Office: 383M

Phone: 723-2226

E-mail: andras "at" math.stanford.edu

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

Course Assistant: Chao Li

E-mail: rchlch "at" math.stanford.edu

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.

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.

- Motivation for Fourier series: Separation of variables.
- Inner product spaces, symmetric operators, orthogonality.
- Convergence of the Fourier series.
- The Fourier transform -- basic properties and the inversion formula.
- Tempered distributions and the Fourier transform.

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 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.

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 Set 1, due Wednesday, January 14: Exercises 1,2,5,6,7,11,14. If you want to get started on Problem Set 2, Exercises 16,26, Problem 1 will be on it and are just a bit beyond where the problems on the first problem set lie in terms of material covered. Solutions thanks to Chao and last year's CA, Jeremy!
- Problem Set 2, due Wednesday, January 21: Ch 1: Exercises 8,15,16,21,22,25,26,28, Problem 1,5. Ex. 17 is postponed to PS #3. Solutions thanks to Chao and last year's CA, Jeremy!
- Problem Set 3, due Friday, January 30 in class (2:15pm): Ch 1: Exercises 4,17,32,33,34,35 Ch. 2: Exercises 3,6,9,10,16. Solutions thanks to Chao and last year's CA, Jeremy!
- Problem Set 4, due Wednesday, February 4: Ch. 2: Exercises 2,8,11,12,15, Problem 3. Exercises 4,19 and Problem 4 are postponed to PS #5. Solutions thanks to Chao and last year's CA, Jeremy!
- Problem Set 5, due Friday, February 13, in class, at 2:15pm. Solutions thanks to Chao and last year's CA, Jeremy!
- Problem Set 6, extended and expanded to Wednesday, February 25 (originally was due Friday, February 20, in class, at 2:15pm). Solutions thanks to Chao and last year's CA, Jeremy!
- Problem Set 7, due Wednesday, March 4. Solutions thanks to Chao and last year's CA, Jeremy!
- Problem Set 8, due Friday, March 13, in class, at 2:15pm. Solutions thanks to Chao and last year's CA, Jeremy!