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

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

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

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.

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.

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,

Strichartz,

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