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

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

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

Your final grade for the course will be roughly determined 25% by assignments, 25% by the midterm, 50% by the final.

Fourier Analysis: An Introduction

Princeton University Press, 2003

For errata in this book, see here and here.

For distributions, you should look at

Osgood,

https://ebookcentral.proquest.com/lib/columbia/detail.action?docID=5683572

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.

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,

Osgood,

Walker,

Tolstov,

Folland,

Körner,

Brown and Churchill,

Dym and McKean,

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