Fourier Analysis: Mathematics GU4032 (Spring 2017)

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

### Assignments

There will be assignments due roughly each week, due in class on Tuesday, mostly taken from the textbook.

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 12 (already done last week...),14,20

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

Syllabus

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.

Tuesday, January 24:
Mean convergence of Fourier series, Parseval's equality.

Thursday, January 26:
Proof of mean convergence.

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

Tuesday, February 7
Harmonic functions, Dirichlet problem

Thursday, February 9
Snow day

Tuesday, February 14
Heat equation, Schrodinger equation

Thursday, February 16
Introduction to the Fourier transform

Tuesday, February 21
Properties of the Fourier transform, Fourier inversion

Thursday, February 23
Plancherel theorem, Heat equation

Tuesday, February 28
Harmonic functions in the upper half plane, Heisenberg uncertainty, Review

Thursday, March 2
Midterm exam

Tuesday, March 7
Poisson summation formula

Thursday, March 9
Theta and zeta functions

Tuesday, March 21
Distributions: definitions and examples

Thursday, March 23
Distributions: differentiation

Tuesday, March 28
Distributions: Fourier transforms

Thursday, March 30
Distributional solutions of differential equations

Tuesday, April 4
Fourier transforms in higher dimensions

Thursday, April 6
More Fourier transforms in higher dimensions, applications to PDEs.

Tuesday, April 11
Heat equation in higher dimension, wave equation in d=1

Thursday, April 13
Wave equation in higher dimensions

Tuesday, April 18
Green's functions.

Thursday, April 20
Fourier analysis on Z(N)

Tuesday, April 25
Fourier analysis for commutative groups

Thursday, April 27

Tuesday, May 2
Review session

### Exams

There will be a midterm exam, and a final exam.  The midterm exam is tentatively scheduled for Thursday March 2.  The final is exam is tentatively scheduled for Tuesday May 9, 1:10 - 4 pm.

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

Office Hours

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, 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