Tuesday and Thursday 1:10-2:25

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. 24):

Chapter 1, Exercises 3,4,5

Chapter 2, Exercises 2,4,6

Assignment 2 (due Tuesday, Jan. 31):

Chapter 2, Problem 2a

Chapter 3, Exercises 2,8,9,12 and Problem 2

Assignment 3 (due Tuesday, Feb. 7):

Chapter 2, Exercises 10,12,15

Chapter 3, Exercises

Assignment 4 (due Tuesday, Feb. 21):

Chapter 1, Exercise 10

Chapter 2, Exercises 13,17,18,19,20

Chapter 4, Exercises 11,12,13

Assignment 5 (due Tuesday, Feb. 28):

Chapter 5, Exercises 2,6,12, Problems 1,7

Assuming Fourier inversion for f(0), prove Fourier inversion for f(x)

Assignment 6 (due Tuesday, March 21):

Chapter 3, Problems 4,5

Chapter 5, Exercises 15,17,18,19,23

Assignment 7 (due Tuesday, March 28):

Strichartz, Chapter 1 Problems 3,4,6,7

Strichartz, Chapter 2 Problems 4,6,13,16

Assignment 8 (due Tuesday, April 4):

Strichartz, Chapter 3 Problems 7,16

Strichartz, Chapter 4 Problems 1,6,9,10,11,12

Assignment 9 (due Tuesday, April 11):

Chapter 6, Exercises 1,4,5,6,7,8

Assignment 10 (due Tuesday, April 18):

Chapter 6, Exercises 7,8,10,11

Chapter 6, Problem 7

Assignment 11 (due at final exam):

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.

Tuesday, January 17:

Overview of the course. Review of complex numbers and Euler's formula. Definition of Fourier series.

Reading: Chapter 1 (for motivation, the topics of this chapter will be treated in detail later in the course).

Thursday, January 19:

Examples of Fourier series. Convolution.

Reading: Sections 2.1 and 2.3.

Tuesday, January 24:

Mean convergence of Fourier series, Parseval's equality.

Reading: Sections 3.1

Thursday, January 26:

Proof of mean convergence.

Reading: Sections 3.1 and 3.2.1

Tuesday, January 31:

Pointwise convergence, Cesaro summability, Fejer kernel

Reading: Sections 3.2.1, 2.2, 2.4, 2.5.1, 2.5.2

Thursday, February 2:

More about pointwise convergence, discontinuous functions

Poisson kernel, Abel summability

Reading: Sections 2.5.2, 2.5.3, 2.5.4

Tuesday, February 7

Harmonic functions, Dirichlet problem

Reading: Sections 1.2.2, 2.5.4

Thursday, February 9

Snow day

Tuesday, February 14

Heat equation, Schrodinger equation

Reading: Sections 4.4

Thursday, February 16

Introduction to the Fourier transform

Reading: Introduction to Chapter 5

Tuesday, February 21

Properties of the Fourier transform, Fourier inversion

Reading: Section 5.1-5.1.5

Thursday, February 23

Plancherel theorem, Heat equation

Reading: Section 5.1.6, 5.2.1

Tuesday, February 28

Harmonic functions in the upper half plane, Heisenberg uncertainty, Review

Reading: Sections 5.2.2, 5.4

Thursday, March 2

Midterm exam

Tuesday, March 7

Poisson summation formula

Reading: Section 5.3

Thursday, March 9

Theta and zeta functions

Reading: Section 5.3

Tuesday, March 21

Distributions: definitions and examples

Reading: Strichartz, Chapter 1

Thursday, March 23

Distributions: differentiation

Reading: Strichartz, Chapter 2

Tuesday, March 28

Distributions: Fourier transforms

Reading: Strichartz, Chapter 4

Thursday, March 30

Distributional solutions of differential equations

Reading: Strichartz, Chapter 5

Tuesday, April 4

Fourier transforms in higher dimensions

Reading: Sections 6.1,6.2,6.4

Thursday, April 6

More Fourier transforms in higher dimensions, applications to PDEs.

Reading: Sections 6.1,6.2,6.4

Tuesday, April 11

Heat equation in higher dimension, wave equation in d=1

Reading: Section 6.3

Thursday, April 13

Wave equation in higher dimensions

Reading: Section 6.3

Tuesday, April 18

Green's functions.

Reading: notes available here.

Thursday, April 20

Fourier analysis on Z(N)

Reading: Section 7.1

Tuesday, April 25

Fourier analysis for commutative groups

Reading: Section 7.2

Thursday, April 27

Gauss sums and quadratic reciprocity

Reading: Quadratic Reciprocity via Theta Functions, Ram Murty and Pacelli

Tuesday, 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

I should be available after class 2:30-4 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.

Other Books and Online Resources

Besides the course textbook, some other textbooks at a similar level that you might find useful are

Walker,

Tolstov,

Folland,

Körner,

Brown and Churchill,

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