Fourier Analysis, MTH 496-001
Schedule and Topics Covered
| Date | Topic/Section | Assignments and Reading |
| 1/9 | Welcome, and DFT Matrices | -Read Syllabus |
| 1/11 | Discrete Convolution & Applications | |
| 1/13 | The Fast Fourier Transform | Homework 1 Due 1/23 the .tex file |
| 1/16 | Holiday - no class | |
| 1/18 | Trigonometric Polynomials | |
| 1/20 | Trigonometric Polynomials, the DFT, and Aliasing | |
| 1/23 | Solving the Poisson and Heat Equations Homework 1 Due in Class Today! | Read Chapter 1 of Folland |
| 1/25 | Fourier Series of Periodic Functions | Homework 2 Due 2/6 the .tex file |
| 1/27 | Fourier Series of Periodic Functions | |
| 1/30 | Convergence Issues: Bessel's Inequality | Read Section 2.1 of Folland |
| 2/1 | Toward Pointwise Convergence of Fourier Series | |
| 2/3 | The Dirichlet kernel | |
| 2/6 | Pointwise Convergence of Fourier Series for Piecewise Smooth Functions Homework 2 Due in Class Today! | Read Section 2.2 of Folland |
| 2/8 | Uniform Convergence of Fourier Series, and Rates of Convergence for Continuous Functions | Homework 3 Due 2/27 the .tex file |
| 2/10 | Fourier Series on Arbitrary Intervals | Read Section 2.3 of Folland |
| 2/13 | Some Final Properties of Fourier Series | Read Section 2.4 of Folland |
| 2/15 | No Class Today - CAPS has a lot of useful counseling resources available for students in need. | |
| 2/17 | No Class Today | |
| 2/20 | Inner Product Spaces | Read Sections 3.1 and 3.2 of Folland |
| 2/22 | Abstract Hilbert Spaces | Read Section 3.3 of Folland |
| 2/24 | The Fourier Basis of L2[-π,π] | |
| 2/27 | The Fourier Transform and Convolutions on R Homework 3 Due in Class Today! | Read Folland Chapter 7 through Section 7.1 |
| 3/1 | Compactly Supported Continuous Functions are Dense in L2(R) | Homework 4 Due 3/22 the .tex file |
| 3/3 | Convolutions and Mollification | |
| 3/6 | Spring Break - no class | |
| 3/8 | Spring Break - no class | |
| 3/10 | Spring Break - no class | |
| 3/13 | Midterm Exam Today!!! | |
| 3/15 | C0∞ is Dense in L2(R) | |
| 3/17 | The Fourier Transform is Invertible | |
| 3/20 | The Fourier Transform is Invertible | Read Sections 7.1 and 7.2 of Folland |
| 3/22 | The Nyquist-Shannon Sampling Theorem Homework 4 Due in Class Today! | Homework 5 Due 4/7 the .tex file |
| 3/24 | Sampling Theorem and Higher Dim. Fourier Series - Project topics (if you want to do one) are due today! - Exam corrections (if you want to do them) are due today! | Read Section 7.3 of Folland |
| 3/27 | The Short-Time Fourier Transform (STFT) | |
| 3/29 | Why Distributions? | Read Section 9.1 of Folland |
| 3/31 | Distribution Basics | |
| 4/3 | Distribution Basics | |
| 4/5 | Weak Convergence | Read Section 9.2 of Folland |
| 4/7 | Distributional Convolution Homework 5 Due in Class Today! | Homework 6 Due 4/21 the .tex file |
| 4/10 | Distributions are Weak Limits of C∞ functions | |
| 4/12 | Support of Distributions | |
| 4/14 | Periodic Distributions and Fourier Series | Read Section 9.3 of Folland |
| 4/17 | Fourier Transform of a Tempered Distribution IF DOING A PROJECT: Turn in a rough draft of your report and presentation materials. | |
| 4/19 | Review | |
| 4/21 | Review Homework 6 Due in Class Today! | |
| 4/24 | - Fourier Methods for Determining Crystal Properties (Kaedon) - Fourier Analysis in Quantum Mechanics (Rana and Jenny) | |
| 4/26 | - Navier-Stokes and Fourier (Sepehr) - Sparse Fourier Transforms and Applications (Xin) | |
| 4/28 | Review | |
| 5/2 | Final Exam Time | 12:45pm - 2:45pm |