M354 Fourier Analysis and its Applications
Winter 2019
Instructor:  Dr. Shixu Meng 
Office:  East Hall 4827 
When:  MoWeFr 10:00AM – 11:00AM 
Where:  4096 East Hall 
Office Hours:  MoWeFr 11:00AM – 12:00PM 
Course Description: This is an introduction to Fourier Analysis geared towards advanced undergraduate students from both pure and applied areas. It should be particular ly suitable for majors in the sciences and engineering. Topics will include properties of complex numbers, Fourier Series, the Dirichlet kernels, approximations by trigonometric polynomials, uniqueness of Fourier coefficients, Parseval’s identity, properties of trigono metric polynomials, convergence of Fourier series, applications of Fourier Series, and the Fourier Transform, including the Poisson summation formula and Plancherel’s identity. While the main effort will be to establish the foundations of the subject, applications may include the Fast Fourier Transform, the heat equation, the wave equation, and the Laplace equation. See more details in Fourier Analysis and its Applications Syllabus.
Weekly  Schedule  Attachments 
Week of 01/09 — 01/11:  Introduction and complex number  
Week of 01/14 — 01/18:  Fourier series of a periodic function  Quiz 1 
Week of 01/21 — 01/25:  Fourier series of a periodic function, Convergence theorem  
Week of 01/28 — 02/01:  Convergence theorem, Derivatives, Integrals  Quiz 2 
Week of 02/04 — 02/08:  Uniform convergence, Fourier series on intervals  Quiz 3 
Week of 02/11 — 02/15:  Abel test, Applications to PDEs  
Week of 02/18 — 02/22:  Convolution, Cesaro summability  Quiz 4 
02/25:  Midterm Exam I  Midterm Exam I 
Week of 02/27 — 03/01:  Cesaro summability, Fejer kernel  Quiz 5 
Week of 03/04 — 03/08:  Spring break  
Week of 03/11 — 03/15:  Vectors, functions, and inner product  
Week of 03/18 — 03/22:  Fourier basis, completeness, and convergence  Quiz 6 
Week of 03/25 — 03/27:  SturmLivioulle problems  Quiz 7 
03/29:  Midterm Exam II  Midterm Exam II 
Week of 04/01 — 04/05:  Fourier transform, convolution  Quiz 8 
Week of 04/08 — 04/12:  Fourier inversion theorem, Plancherel Theorem  Quiz 9 
Week of 04/15 — 04/19:  Applications of Fourier transform  
04/22:  Review  Brief Review 
05/01:  Final Exam (4:00pm — 6:00pm)  Final Exam 
Homework and Solutions
