Week 2

You will investigate jump discontinuities and their relation to Fourier series coefficients.

Reading Assignments

• Finish reading Chapter 2 (Fourier Series) of Fourier Analysis and its Applications by Folland.
• Finish reading Chapter 3 (MR Overview) from Principles of Magnetic Resonance Imaging by Nishimura.
  • Read Chapter 9 (specifically, Section 9.2 on Filtering) from Spectral Methods for Time-Dependent Problems by Hesthaven et al.

Pen and Paper Exercises

• Complete Exercise A on Fourier series from Week 1.
• Derive a relation between Fourier coefficients and the jump locations and "heights" of a piecewise-smooth function (you will need to use integration by parts for this). Start by assuming the function has a single jump discontinuity; then generalize to the case of multiple jumps.
• Given Fourier coefficients, devise a method of finding/estimating jump locations and "heights". You may start by assuming that you know the jump "heights" apriori. Next, generalize to the case of jointly estimating jump locations and "heights".

Matlab Coding Exercises

• Complete (in Matlab) Exercise B on Fourier series.
• Complete (in Matlab) the following exercise on Fourier coefficients and jumps.

Documents to Prepare

• (due Wed., 6/1) Write down (as a group) an abstract summarizing this research project. You will be presenting your research at the end of this REU program at <>. This abstract will serve to introduce and advertise your research.
• (due end of day Thurs., 6/2) Prepare a 15 minute Beamer/LaTeX presentation summarizing your research findings and readings from Weeks 1 and 2. You will be presenting (as a group) to your fellow REU students/groups during the Friday meeting/discussion at the Holmes Hall seminar room.