18.103 Fall 2009

Fourier Analysis -- Theory and Applications

General Information

Time : MWF 10-11 am
Place : 2-102
Instructor : Ju-Lee Kim
Office : 2-281
Phone : 253-7576
e-mail: julee at math.mit.edu
Office hours: MW 11:00am-12:00pm or by appointment

TAs

Textbook

M. Adams and V. Guillemin, Measure Theory and Probability, Boston, Springer-Verlag, 1996

We will basically go through the above textbook. However, we will not be able to go over all the details of the textbook. Each lecture will cover important points of the material and you are responsible for reading the details.

More Reference

E. M. Stein and S. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003.

Course Outline

Measure theory, Lebesgue measures, Lebesgue integrals, convergence theorems, Hilbert space, Fourier analysis, applications to probability.

Prerequisite

18.100A or 18.100B

Homeworks

Weekly HW will be due each Friday 11am. Most problems in HWs will be from the abovementioned textbook.
The lowest grade HW will be dropped. However, no late HW will be accepted .
I would not require to type your HW. However, your handwriting should be legible, it should be stapled and the edges of the HW paper should be straight.
You are encouraged to work together on the problem sets, but you MUST write up your own solutions.
If you do work with others in the class, please list them somewhere at the cover of your homework.

Exams

There will be four one-hour in-class exams (scheduled on Oct 2, Oct 30, Nov 23, Dec 4). There will be no final exam. All exams are cumulative, but will have more emphasis on the topics not covered by previous exams. The exams will be closed book and no calculators are allowed.

Policy on make-up exams

Only early make-up exams are allowed. There will not be any make-up exam after the actual one. In case that you need to take a make-up exam, you should inform the instructor as soon as possible.

Grades

HW: 100 pts
Each exam: 100 pts

Online Gradebook

Your grades are available at Stellar. The Stellar website for this course is maintained only for grading purpose.

Schedule and announcements

Week	Lecture	Practice problems	Announcement
1. Sept 9-11	Introduction and Lebesgue measures	Section 1.1 #1,4,5,6,10,15,16,17,21	HW1 Section 1.1 #1,4,5,6,10, Section 1.3, #1,4,5 (8 problems) (due Sept 18) HW 1 Solution
2. Sept 14-18	Measure Theory	Section 1.3 #1,2,3,5,10,11,12,14,15,19	HW2 Section 1.1 #12, Section 1.3, #2,3,11,12,14,19,20 (8 problems) (due Sept 25) HW 2 Solution
3. Sept 21-25	Applications to Probability	Section 1.4: 1,6,7,8,10,11,15,18,19	Extra Credit HW Section 1.4 #17 (due Oct 2)
4. Sept 28-Oct 2	Measurable functions and Exam 1	Section 2.1. #1,3,4,8,10	Exam 1 on Oct 2 Exam 1 Solution (Ave: 74, A>80 B>60 C<60) HW3 Sec 2.1. #1,2,8,9,10,11, Sec 2.2. #1,3(due Oct 9) HW 3 Solution
5. Oct 5-Oct 9	The Lebesgue Integral	Section 2.2. #1,3,6,7,8,10	Add date: Oct 9 HW4 Sec 2.2. #6,7,8,10, Sec 2.3. #2,3,5,9 (due Oct 16) HW 4 Solution
6. Oct 12-Oct 16	Convergence theorems	Section 2.3 #2,3,5,9,12,13,14,15,16, Section 2.4 #2,5,7	Monday class is held on Tuesday. HW5 Section 2.5. #2,3,5,6,7,8,12,13(Due Oct 23) HW 5 Solution
7. Oct 19-Oct 23	Fubini Theorem	Section 2.5 #2,3,6,7,8,12,13	Exam 2 is open book. It covers up to Section 2.5.
8. Oct 26-Oct 30	Fourier Analysis, L^1 theory	Section 3.1. #1--9	Exam 2 on Oct 30 (Average: 62 out of 90) Exam 2 Solution HW6 Section 3.1. #1-9 (Due Nov 6, you may refer to handouts) HW 6 Solution
9. Nov 2-Nov 6	Hilbert space		HW7 Section 3.2. #4-10 (Due Nov 13, you may refer to handouts) HW 7 Solution
10. Nov 9-Nov 13	Fourier Series and Fourier Integrals	Section 3.3, #4,5,6,9,10, Section 3.4, #2,3,4,5,7,8,9	No class on Wednesday (Veteran's day). HW8 Section 3.3. #5,6,9 Section 3.4 #4,8,9(Due Nov 20)
11. Nov 16-Nov 20	Fourier integrals, Applications	Section 3.5, #1--9	Drop date:Wed Nov 18
12. Nov 23-Nov 27			Exam 3 on Nov 23. No class on Friday (Thanksgiving).
13. Nov 30-Dec 4			Exam 4 on Dec 2--4. This will be a take-home exam. You can pick up the exam during the class on Dec 2 and bring it back on Dec 4. There will be classes on both days.
14. Dec 7-Dec 9
15. Dec 14-18	Final Exam period