TMA362 / MMG710 Fourier analysis Autumn 20
Concerning the re-exam on January 7, 2021:
On your Canvas dashboard you should have a link to:
Omtentamen 1 Modul: 0101, TMA362 / MMG710
This is where you will find the exam, the zoom-links, and further information.
October exam:
January exam:
Due to the Covid-19 pandemic, this course will be given online (via zoom).
With the exception of the first and the last week, there are 4 weekly scheduled meetings (28 meetings in total). Zoom-links to these meetings will be posted below. The schedule can be found on TimeEdit.
The meetings will NOT be recorded, but I will try to upload my written comments after each meeting.
Since I don't believe that regular lectures (45 + 45 min), delivered over zoom, would be efficient for this course, I have chosen to implement a more problem-based learning approach.
The skeleton of the course are 6 (optional) home assignments (you can get bonus points for the exam if you hand them in). The teaching will be conducted as "office hours", during which we discuss some aspects of the theory that you will need in order to solve the problems in the assignments. I will be ready with an Ipad and a shared screen during these meetings.
If you have a question and don't feel like waiting until the next meeting (or if you simply prefer not to ask the question over zoom), please email me the question, and I will write a reply asap. If I believe that the question is of general interest, I might present a more detailed reply/discussion that I upload on Canvas.
There is no course book. Instead I have prepared a series of "vignettes" ( = short pieces that expose either a little bit of theory, illuminating examples or some computational tricks). The dependencies between these vignettes will be clearly marked (you might have to read several of them together).
Several exercise sheets (as well as detailed solutions to some of them), and old exams (with solutions) can be found below. Recommended exercises for every week will be posted below.
Course schedule
Week 1 (Aug 31-Sep 6, 3 meetings) Fourier series in 1 dimension (convergence)
Tuesday 1/9, 13.15-15.00: https://chalmers.zoom.us/j/64966762514
Brief overview of the course, introduction to HA1 (Short summary of the vignettes: Dirac families [Fejer's Theorem] and Weyl's Equidistribution Criterion).
Wednesday 2/9, 15.15-17.00: https://chalmers.zoom.us/j/62792838105
Some explicit computations with Fourier series (Reflection principle).
Thursday 3/9, 8.00-09.45: https://chalmers.zoom.us/j/63786106612
Continuation of computations, questions/discussions from the class.
What you should be able to do after this week: compute Fourier coefficients and limits of Fourier series (and their Fejer means).
Recommended exercises for this week:
Exercise sheet A: 1,3,6,7
Exercise sheet B: 8,9,11,12,13,16
Exam October 2019: Problem 2 [A,C,D,E], Problem 7.
Exam August 2020: Problem 3, Problem 6.
Week 2 (Sep 7-Sep 13) Parseval's Theorem, inner product spaces, best approximation.
Monday 7/9, 10:00-11:45: https://chalmers.zoom.us/j/63123986463
Convolution of periodic functions, Parseval's Theorem.
Tuesday 8/9, 13:15-15:00: https://chalmers.zoom.us/j/65756978547
Inner product spaces, Cauchy-Schwartz and Bessel's inequalities.
Wednesday 9/9: 15:15-17:00: https://chalmers.zoom.us/j/68869360441
Continuation: inner product spaces, Cauchy-Schwartz and Bessel's inequalities.
Thursday 10/9: 8:00-9:45: https://chalmers.zoom.us/j/62709547977
The best approximation lemma, Gram-Schmidt.
What you should be able to do after this week: be aware of the convolution operation, use Parseval's formula, estimate integrals and sums using ideas from inner product spaces. Use Best Approximation Lemma.
Recommended exercises for this week:
Exercise sheet A: 8,9,10,11,12 [see also HA2], 13, 15.
Exercise sheet B: 39, 41, 44.
Exam October 2019: Problem 2[B], Problem 6.
Exam January 2020: Problem 6, Problem 8.
Exam August 2020: Problem 5, Problem 9.
Model Exam I: Problem 6.
Model Exam II: Problem 5, Problem 7.
Week 3 (Sep 14-Sep 20) Poisson's summation formula, Fourier transform, Fejer's Theorem.
Monday 14/9, 10:00-11:45: https://chalmers.zoom.us/j/68257978240
Periodizations, Poisson's summation formula, a brief introduction to Fourier transform.
