Introduction to Fourier Analysis

**Instructor: **Terence
Tao,
MS 5622, ph. 206-4844 (tao@math.ucla.edu)
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

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

**Office hours: **TBA

**TA: **Monica Visan,
MS
6147, mvisan@math.ucla.edu

**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 http://www.math.ucla.edu/~tao/resource/general/133.1.04w/schedule.html.
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

http://www.math.ucla.edu/~tao/resource/general/133.1.04w

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.