Math 172: Lebesgue Integration and Fourier Analysis

Syllabus and Course info

This course is being offered for the first time in Spring 2009. It is designed for students who enjoyed 171, and want a natural continuation. It also serves to prepare students who want to later take the 205 sequence, but feel the might not have the background currently. Finally, in future years a PDE course for math majors (173, replacing 131M) will be designed as a continuation of 172.

This course is similar to 205A, but designed for undergraduate Math majors and graduate students in other disciplines. Some topics covered will include Lebesgue measure and integration on Euclidean space, L^p spaces, the Fourier transform, the Hardy-Littlewood maximal function and Lebesgue differentiation.

The difficulty of this course will be somewhere between that of 171 and 205. Explicitly, difficulty(171) < difficulty(172) ≤ difficulty(175) < difficulty(205A). To emphasize, the first and last inequalities are strict. The middle one *might* not be.

The prerequisites are Math 171. Though strong 115 students, with additional preparation and a strong heart can take this class.

If you're contemplating taking 205A, then read this for a few words of advice.

Schedule info.

You can alternately view exam dates/times in Google calendar here

Email list and contact information

All registered students should be automatically subscribed to the students mailing list. If by error you are NOT subscribed to this list, then subscribe yourself to the at guests mailing list. Any email announcement sent by me will also be sent to this list.

Requests via email to subscribe / remove people from the class mailing lists will be ignored. You can remove yourself from these class mailing lists by following the link at the bottom of list messages.

Grading

Homework 30%, Midterm 30% and final 40%, or Homework 30%, final 70% whichever is a better grade.

Homework

Homework will be assigned every week on Tuesday, and due the following Tuesday in class, at the beginning of class. Late homework will NEVER be accepted, however to accommodate special circumstances your lowest homework score will not count towards your grade.

Note: Homework is probably the most important part of this course. Trying such problems on your own is the only way to get a good conceptual understanding of this material. You're encouraged to work in groups, however blind plagiarism is the easiest way to ensure miserable performance on exams (not to mention a violation of the Honor Code).

Some problems on the homework are supposed to be hard. Please do not feel discouraged if you can not do a few of the problems on the homework. You should think about these problems for as long as you can afford, and then solicit help from me/Ian. The only way to understand this material is by trying hard problems yourself, and sometimes a long concerted attempt (even if unsuccessful) can enhance your understanding better than anything else.

Optional problems: Your assignments will frequently contain optional problems. These problems are helpful to think about, but you should not turn them in with your regular homework. Problems are made optional for a variety of reasons: Some problems are optional because they are (easy?) standard facts which I did not have time to do in class. Others are optional because they are interesting `challenge' problems, which may or may not have a tractable solution in the scope of this course. You're welcome to discuss any optional problem with me / your TA, but don't turn it in with your homework. It won't be graded.

1. Page 1: Assignment 1
2. Page 2: Assignment 2
3. Page 3: Assignment 3
4. Page 4: Assignment 4
5. Page 5: Assignment 5
6. Page 6: Assignment 6
7. Page 7: Assignment 7
8. Page 8: Assignment 8
9. Page 9: Assignment 9 (last ever)

Handouts

1. Lecture schedule. A rough indication of what you will see in lectures. This will be updated as the quarter progresses.
2. Solutions. Perfect / nearly perfect solutions by students have been scanned in and put online here for the benefit of others. (If for some reason you would not like your solutions to be hosted here, drop me an email and I will ensure that your solutions are never scanned in.)
3. Your midterm, and solutions.
4. Optional problems. Looks like there's no place on the homework assignments to include optional problems anymore. So here are a list of facts, and 'fun' problems which I say/use in class but don't fully proves.
5. Hints for problems on 3(a) and (d) on HW7.
6. The Ergodic theorem and an application to continued fractions. I did not do all details of this in class, so wrote it up for your reference. This is optional material, and will not be on your final.
7. Your final, and solutions.

Textbook/References

1. Lebesgue Integration on Euclidean Space by Frank Jones. (This will be your text book, and the course will roughly follow it).
2. Real Analysis by H. L. Royden
3. Real and Complex Analysis by Walter Rudin. (This more towards the level of 205A.)
4. Probability and Measure by Patrick Billingsley. (This is a great supplement to this course, since your text doesn't really touch on probability. If you're inclined towards Economics, Finance, Statics I recommend skimming this as the quarter progresses.)

Feedback

Feedback at any time (either anonymous or signed) is always appreciated. You can use this form to send me (or your course assistant) anonymous (or signed) feedback. [Note: Evil spammers have been using this form to clutter my INBOX. Thus I have restricted access to this form to within stanford.edu domain.]