Math 172 Homepage, Winter 2014-2015

Lebesgue integration and Fourier analysis

Instructor: András Vasy

Office: 383M

Phone: 723-2226

E-mail: andras "at"

Tentative office hours: MW 3:15-3:45, Th2-3, TW 10:30-11:30.

No office hour on Monday-Wednesday, March 9-11.

On Thursday, March 12, office hour is extended to 1:30-4pm.

Course Assistant: Chao Li

E-mail: rchlch "at"

Office: 381B

Office hours: M4-6, T6-9, F4-5.

Class location: MWF 2:15-3:05 pm, 380-380D. Due to an emergency, the last lecture of the quarter, on Friday, March 13, will be given by another instructor.

There were two make-up classes, Thursday, Feb 12, and Tuesday, March 3, to replace the March 9, 11 classes (when there will be no lectures). These classes were both in GESB131 (Green Earth Sciences Building), 1:15-2:05pm.

Required textbook: Stein and Shakarchi: Real Analysis.

Recommended textbook: Stein and Shakarchi: Fourier Analysis

For topics covered in the recommended textbook, the instructor will provide his own lecture notes.

Lecture notes:

(All might be updated as the lectures progress.)

The running syllabus may change somewhat, but should give an indication of the scope and speed of the course.

This course is similar to 205A, but designed for undergraduate students, and for graduate students in other departments. It also includes basic Fourier analysis. It is the continuation of the honors analysis course 171, emphasizing rigorous (i.e. logically careful) proofs, in the spirit of 171.

Grading policy: The grade will be based on the weekly homework (25%), on the in-class (expected in the usual classroom, at the usual class time) midterm exam (30%) and on the in-class (but of course not in the usual classroom, or at usual class time) final exam (45%).

The final exam is on Monday, March 16, 12:15-3:15pm. It will be supervised by Prof. Soundararajan.

A practice exam with Solutions is available.

I suggest that you first study for the final, and only then attempt to solve the problems on the practice exam. I suggest that you try to write the practice exam in timed conditions so that it is more similar to the actual final exam.

The exam covers all of the measure theory material, Chapter 1-2 of the text, as well as the Fourier analysis, as in the 5 handouts available for the course web page. There will be little emphasis on the last topic, distributions, but even when not explicitly asked, thinking about them might help your understanding of the material in a way relevant to the exam (for instance, it places the Fourier transform on L^2 into a better context). There will be an emphasis on Fubini and Tonelli since these were not covered on the midterm. In particular, you should always use careful arguments to check the hypotheses of Fubini's theorem; this often involves a use of Tonelli's theorem (checking the hypotheses again).

The midterm is on Friday, February 6, in 380D, 2:15-3:30pm. Please come a few minutes early so that we can start on time.

Solutions are now available!

It is a closed book, closed notes, no calculators/computers, etc. exam.

A practice exam is available, as are Solutions.

In the additivity part of the original version of 2(i) solutions, <= should have been = (typo).

Recommendations: please read through the topics covered in the textbook (Chapters 1 and 2, except Fubini's theorem, Section 2.3), and your course notes, and make sure you know how to solve the homework problems. In the exam, the instructions will state: "You may quote any theorem from the textbook or the lecture provided that you are not explicitly asked to prove it, and provided you state the theorem precisely and concisely (make sure to check the hypotheses when you quote a theorem)".

There will be at least one problem in which you will be asked to state a definition or a theorem in the first part (and then solve some problem related to it in the second half), so make sure you know all the definitions (exterior (Lebesgue) measure, measurability, Lebesgue measure, sigma algebra, measurable functions, simple functions, integral under various hypotheses, definition of L1 etc.) and major theorems (such as countable additivity of the measure, approximability of measurable sets by open, closed, compact sets and finite unions of rectangles, sigma algebra properties of measurable sets, properties of measurable functions under algebraic operations and limits, infs, sups, bounded convergence, monotone convergence, dominated convergence theorems, Fatou's lemma, completeness of L1, density in L1, etc.). Most exam problems will be similar to homework problems.

The homework will be due either in class or by 9pm in the instructor's mailbox on the designated day, usually Wednesdays. You are allowed to discuss the homework with others in the class, but you must write up your homework solution by yourself. Thus, you should understand the solution, and be able to reproduce it yourself. This ensures that, apart from satisfying a requirement for this class, you can solve the similar problems that are likely to arise on the exams.

Problem Sets

The problem numbers refer to those in the textbook.