Fourier Analysis: Mathematics GU4032
(Spring 2019)
Tuesday and Thursday
10:10-11:25
417 Mathematics
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
Tuesday, mostly taken from the textbook.
Assignment 1 (due Tuesday, Jan. 29):
Chapter 1, Exercises 4 (parts b-i), 5
Chapter 2, Exercises 2, 4, 6
Assignment 2 (due Tuesday, Feb. 5)
Chapter 2, Exercises 10,13,15,17, Problem 2a
Chapter 3, Exercise 20
Assignment 3 (due Tuesday, Feb. 12)
Chapter 2, Exercises 18,19,20
Chapter 3, Exercises 8,9,12
Assignment 4 (due Tuesday, Feb. 19)
Chapter 3, Problems 4,5
Chapter 4, Exercises 11,12,13
Assignment 5 (due Tuesday, Feb. 26)
Chapter 5, Exercises 2,6,12, Problems 1,7
Assignment 6 (due Tuesday, March 12)
Chapter 5, Exercises
15,17,18,19,23, Problem 3a
Assignment 7 (due Tuesday, March 26)
Strichartz, Chapter 1 Problems 3,7
Strichartz, Chapter 2 Problems 4,6,13
Osgood, Problems 4.7,4.8
Assignment 8 (due Tuesday, April 2)
Osgood, Problems 4.5,4.12,4.13,4.18
Strichartz, Chapter 4, problems 1,6
Assignment 9 (due Tuesday, April 9)
Chapter 6, Exercises 1,4,5,6
Assignment 10 (due Tuesday, April 16):
Chapter 6, Exercises 7,8,10,11
Chapter 6, Problem 7
Assignment 11 (due Tuesday, April 23):
Chapter 7, Exercises
1,3,4,5,6,7,13
Assignment 12 (due Thursday, May 2):
Chapter 7, Exercise 11
Chapter 8, Exercises 3,6,8,9,11
Syllabus
For each class, see here for what will be covered, and for which
sections of the textbook you should be reading.
Tuesday, 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
Thursday, January 24:
Uniqueness of Fourier series. Convolution.
Reading: Chapter 2, sections 2 and 3
Tuesday, January 29:
Pointwise convergence of Fourier series, Dirichlet kernel. Gibbs
phenomenon.
Reading: Sections 3.2.1, 2.4
Thursday, January 31:
Cesaro summability, Fejer kernel.Poisson kernel, Abel summability
Reading: Sections 2.5
Tuesday, February 5:
Mean convergence of Fourier series, Parseval's equality.
Reading: Chapter 3, section 1
Thursday, February 7:
Harmonic functions, Dirichlet problem
Reading: Sections 1.2.2, 2.5.4
Tuesday, February 12:
Heat equation and Schrödinger equation on a circle
Reading: Section 4.4
Thursday, February 14
Introduction to the Fourier transform
Reading: Introduction to Chapter 5, Sections 5.1.1-5.1.3
Tuesday, February 19
Properties of the Fourier transform, Fourier inversion
Reading: Sections 5.1.4-5.1.5
Thursday, February 21
Plancherel theorem, Heat equation, Schrödinger equation
Reading: Section 5.1.6, 5.2.1
Tuesday, February 26
Harmonic functions in the upper half plane, Heisenberg uncertainty,
Review
Reading: Sections 5.2.2, 5.4
Thursday, February 28
Midterm exam
Tuesday, March 5
Poisson summation formula
Reading: Section 5.3
Thursday, March 7
Theta and zeta functions
Reading: Section 5.3
Tuesday, March 12
Distributions: definitions and examples
Reading: Strichartz, Chapter 1 and Osgood, Chapter 4.4
Thursday, March 14
Distributions: differentiation
Reading: Strichartz, Chapter 2 and Osgood, Chapter 4.6
Tuesday, March 26
Distributions: Fourier transforms
Reading: Strichartz, Chapter 4 and Osgood, Chapter 4.5
Thursday, March 28
Distributions: Convolution and solutions of differential equations
Reading: Strichartz, Chapter 5 and Osgood, Chapter 4.7
Tuesday, April 2
Fourier transforms in higher dimensions
Reading: Sections 6.1,6.2,6.4
Thursday, April 4
More Fourier transforms in higher dimensions, applications to PDEs.
Reading: Sections 6.1,6.2,6.4
Tuesday, April 9
Heat equation in higher dimension, wave equation in d=1
Reading: Section 6.3
Thursday, April 11
Wave equation in higher dimensions
Reading: Section 6.3
Tuesday, April 16
Fourier analysis on Z(N)
Reading: Section 7.1
Thursday, April 18
Fourier analysis for commutative groups
Reading: Section 7.2
Tuesday, April 23
Some number theory
Reading:
Thursday, April 25
Dirichlet's theorem
Reading: Chapter 8
Tuesday, April 30
Dirichlet's theorem
Reading: Chapter 8
Thursday, May 2
Review session
Exams
There will be a midterm exam, and a final exam. The final is
exam is tentatively scheduled for Thursday, May 16, 9am-noon.
Grading
Your final grade for the course will be roughly determined 25% by
assignments, 25% by the midterm, 50% by the final.
Textbook
Elias Stein and Rami Shakarchi
Fourier
Analysis: An Introduction
Princeton University Press, 2003
For errata in this book, see here
and here.
Office Hours
I should be available after class 11:30-12:30 in my office (Math
421). 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 Assistants
The TAs for the course and their office hours in the Math 406 help
room are:
Amy Lee, al3393@columbia.edu (Wednesday, 4-6)
Josh Zhou, zz2397@columbia.edu (Thursday, 4-6)
Other Books and Online
Resources
Besides the course textbook, some other textbooks at a similar level
that you might find useful are
Osgood, Lectures on the Fourier Transform and its Applications
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