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.

Wednesday, March 11
Theta and zeta functions
Reading: Section 5.3
Lecture notes

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
Lecture notes

Wednesday, April 1
Distributions: differentiation
Reading: Strichartz, Chapter 2 and Osgood, Chapter 4.6
Lecture notes

Monday, April 6
Distributions: Fourier transforms
Reading: Strichartz, Chapter 4 and Osgood, Chapter 4.5
Lecture notes

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
Final Exam 9am-noon

Exams

There will be a midterm exam, and a final exam.  The final is exam is tentatively scheduled for Monday May 11, 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.

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