Tuesday 15/9, 13:15-15:00: https://chalmers.zoom.us/j/69719307987
Fejer's Theorem for the Fourier transform. A few computations.
Wednesday 16/9: 15:15-17:00: https://chalmers.zoom.us/j/69711913536
Plancherel's Theorem. Some discussions about HA3. Due to technical difficulties, this lecture is cancelled.
Thursday 17/9: 8:00-9:45: https://chalmers.zoom.us/j/68747769789
What you should be able to do after this week:
Use Poisson's summation formula, compute Fourier transform of Gaussians and Cauchy kernels, as well as other simple examples. Use the Plancherel's Theorem to compute various integrals.
Recommended exercises for this week:
Exercise sheet A: 17-25.
Exercise sheet B: 55,59,60,63,66.
Exam October 2019: Problem 1, Problem 3, Problem 9.
Exam January 2020: Problem 1, Problem 3, Problem 10.
Exam August 2020: Problem 1, Problem 8.
Model Exam I: Problem 1, Problem 3, Problem 5, Problem 8, Problem 9.
Model Exam II: Problem 2.
Week 4 (Sep 21-Sep 27) Higher dimensional Fourier analysis.
Monday 21/9, 10:00-11:45: https://chalmers.zoom.us/j/66107006500
An overview of HA3 - repetition of the previous week.
Tuesday 22/9, 13:15-15:00:https://chalmers.zoom.us/j/67496277454
Fourier series and Fourier transform in arbitrary dimensions. Subordination principle.
Wednesday 23/9: 15:15-17:00: https://chalmers.zoom.us/j/68696941528
Fourier transform of radial functions, Bessel functions.
Thursday 24/9: 8:00-9:45: https://chalmers.zoom.us/j/65610917379
The Radon transform and tomography.
What you should be able to do after this week:
Use Poisson's summation formula, compute Fourier transform of Gaussians and Cauchy kernels, in higher dimensions. Estimate some simple Bessel functions. Understand the Radon transform, and how it relates the higher dimensional Fourier transform with the one-dimensional Fourier transform.
Recommended exercises for this week:
Exercise sheet A: 26, 27
Exam October 2019: Problem 5
Exam January 2020: Problem 5
Week 5 (Sep 28-Oct 4) Laplace transform
Monday 28/9, 10:00-11:45: https://chalmers.zoom.us/j/67172067713
Introduction to the Laplace transform.
Tuesday 22/9, 13:15-15:00: https://chalmers.zoom.us/j/63695969093
Examples of Laplace transforms.
Wednesday 23/9: 15:15-17:00: https://chalmers.zoom.us/j/68161734550
Further examples.
Thursday 24/9: 8:00-9:45: https://chalmers.zoom.us/j/62869904279
Brief overview of HA4 and HA5+6.
What you should be able to do after this week:
Compute Laplace transforms of poly-exponential expressions, know how to use Ikehara's Tauberian Theorem, solve differential equations with constant coefficients using Laplace transforms.
Recommended exercises for this week:
Exercise sheet B: 76-81.
Exam October 2019: Problem 4, Problem 8.
Exam January 2020: Problem 4, Problem 9.
Exam August 2020: Problem 4, Problem 7.
Model Exam II: Problem 4, Problem 9.
Week 6+7+8 (Oct 5-Oct 25) Deeper discussions/repetition
Monday 5/10, 10:00-11:45: https://chalmers.zoom.us/j/62563232438
Tuesday 6/10, 13:15-15:00: https://chalmers.zoom.us/j/61788902372
Wednesday 7/10, 15:15-17:00: https://chalmers.zoom.us/j/69653594839
Thursday 8/10, 8:00-10:00: https://chalmers.zoom.us/j/67616387998
Monday 12/10, 10.00-11.45: https://chalmers.zoom.us/j/67418710168
Tuesday 13/10, 13.15-15.00 https://chalmers.zoom.us/j/66724881264
Wednesday 14/10, 15:15-17.00 https://chalmers.zoom.us/j/61347264743
Thursday 15/10, 8:00-9.45 https://chalmers.zoom.us/j/68339285244
Monday 19/10, 10.00-11.45 https://chalmers.zoom.us/j/68635842650
Examination
Written exam (most likely online). Total: 50 points.
Grades:
Chalmers (TMA362): 3 = [23-32 points], 4 = [33-42 points], 5 = [43-50 points].
GU (MMG710): G = [23-37 points], VG = [38-50 points].
