Mathematics 133 – Winter 2004

Introduction to Fourier Analysis

Instructor: Terence Tao, MS 5622, ph. 206-4844 (  Note: I get a lot of spam e-mails these days, so please mark your e-mails with "Math 133" or something similar in the subject line, or send the e-mails from an UCLA address, otherwise it may get lost in the spam.

Lectures: MWF 9-9:50am, at MS 6221

Section: Th 9-9:50 am, at MS 6221

Office hours: TBA

TA: Monica Visan, MS 6147,

TA Office hours: TBA

Textbook: Stein and Shakarchi, Princeton Lectures in Analysis, Book I: Fourier Analysis and Partial Differential Equations, Princeton University Press.   The topics to be covered can be found at  We will follow the textbook closely; it is strongly recommended that you read this text concurrently with the course (and not just when it is time to do homework!).  On occasion I may supplement the textbook with some additional handouts on my own; these will be available on the class web page.

Homework: Homework will be due on Thursdays (in the TA mailbox or handed to the TA directly) and returned every Thursday in section meetings, starting on Thursday, Jan 22; there will be seven assignments. Each homework will consist of about ten problems of varying difficulty, about half of which will be from the textbook. The exact questions will be available on the Web, and also handed out in lectures. Only three of the questions, chosen at random, will actually be graded, however it is strongly recommended that you attempt all the questions in the assignment.  These questions are a mix of theoretical and computational problems, with perhaps a slight bias towards the theoretical side. You may use whatever resources you wish to do the homework, including calculators, textbooks, friends, TAs, etc.  However, you should not just be content with copying down someone else's answer to a homework question; it is important that you understand why that answer is correct, and how you would go about it if you had to do the problem on your own.  Indeed, if you cannot do this, you may have severe problems with the midterms and finals, so when reviewing for those exams it is important to be sure that you can do most of the homework problems without external assistance.

Solutions to selected questions from each homework may be available on the Web after the due date. Late homework will not be accepted.  The lowest score of your eight assignments will be automatically dropped

Java quiz:  A Java-based multiple choice quiz is available on the class web page.  This applet is designed to test your basic knowledge of concepts, both on course material and on more foundational material (such as logic, functions, the real numbers, set theory, etc); questions will be added to this quiz as the course progresses.  This quiz is anonymous, and has no effect on your grade.  However, it should be a useful tool for discovering any weak spots in your knowledge of course material.   Please let me know if there are any technical difficulties in loading or running the applet.

Examinations: There is one mid-term on Wed Feb 11, 9-9:50, at MS 6221, and a final on Wed Mar 24 3pm-6pm (exam code 02), at a room to be announced.   You may choose an optional "nickname" to go on your exams; this nickname will then be used when the exam scores are posted (otherwise, your scores will be anonymous).   More information about these exams will be given later in the course.

Grading: The final grade is based on the homework (20%, with the lowest homework score dropped), one midterm (40%), and the final examination (40%).  Make-up exams for the midterm and final will be available, but be warned that the questions on the make-up exam may be slightly more difficult than the ones given on the actual mid-term, especially if the make-up occurs after the actual mid-term. 

If you cannot make one of the examinations, contact me as soon as possible, preferably one week in advance of the exam. Retroactive, or last-minute requests for a make-up, will most likely be denied. Requests to reweight the exam grading after a bad midterm score will only be granted on cases of genuine emergency (e.g. medical emergency).

Calculators and written materials: The exams will be closed book.  However, you may bring one A4 size piece of paper to each exam.  You may write on this paper whatever you please, but please note that merely microfilming the book onto this A4 piece of paper may not be as useful to you as you might think; instead, I recommend using the opportunity to organize and condense your notes and thoughts on the subject in a succinct and clear manner.  The examinations will be a mix of theory and computational questions, but you will not be asked to regurgitate a proof of a theorem given in lecture or in the textbook.  Since the level of arithmetic computation will rarely rise above the level of manipulation of single-digit numbers, no calculators will be allowed.

Mathematical level: This upper-division course is of intermediate difficulty; it does contain some difficult concepts and theory, but also has a large computational component.  The theoretical material consists primarily of understanding various concepts (e.g. convergence, continuity, square-integrability) and recognizing that some results in Fourier analysis only work under certain assumptions.  There will be the occasional proof-type question, but this will not be a main focus of this course (unlike, say, Math 131, or to a lesser extent Math 115).

Prerequisites: This course assumes that you have taken Math 33A and 33B and are comfortable with the concepts in those courses.  Experience with other upper division Math courses, notably Math 115, Math 131, Math 132, Math 135, and Math 136, will be helpful (as there is some overlap in concepts) but is not essential, as we will review any such material which comes up in the course.  In your homework you are free to use any results from any of your courses, including those from Physics or Engineering courses; bear in mind, though, we will be grading your work based on mathematical standards of rigor, which means that you should only use an identity or other result if you can verify all the hypotheses needed for that identity to be valid (for instance, you cannot apply a convergence result on Fourier series to a discontinuous function such as a square wave if that convergence result required the function to be continuous).

World-Wide Web: You are encouraged to visit the web-page for this section at

This page will contain all the official information for the course, the latest homework, lecture notes, handouts, Virtual Office Hours, solutions to previous homework, sample exams and quizzes, updates, and other pieces of information.