# MAT 180-002 Fourier Analysis and Its Applications Spring Quarter, 2000

CRN: 74513
Class: MF 10:00am-11:20am, 693 Kerr

Instructor: Naoki Saito
Office: 675 Kerr
Email: saito@math.ucdavis.edu
Office Hours: MW 1:15pm-2:15pm or by appointment via email

Course Description:
The Fourier analysis was originally developed to solve a particular PDE, namely, the heat equation, about 200 years ago. However, over the years, the 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 generalized to various different forms. Its philosophy is still the same: analyze a function by decomposing it into a linear combination of elementary building blocks (e.g., sines and cosines). In this course, first I will discuss the basics of the Fourier series and integrals in an intuitive manner. Then, I would like to discuss their applications, in particular, approximation theory, signal analysis, probability, statistics, and computational applications. Topics will be chosen from the following list:
• Gibbs's phenomena
• Trigonometric approximations
• Chebychev polynomial and the best approximations
• Random walks and Brownian motions
• The central limit theorem
• Shannon's sampling theorem and discretization of signals and functions
• Heisenberg's uncertainty principle
• Discrete Fourier Transforms
• Fast Fourier Transforms
• Gaussian quadratures for numerical integrations
• Spherical harmonics
Prerequisite:
Students who want to enroll this course should have:
• Strong scientific curiosity.
• Basic understanding of linear algebra, such as MAT 22A, 167, or equivalent.
• Basic understanding of undergraduate analysis, such as MAT 121AB, 127ABC, or equivalent.
• This course is designed for undergraduate students, but the graduate students of all science and engineering disciplines are also very welcome! (In particular, those who want to clarify the idea of the Fourier transforms, or felt strange about negative frequencies.)
Text:
There are many good textbooks in Fourier Analysis. I will list several of them with comments. The first two are available at UCD bookstore. They are all optional books, not mandatory to buy.
• H. Dym and H. P. McKean: Fourier Series and Integrals, Academic Press, 1972
This book contains numerous applications of Fourier analysis. Strongly recommended for anyone who is interested in applications and wants to deepen their understanding of Fourier analysis. It also includes a nice description of Lebesgue integration and group theory.
• T. W. Körner: Fourier Analysis, Cambridge University Press, 1988
This is a monumental work on Fourier analysis, consisting of a bunch of interrelated essays. Read one section per day! You will gain a lot. Highly recommended.
• J. S. Walker: Fourier Analysis, Oxford University Press, 1988
A well-written and solid book on Fourier analysis with applications on optics, computer-aided tomography, spherical harmonics, etc.
• G. B. Folland: Fourier Analysis and Its Applications, Brooks/Cole Publishing Co., 1992
An introductory but extremely well-written textbook on Fourier analysis. Contains chapters on special functions, generalized functions (distributions), and Greens functions. Applications are mainly for differential equations. Expensive but worth buying it.
• J. M. Ash (ed.): Studies in Harmonic Analysis, Mathematical Association of America, 1976
This is a collection of conference talks by the authorities held in Chicago in 1975. Most of the chapters are as if these authorities are directly talking to you in a friendly manner about the essence of the ideas in harmonic analysis without much detailed proofs. Contains really deep mathematics.
• S. G. Krantz: A Panorama of Harmonic Analysis, Mathematical Association of America, 1999
This book gives a historical perspective of harmonic analysis ranging from classical to modern, from elementary to advanced. One can see how subtle it is to sum multiple Fourier series. This also includes short description on wavelets. Highly recommended.
• E. M. Stein and G. Weiss: Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971
A classic of the multidimensional Fourier analysis. Includes detailed discussions on the invariance properties of Fourier transform.
• A. Zygmund: Trigonometric Series (2nd Ed., Volume I & II combined), Cambridge University Press, 1959
An ultimate bible on Fourier series and integrals for hard analysts. This is basically a dictionary. Almost no applications are treated here.
• R. N. Bracewell: The Fourier Transform and Its Applications (2nd Ed., Revised), MacGraw-Hill, 1986
Another bible for engineers. Contains an excellent pictorial dictionary of many functions and their Fourier transforms.
• G. P. Tolstov: Fourier Series, Dover, 1972.
The most cost effective book (about \$12). Very well written. Highly recommended.
• G. H. Hardy and W. W. Rogosinski: Fourier Series, Dover, 1999.
This is a prelude to Zygmund's book. Spirit of pure mathematics. No applications included. Economical (\$7).
• W. L. Briggs and V. E. Henson: The DFT: An Owner's Manual for the Discrete Fourier Transform, SIAM 1995
This is a very useful book on DFT. Includes many practical applications, such as tomography, seismic migrations, difference equation solvers. Detailed analysis on the error of the DFT. A nice book to keep on your desk.
• A. Terras: Fourier Analysis on Finite Groups and Applications, Cambridge University Press, 1999.
Another type of Fourier analysis. A more detailed version of the first half of Chapter 4 of Dym and McKean plus many more examples and applications of that aspect of Fourier analysis.
Attendance:
• Regular attendance to the lectures is strongly advised. I will check the attendance at every class since there is no homework in this course.