed as distributions. (See Lecture 8 Part 2)
Lecture 9 Part 2: The Fourier transform of an array of delta functions (alias Poisson summation) Lecture 9 Part 2
Problem Class: The wave equation on the line in one dimension. Sheet 9       Solution

Week 10
Lecture 10 Part 1: Proof of Lemma 3.1: Schwartz space is invariant under the FT. Lecture 10 Part 1
Lecture 10 Part 2: Digital music technology, I: Shannon sampling. Lecture 10 Part 2
Problem Class: Undersampling. Sheet 10       Solution
Original     Undersample 15    Undersample 35

Week 11
Lecture 11 Part 1: Digital music technology II: Undersampling; Sound examples Lecture 11 Part 1
Lecture 11 Part 2: Calculating with distributions: convolution, multiplication, differentiation Lecture 11 Part 2
Problem Class: Fourier transform of a crystal lattice. Sheet 11       Solution

Week 12
Lecture 12 Part 1: Calculating with distributions: proof of Fourier inversion and of the calculus rules (See Lecture 11 Part 2)
Lecture 12 Part 2: Quasicrystals. Lecture 12 Part 2
Problem Class: Distribution calculus and phase problem. Sheet 12       Solution

Week 13
Lecture 13 Part 1: Wave equation in n dimensions. Lecture 13
Lecture 13 Part 2: Course summary. Brief discussion of the exam. Summary
Problem Class: Free Schrödinger equation Sheet 13       Solution

Lecture Appendices