Exams from previous years can be found below.
Home assignments
There will be (optional) 6 home assignments (each worth 3 bonus points for the exam = 18 in total). The aim of these assignments is to put the theory to work and solve a large collection of seemingly unrelated problems.
HA 1: First digit statistics and Benford's Law [Fourier series, approximation and convergence].
Tools needed: The notion of an equidistributed sequence, van der Corput's lemma (see the vignette: Weyl's Equidistribution Criterion).
Deadline: Monday September 14, at 12.00 (email your solution to micbjo@chalmers.se)
Comments (concerning the usage of van der Corput)
HA2: Dido's Problem. [Fourier series, Parseval's Theorem]
Tools needed: Parseval's Theorem.
Deadline: Monday September 28, at 12:00 (email your solution to micbjo@chalmers.se)
HA3: The Central Limit Theorem. [Fourier transform in 1 dimension]
Tools needed: Fourier transform of the Gaussian, Fourier inversion, Plancherel's Theorem.
Deadline: Monday October 12, at midnight (email your solution to micbjo@chalmers.se)
HA4: Gauss Circle Problem. [Fourier transform in 2 dimensions]
Tools needed: Fourier transform of a ball, Poisson's Summation Formula in 2 dimensions.
Deadline: Monday, October 19, at midnight (email your solution to micbjo@chalmers.se)
HA5+6: The Prime Number Theorem [Laplace transform]
Tools needed: Ikehara's Tauberian Theorem.
Deadline: Sunday October 25, at midnight (email your solution to micbjo@chalmers.se)
Vignettes
Note: the vignettes contain a lot of stuff, far from everything is really needed in order to pass the course. You should understand the main results (Fejer's Theorem, Parseval's Theorem, etc), know how to apply them, and be aware of some pitfalls. I will never ask you to recite a proof of a theorem or lemma from the lecture notes on the exam, nor in the home assignments. Feel free to use the results in the vignettes as black boxes when you present your solutions.
- Stuff you should already know [updated regularly]
Fourier series (convergence)
Dirac families are extremely useful when you want to prove that Fourier series (or related objects) have good approximation properties. In this vignette, I outline the general theory, and prove Fejer's Theorem (Fourier series approximation) and Weierstrass Theorem (approximation by polynomials). It's a long note; the most important results that you will need are Corollary 2.5, Corollary 2.6 and Corollary 2.7.
- Weyl's Equidistribution Criterion
This result (together with the accompanying lemma of van der Corput [Lemma 1.4]) plays a key role in HA I. Also this is a long note, and you don't need to read everything. The key goal of this note is to explain how Fourier series can be used to decompose complicated objects (such as the frequency counting of first digits in HA1) into simple objects (trigonometric sums) - this is outlined in (I-IV) on p. 2-4.
One of Fourier's original motivations for developing the theory Fourier series was to solve some partial differential equations which show up in mathematical physics, for instance the heat equation. In this vignette I give a few examples of how the heat equation can be solved using Fourier series.
I present a concrete 1-periodic continuous function (due to Fejer) whose Fourier series diverges at one point. This explains why it is really necessary to introduce the weighted Fejer sums to get (everywhere) convergence.
- (Optional) Hardy's Tauberian Theorem
I show that if the Fourier coefficients of a Riemann-integrable function are O(1/m), then the Fourier sums converges at every continuity point [Fejer's Theorem only tells us that the Fejer sums converge.]
A consequence of the Weierstrass Theorem (see "Dirac families") is that continuous functions on a bounded and closed interval are determined by their integrals against polynomials. This is extremely important in probability theory (method of moments). The aim of this note is to show that one needs to be very cautious if one wants to use this technique for continuous functions on an unbounded interval. A concrete example (due to Stieltjes) of a non-zero function on the positive real line which integrates to zero against every polynomial is presented.
Fourier series and inner product spaces
A brief introduction to inner product spaces, and a proof of Cauchy-Schwarz inequality, and some immediate consequences thereof.
- Bessel's inequality and Parseval's Theorem
I prove Bessel's inequality for a general inner product space, and specialise to the setting of Fourier series and deduce Parseval's Theorem.
Fourier transform (1 dimension)
- Fejer's Theorem for the Fourier transform
I introduce the Fourier transform and prove an analogue of Fejer's Theorem for it.
I prove an analogue of Parseval's Theorem for the Fourier transform.
