MATH 322 Harmonic Analysis

Spring 2007 

Description

In 1807, the French mathematician and physicist Joseph Fourier submitted a paper on heat conduction to the Academy of Sciences of Paris. In this paper Fourier made the claim that any function can be expanded into an infinite sum of trigonometric functions. The paper was rejected after being read by some of the leading mathematicians of his day. They objected to the fact that Fourier had not presented much in the way of proof for this statement, and most of them did not believe it.

In spite of its less than glorious start, Fourier's paper was the impetus for major developments in mathematics and in the application of mathematics. His ideas forced mathematicians to come to grips with the definition of a function. This, together with other metamathematical questions, caused nineteenth-century mathematicians to rethink completely the foundations of their subject, and to put it on a more rigorous foundation. Fourier's ideas gave rise to a new part of mathematics, called harmonic analysis or Fourier analysis.

We will study harmonic analysis in three settings.

The applications of Fourier analysis outside of mathematics continue to multiply. We will present some of those. However, we will also discuss some of the applications to other parts of mathematics. These will include the isoperimetric inequality from geometry, Weyl's equidistribution theorem from dynamics, the Radon transform, and, if there is enough time, Dirichlet's theorem on the infinitude of primes in arithmetic sequences.

Staff

Instructor

John C. Polking
Office: HB 450. Office hours: 1:30 to 3:00 Wednesdays and Thursdays
Email: polking@rice.edu
Telephone: ext 4841
Teaching Assistant

Ryan Dunning
Office: HB 45. Office hours:
Email: rdunning@rice.edu
Telephone: ext 2785

Text and Supplementary Material

The text for this course is Fourier Analysis, By Elias M. Stein and Rami Shakarchi.

Fourier series and its applications to partial differential equations are discussed in Differential Equations with Boundary Values by Polking, Boggess, and Arnold. This treatment is at a more elementary level and lacks rigor, but you might find it interesting.

There is a very interesting treatment of harmonic analysis, its applications, and its history in Fourier Analysis by Thomas Körner.

Computer and Owlnet information

Computers will not be widely used in this course, but there is one program which you will find useful. It is called fseries and it runs under MATLAB. It is discussed here. The picture at the beginning of this page exemplifies what fseries can do.

 In order to use fseries you will need to download the file fseries.m.

Grading

Half of your final grade for the course will be determined by your performance on the homework. Working problems is the best way to learn the material in this course. In addition there will be either two projects, or a midterm and a final exam.

Homework

There will be a homework assignment each week. The lowest homework grade will not be counted in determining the final grade.

All homework is due in class on the date announced. This will typically be about a week after the assignment is posted. Each student will be allowed to have at most one late homework assignment during the semester. The one late homework will be accepted up to seven days after the due date, with or without excuse, and without penalty. No other late homeworks will be accepted even with an excuse. There will be absolutely no exceptions to these rules.

Many of the Exercises in the book extend the material in the text. Therefore they are as important as the text itself. Many homework assignments will contain more problems to be done than are to be turned in. It should be emphasized that a person learns mathematics by doing problems. You are encouraged to at least look at all of the exercises in the book.

A homework assignment is meant to convince the grader that you understand the material. The best way to do that is to use complete sentences and to organize your work in paragraphs. In your writing, attempt to achieve the same clarity you find in textbooks. The grader has instructions that if he cannot understand your writing, he is not to grade your paper.

The homework is not pledged. You are encouraged to discuss the homework, and to work together on the problems. However each student is responsible for the final preparation of his or her own homework papers.

Most mathematicians and scientists prepare their papers using the mathematical typesetting language TeX. It is not required, but you are encouraged to use TeX for your assignments. This is especially true for the project reports.

Homework Assignments and Projects:

John C. Polking <polking@rice.edu>

Last modified: Thursday, February 9, 2007