# M354

##### 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: Sturm-Livioulle 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

 HW 1 Solution 1 HW 2 Solution 2 HW 3 Solution 3 HW 4 Solution 4 HW 5 Solution 5 HW 6 Solution 6