- Integrability of Fourier transforms
I show that a differentiable function with an absolutely integrable derivative has an absolutely integrable Fourier transform. The proof uses Plancherel's Theorem and Cauchy-Schwarz inequality. Under the stronger assumption that the function is twice differentiable with an absolutely integrable second derivative, I give an elementary proof of the absolute integrability of its Fourier transform.
A few basic calculations of Fourier transforms: Gaussian and Cauhcy kernels.
A very useful relation between sums of functions and their Fourier transforms.
- Laplace equation in the upper half plane
I show how the Fourier transform can be used to solve the Laplace equation in the upper half plane.
Fourier analysis in higher dimensions
- Fourier analysis in higher dimensions
In this vignette I summarise the higher dimensional versions of Fejer's Theorem, Parseval's Theorem, Plancherel's Theorem and Poisson's Summation Formula (which will be needed in HA4).
I show how one computes the Fourier transform using a very simple version of subordination.
The ideas behind this transform are used in medical tomography when you want to reconstruct a body from planar slices.
- The Fourier transform of a ball
In this vignette I establish some decay estimates for the Fourier transform of a two-dimensional ball. These estimates are used in HA4.
The Laplace transform
Basic theory of the Laplace transform, elementary properties with respect to translations, differentiations, phase shifts etc. Applications to ordinary differential equations with constant coefficients.
In this vignette we relate the asymptotic behaviour of a non-decreasing function to the behaviour of its Laplace transform close to the abscissa of convergence.
Exercises and solutions
Sample solutions: Exercises 1,3,6, Exercises 9,10,13,15, Exercises 18-23.
These are some additional exercises from Folland's book "Fourier analysis and its applications". There are some overlaps between the sheets.
Sample solutions:
Model exams from when the new exam format was introduced
(No solutions will be given)
Notes from the lectures
Lecture I:
Zoom-notes: Lecture I
Lecture II:
Pre-notes: Lecture II
Zoom-notes: Lecture II
Lecture III:
Pre-notes: Lecture III
Zoom-notes: Lecture III
Lecture IV:
Pre-notes: Lecture IV
Zoom-notes: Lecture IV
Lecture V:
Pre-notes: Lecture V
Zoom-notes: Lecture V
Lecture VI:
Pre-notes: Lecture VI
Zoom-notes: Lecture VI
Lecture VII:
Pre-notes: Lecture VII
Zoom-notes: Lecture VII
Lecture VIII:
Pre-notes: Lecture VIII
Zoom-notes: Lecture VIII
Lecture IX:
Pre-notes: Lecture IX
Zoom-notes: Lecture IX
Lecture X:
Pre- notes: Lecture X
Zoom-notes: Lecture cancelled!
Lecture XI:
Pre-notes: Lecture XI
Zoom-notes: Lecture XI
Lecture XII:
Pre-notes: Lecture XII [New scan!]
Zoom-notes: Lecture cancelled!
Lecture XIII:
Pre-notes: Lecture XIII
Zoom-notes: Lecture XIII
Lecture XIV:
Pre-notes: Lecture XIV
Zoom-notes: Lecture XIV
Lecture XV:
Pre-notes: Lecture XV
Zoom-notes: Lecture XV
Lecture XVI:
Pre-notes: Lecture XVI
Zoom-notes: Lecture XVI
Lecture XVII:
Pre-notes: Lecture XVII
Zoom-notes: Lecture XVII
Lecture XVIII:
Pre-notes: not prepared.
Zoom-notes: Lecture XVIII
Lecture XIX:
Pre-notes: Lecture XIX
Zoom-notes: Lecture XIX
Lecture XX:
Pre-notes: Lecture XX
Zoom-notes: Lecture XX
Lecture XXI:
Pre-notes: Lecture XXI
Zoom-notes: Lecture XXI
Lecture XXII:
Pre-notes: Lecture XXII
Zoom-notes: Lecture XXII
Lecture XXIII:
Pre-notes: Lecture XXIII
Zoom-notes:
Lecture XXIV:
Pre-notes: Lecture XXIV
Zoom-notes: Lecture XXIV
Lecture XXV:
Pre-notes: Lecture XXV
Zoom-notes: Lecture XXV
Lecture XXVI:
Pre-notes: Lecture XXVI
Zoom-notes: Lecture XVI
Lecture XXVII:
Pre-notes: Lecture XXVII
Zoom-notes: Lecture XXVII
Course summary:
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