Course instructor: Edmund Y.-M. CHIANG
Lecture hours: Monday 15:00--16:20 (room  4504) and Friday 10:30--11:50.
Email: machiang@ust.hk

Overview: This is an introductory course of Fourier analysis and its applications in solving some linear partial differential equations, such as the heat equation, wave equation, Laplace equation etc. Besides, the main topics will cover Fourier series, orthogonal systems, Fourier convergence theorems, boundary value problems, special functions, Fourier transforms, and other topics when time allows.

Prerequisite: We assume some knowledge of basic mathematical analysis, multivariable calculus (mostly partial derivatives). Previous knowledge on differential equations and/or complex analysis will be an advantange but students who have not done these courses may still take this course subject to the approval of the course instructor.

Main reference books:

1. G. P. Tolstov, ``Fourier Series", Dover publication, 1976
2. G. B. Folland, ``Fourier Analysis and its Applications", Brooks/Cole Publishing Company, 1992. Republished by American Mathematical Society.
3. D. M. Bressoud, ``A Radical Approach to Real Analysis", 2nd Ed., Mathematical Association of America, Washington, DC, 2007

Distribution of marks (tentative): Homework (25%), Presentation (25%) and Final examinaiton (50%).

Lecture notes: Will be uploaded roughly every week. The notes will be mainly based on Tolstov and Folland's books.

   Chapter 1 Introduction (2nd February 2015)
   Chapter 2 Convergence (5th February 2015) Convergence (9th February 2015: corrected numbering)
   Chapter 3 Fourier Series part I and II (13th February 2015)
   Chapter 4 Orthogonal systems (16th, 23rd Februday 2015)
   Chaper 5 Convergence of Fourier series part I (23rd February 2015), Part II (26th February 2015)
   Chapter 6 Properties of Fourier series Part I (2nd March 2015) Part II (5th, 9th March 2015)
   Chapter 7 Summability of Fourier series Part I (9th March 2015), Part II (13th March 2015)
   Chapter 8 Eigenfunction Method and its Applications to PDEs Part I  (16th March 2015) Part II (19th March 2015) Part III (23rd March 2015), Part IV (27th March), Part V (09th April)
   Chapter 9 Fourier-Bessel series Part I (13th April) Part II (17th April) Part III (20th April)
   Chapter 10 Fourier transforms (Part I), Part II (26th April) Part III (4th May) Part IV (9th May)


Facebook: You may find more information about Fourier from a FB page under the title Fourier math4822e

Remarks: Fourier analysis tools are used extensively in, but not limited to, physcial sciences and engineering subjects. It also plays important roles in pure mathematics research. In particular, the subject matter put many mathematical analysis results learned in earlier undergraduate years in concrete forms. Thus this is an ideal course to reinforce/clarify basic mathematical analysis theories students learned in previous analysis courses. Apart from a few basic reference books, there will be a set of acompany lecture notes provided that serve as learning material. The grade of this course will be based on homework and a final examination. Presentation may also be included depending on the number of students enrolled in the course.