Fourier Analysis: Mathematics GU4032
(Spring 2020)
Peter Woit (woit@math.columbia.edu)
Monday and Wednesday
11:40-12:55
Mathematics 520
This course will cover the theory and applications of Fourier series
and the Fourier transform.
Topics to be covered will include the following:
Fourier series: basic theory
Fourier series: convergence questions
Fourier series: applications
The Fourier transform: basic theory
The Fourier transform: distributions
The Fourier transform: applications
Applications to partial differential equations of physics
Representation theory of Abelian groups
Applications to number theory
Assignments
There will be assignments roughly each week, due in class on
Wednesday, mostly taken from the textbook.
Assignment 1 (due Wednesday, Jan. 29):
Chapter 1, Exercises 4 (parts b-i), 5
Chapter 2, Exercises 2, 4, 6
Assignment 2 (due Monday, Feb. 10)
Chapter 2, Exercises 10,13,15,17, Problem 2a
Chapter 3, Exercise 20
Assignment 3 (due Monday, Feb. 17)
Chapter 2, Exercises 18,19,20
Chapter 3, Exercises 8,9,12
Assignment 4 (due Monday, Feb. 24)
Chapter 3, Problems 4,5
Chapter 4, Exercises 11,12,13
Assignment 5 (due Monday, March 2)
Chapter 5, Exercises 2,6,12, Problems 1,7
Assignment 6 (due Monday, March 30)
Chapter 5, Exercises
15,17,18,19,23, Problem 3a
Assignment 7 (due Monday, April 6)
Strichartz, Chapter 1 Problem 11
Strichartz, Chapter 2 Problem 13
Osgood, Problems 4.3, 4.4,4.7,4.8
Assignment 8 (due Monday, April 13)
Osgood, Problems 4.5,4.12,4.13,4.18
Strichartz, Chapter 4, problems 1,6
Assignment 9 (due Monday, April 20)
Chapter 6, Exercises 1,4,5,6
Assignment 10 (due Monday, April 27):
Chapter 6, Exercises 7,8,10,11
Chapter 6, Problem 7
Assignment 11 (due Monday, May 4):
Chapter 7, Exercises
1,3,4,5,6,7,13
Syllabus
For each class, see here for what will be covered, and for which
sections of the textbook you should be reading.
Wednesday, January 22:
Overview of the course. Definition of Fourier series, examples.
Reading: Chapter 1 (for motivation, the topics of this chapter
will be treated in detail later in the course). Section 1 of
Chapter 2
Monday, January 27:
Uniqueness of Fourier series. Convolution.
Reading: Chapter 2, sections 2 and 3
Wednesday, January 29:
Pointwise convergence of Fourier series, Dirichlet kernel. Gibbs
phenomenon.
Reading: Sections 3.2.1, 2.4
Monday, February 3:
Cesaro summability, Fejer kernel. Abel summability, Poisson kernel,
Reading: Sections 2.5
Wednesday, February 5:
Mean convergence of Fourier series, Parseval's equality.
Reading: Chapter 3, section 1
Monday, February 10:
Harmonic functions, Dirichlet problem
Reading: Sections 1.2.2, 2.5.4
Wednesday, February 12:
Heat equation and Schrödinger equation on a circle
Reading: Section 4.4
Monday, February 17
Introduction to the Fourier transform
Reading: Introduction to Chapter 5, Sections 5.1.1-5.1.3
Wednesday, February 19
Properties of the Fourier transform, Fourier inversion
Reading: Sections 5.1.4-5.1.5
Monday, February 24
Plancherel theorem, Heat equation, Schrödinger equation
Reading: Section 5.1.6, 5.2.1
Wednesday, February 26
Harmonic functions in the upper half plane, Heisenberg uncertainty,
Review
Reading: Sections 5.2.2, 5.4
Monday, March 2
Midterm exam
Wednesday, March 4
Poisson summation formula
Reading: Section 5.3
Monday, March 9
Class canceled by university.
For material covered in the classes from this point on, lecture
notes are at
Fourier
Analysis Notes, Spring 2020
Wednesday, March 11
Theta and zeta functions
Reading: Section 5.3
Monday, March 23 and Wednesday, March 25
Classes canceled by university.
Monday, March 30
Distributions: definitions and examples
Reading: Strichartz, Chapter 1 and Osgood, Chapter 4.4
Wednesday, April 1
Distributions: differentiation
Reading: Strichartz, Chapter 2 and Osgood, Chapter 4.6
Monday, April 6
Distributions: Fourier transforms
Reading: Strichartz, Chapter 4 and Osgood, Chapter 4.5
Wednesday, April 8
Distributions: Convolution and solutions of differential equations
Reading: Strichartz, Chapter 5 and Osgood, Chapter 4.7
Monday, April 13
Fourier transforms in higher dimensions
Reading: Sections 6.1,6.2,6.4
Wednesday, April 15
More Fourier transforms in higher dimensions, applications to PDEs.
Reading: Sections 6.1,6.2,6.4
Monday, April 20
Heat equation in higher dimension, wave equation in d=1
Reading: Section 6.3
Wednesday, April 22
Wave equation in higher dimensions
Reading: Section 6.3
Monday, April 27
Fourier analysis on Z(N)
Reading: Section 7.1
Wednesday, April 29
Fourier analysis for commutative groups
Reading: Section 7.2
Monday, May 4
Some number theory, Dirichlet's theorem
Reading: Chapter 8
Monday, May 11
Take home exam due
Exams
There will be a midterm exam, and a take home final exam.
Grading
Your final grade for the course will be a pass fail grade roughly
determined 25% by assignments, 50% by the midterm, 25% by the take
home final.
Textbook
Elias Stein and Rami Shakarchi
Fourier
Analysis: An Introduction
Princeton University Press, 2003
For errata in this book, see here
and here.
For material covered in the classes after class moved to online
only, lecture notes are at
Fourier
Analysis Notes, Spring 2020
For distributions, you should look at
Osgood, Lectures on the Fourier Transform and its Applications
which is available through the Columbia library system, should
be here
https://ebookcentral.proquest.com/lib/columbia/detail.action?docID=5683572
Office Hours
I should always be available after class (after a lunch break) in my
office (Math 421), so 2-3pm. Feel free to come by Math 421 at
any time and I will likely have some time to talk, or make an
appointment by emailing me.
Teaching Assistant
The TA for the course is Maithreya Sitaraman
(maithreya@math.columbia.edu)
Other Books and Online
Resources
Besides the course textbook, some other textbooks at a similar level
that you might find useful are
Howell, Principles of Fourier Analysis
Osgood, Lectures on the Fourier Transform and its Applications
Kammler, A First Course in Fourier Analysis
Strichartz, A Guide to Distribution Theory and Fourier
Transforms
Walker, The Theory of Fourier Series and Integrals
Tolstov, Fourier Series
Folland, Fourier Analysis and its Applications
Körner, Fourier Analysis
Brown and Churchill, Fourier Series and Boundary Value Problems
Dym and McKean, Fourier Series and Integrals
Vretblad, Fourier
Analysis and its Applications
Dyke, An
Introduction to Laplace Transforms and Fourier Series
Duistermaat and Kolk, Distributions
Some lecture notes available online are
Körner, Part
III Lecture notes
Asadzadeh, Lecture
notes in Fourier analysis