Fourier Analysis
The course will be implemented as remote teaching due to the pandemic situation.
REMOTE Exam on March 10, between 8.00 and 12.15. Exam takes place in the Koppa system. You should've received a link from the lecturer by e-mail; if this is not the case, and you would like to participate in the exam, contact the lecturer asap.
There will be lectures on Tuesdays (14-16), first lecture on the 12th of January.
There will be exercises on THURSDAYS (14-16), first session on the 21th of January.
There will be NO lectures on Thursdays. Instead, the lecture material has been recorded beforehand (links below). The idea is to view certain recordings (details later) instead of the Thursday lectures. Any questions on the "Thursday recordings" can be discussed on the subsequent Tuesday lectures (I'd be happy to receive questions beforehand, for example by Monday morning, but naturally ex tempore discussion is also welcome.)
In fact, all of the material is contained in the recordings. So, it's up to you if you prefer "live" Zoom lectures, or the recordings.
Prerequisites
It will be useful to have an understanding of the following topics:
Some measure theory (necessary for Lebesgue measure, useful but not necessary for other measures)
Banach and Hilbert spaces (the definitions, and what an orthonormal basis in a Hilbert space means)
Lp-spaces (Hölder's and Minkowski's inequalities at least). You also know what "convolution" means.
Complex numbers (up to the point where you understand the meaning of e^{ix} = cos(x) + i sin(x)).
Regarding some of these preliminaries, here's a 30-minute (not very professional!) recording where I skim through Banach spaces, Hilbert spaces, and Lp-spaces. If you're a student on the course, and would like to discuss some of these topics more carefully in class, I'd be happy to know, as early as possible. I only started working at JYU in the fall, and don't (yet) fully know, what topics are covered, and where.
Lecture notes and recordings
The material on the following lecture notes is gathered from the lecture notes of Mikko Salo, plus a few books books, notably Tom Wolff's Lectures in Harmonic Analysis and Walter Rudin's Functional Analysis. If you prefer reading LaTeX over the lecture notes, you'll find most of the proofs in Mikko Salo's notes (although the proofs differ more and more towards the end of the course).
The lecture recordings are available on Youtube, or see links below. Youtube arguably has a nicer viewing interface.
An introductory lecture, (video) briefly introducing and motivating Fourier series and transforms, and applying the former to solve the 1d heat equation.
L2 theory of Fourier series (Tiivistelmä) (First part as video, Second part as video, Third part as video). The topics here are the convergence in L2 of Fourier series, Fejér and Dirichlet means, and the Lp convergence of Fejér means
Additional topics on Fourier series (Tiivistelmä) (First part as video, Second part as video, Third part as video, Fourth part as video, Fifth part as video). The topics here are pointwise and Lp convergence of Fourier series, smoothness and decay of Fourier coefficients, an application to elliptic regularity theory
Fourier transform I (Tiivistelmä I, Tiivistelmä II) (First part as video, Second part as video, Third part as video, Fourth part as video, Fifth part as video). The topics are: definitions, and invertibility questions for L1, L2, and Schwartz functions
Fourier transform II (Tiivistelmä I, Tiivistelmä II) (First part as video, Second part as video, Third part as video, Fourth part as video, Fifth part as video, Sixth part as video, Seventh part as video, Eighth part as video). The topics are: Fourier transform of Lp functions, tempered distributions
The Hilbert transform (First part as video, Second part as video). These notes contain an application of the theory of Fourier transforms and tempered distributions to the theory of singular integral operators. These are the last notes for the course.
Weekly progress of the lectures:
First week: Introductory lecture + First and Second parts of the L2 theory of Fourier series.
Second week: Lectures 4-7 on Youtube, or everything up to the Third part of the second lecture notes.
Third week: Lectures 8-11 on Youtube, or everything up to the Second part of the third lecture notes (aka Fourier transform I)
Fourth week: Lectures 12-15 on Youtube, or everything up to the First part of the fourth lecture notes (aka Fourier transform II)
Fifth week: Lectures 16-18 on Youtube, or everything up to the Fourth part of the fourth lecture notes (aka Fourier transform II)
Sixth week: Lectures 19-22 on Youtube, or everything up to the end of the Fourier transform II lecture notes.
Seventh week: Lectures 23-24 on Youtube, or the two videos on the Hilbert transform.
Exercises
Zoom link for the exercises:
https://jyufi.zoom.us/j/61362689690
(Same e-passcode as for the lectures; ask Michele or Tuomas if you don't know it.)
The exercise point system is explained in the first set of exercises, see below. You need to complete at least 30% of the exercises to take part in the course exam.
First set of exercises (due on Thursday, Jan 21)
Second set of exercises (due on Thursday, Jan 28)
Third set of exercises (due on Thursday, Feb 4)
Fourth set of exercises (due on Thursday, Feb 11)
Fifth set of exercises (due on Thursday, Feb 18)
Sixth set of exercises (due on Thursday, Feb 25, these are the last exercises)
Some rehearsal questions for the exam (no due date; these are entirely voluntary)
Passing the course
There will be an exam, and an exercise session (assistant: Michele Villa). The exercises will give you some points for the exam (details to be decided later).