Course: MAT 129-001: Fourier Analysis
CRN: 69704
Class: MWF 11:00am-11:50am, GIEDT 1006

Instructor: Naoki Saito
Office: 2142 Math. Sci. Bldg.
Email: saito@math.ucdavis.edu
Office Hours: MW 12:30pm-1:30pm or by appointment

TA: Wenjing Liao
Office: 3202 Math. Sci. Bldg.
Email: wjliao@math.ucdavis.edu
Office Hours: T 1:50pm-2:50pm; R 1:30pm-2:30pm; or by appointment

Students in math, science, and engineering departments who want to understand and learn the Fourier techniques and their vast applications in science and technology.

Fourier analysis was originally developed about 200 years ago to solve a particular PDE, the heat equation. However, over the years Fourier analysis has been shown to be an indispensable tool not only for mathematics but also for many different fields of science and technology, and has been generalized to various different forms. Its philosophy is still the same: analyze a function or data by decomposing them into a linear combination of elementary building blocks (e.g., sines and cosines) and how the original function or data can be synthesized from such elements. In this course, we will first discuss the basics of Fourier series, Fourier integrals, and orthogonal sets of functions in an intuitive manner. Then, we will discuss their applications selected from a variety of fields such as approximation theory, signal analysis, probability, statistics, and differential equations.

• MAT 21D (basic understanding of vector calculus)
• MAT 22A or 67 (basic understanding of linear algebra)
• MAT 22B (basic understanding of ordinary differential equations)
• MAT 25 or consent of instructor (basic understanding of introductory analysis)

• Required Text: G. B. Folland: Fourier Analysis and Its Applications, Amer. Math. Soc., 1992. Errata can be downloaded from http://www.math.washington.edu/~folland/Homepage/index.html (look for the appropriate printings of your textbook).
• Optional Texts/References:
• H. Dym and H. P. McKean: Fourier Series and Integrals, Academic Press, 1972.
• T. W. Körner: Fourier Analysis, Cambridge Univ. Press, 1988.
• E. M. Stein and R. Shakarchi: Fourier Analysis, Princeton Univ. Press, 2003.
• J. S. Walker: Fourier Analysis, Oxford Univ. Press, 1988.

We will cover the following sections of the textbook:
• 1st week: Chap.1, Sec.2.1, 2.2: Fourier series of a periodic function; A convergence theorem
• 2nd week: Sec.2.3, 2.4, 2.6: Derivatives, integrals, and uniform convergence; Fourier series on intervals; Remarks including the Gibbs phenomenon
• 3rd week: Sec.3.1-3.3: Orthogonal sets of functions;Inner products; Convergence and completeness
• 4th Week: Sec.3.4, 3.5: L² space; Regular Sturm-Liouville problems
• 5th Week: Sec.4.1-4.3: Some boundary value problems; 1D heat flow and wave motion
• 6th Week: Sec.4.4, 4.5: The Dirichlet problem; Multiple Fourier series;
• 7th Week: Sec.7.1,7.2: The Fourier transform; Convolution
• 8th Week: Sec.7.3: Applications of Fourier transforms
• 9th Week: Sec.7.5; Other applications, The Fourier transform of several variables; Various applications
• 10th Week: Other applications; will choose from Sec. 2.5, 6.1-6.2; Sec.6.6; Sec.7.4; Sec.7.6; Sec.8.1-8.3, i.e., Fourier series and boundary value problems; Orthogonal Polynomials; Haar and Walsh functions; Fourier transforms and Sturm-Liouville problems; Laplace transform and its inversion, etc.

Formal attendance will not be taken. However, this is a small class, and your attendance (or lack thereof) will be noted as the quarter progresses. Whether you are able to attend class or not, you are responsible for all material presented in class as well as any work that may be due. LATE HOMEWORK IS NEVER ACCEPTED. While I will try to post class announcements via email or on the class web pages, it is your responsibility to find out what happened if you miss class.

I will maintain the Web pages for this course (one of which you are looking at now). All homework assignments and important announcements will be posted on these pages. Please check these pages regularly.
You can access the MAT 129 home page at https://www.math.ucdavis.edu/~saito/courses/129.s10/ from which you can access to this syllabus page and the homework page.

The MAT 129 Mailing List was created. I will use this list to announce some important information. You can also submit your emails (must be related to the class) to this mailing list. If you send your email to this list, then everyone will receive it. So, please use this wisely and politely. Once I had two students in the class who started discussing certain aspects of the exercise problem and sent back and forth about 10 emails to the mailing list in a few hours, and everyone else got fed up... The mailing list name is: mat129-s10@smartsite.ucdavis.edu .

• 30% Homework
• 25% Midterm Exam (in class, tentatively scheduled for Friday, May 7, 2010)
• 45% Final Exam (8:00am-10:00am, Saturday, June 5, 2010)

I will assign homework problems after each lecture, which can be seen at Homework Page. The due date of each homework set is one week after the assigned date (except around Thanksgiving holidays; see the Homework Page above for the details). So, I will collect the homework in the beginning of each lecture. LATE HOMEWORK WILL NOT BE ACCEPTED. All homework must be neat, accurate, and legible. You must show your work, and you are encouraged to write in complete sentences. The reader has explicit instructions to penalize you severely if your work cannot be followed. Getting the correct answer is only part of the problem. You must show work that is legible and logical. A subset of these problems will be graded, and returned on the following Friday at the end of class. I will not include the score of the worst performed homework when computing your grade.
Note: This is a 4 unit course! In practical terms, that means you are expected to work 3 hours at home for each hour of lecture. In other words, expect to have 9 to 10 hours of homework each week.

There will be one midterm and a final examination. The midterm is tentatively scheduled for Friday, May 7 in class. The coverage of the midterm is the sections from Chap.1 to Chap.4 covered by the lectures. The final exam will be 8:00am-10:00am, Saturday, June 5, 2010 at GIEDT 1006. The coverage of the final exam is cumulative. Also, be sure to note the following policies:
• All exams are closed book. You may not use the textbook, crib sheets, notes, or any other outside material. Do not bring your own scratch paper. Do not bring blue books.
• You are not allowed to use calculators/laptop computers/cell phones in the exam. The exam is to test whether you know the material.
• Everyone works on their own exams. Any suspicions of collaboration, copying, or otherwise violating the Student Code of Conduct will be forwarded to the Student Judicial Board.
• The answer is yes: the final exam is cumulative, i.e., it covers the whole course material.
• There will be NO MAKE-UP MIDTERM EXAM. If you miss the midterm exam due to catastrophic events such as serious illness of yourself or death of your immediate family, you must provide me with a written proof (e.g., a report or a letter written by a medical doctor with signature). Only then I will readjust the weight (i.e., Homework 40%; Final 60%).
• If you miss the final exam due to catastrophic events such as serious illness of yourself or death of your immediate family, you will receive "Incomplete" grade, provided that you give me a written proof (e.g., a report or a letter written by a medical doctor with signature). Then you must take a make-up exam in the following quarter to receive a letter grade.