Courses:

When you click the Amazon logo to the left of any citation and purchase the book (or other media) from Amazon.com, MIT OpenCourseWare will receive up to 10% of this purchase and any other purchases you make during that visit. This will not increase the cost of your purchase. Links provided are to the US Amazon site, but you can also support OCW through Amazon sites in other regions. Learn more.

Fourier Analysis - Theory and Applications (18.103)

Rudin, W. Real and Complex Analysis. 3rd ed. New York, NY: McGraw-Hill, 1987. ISBN: 0070542341.

The course materials were presented according to the following breakdown of lectures:

• Five lectures on measure and integration - Leading to Riesz representation theorem
  
  
• Seven lectures on distributions and Fourier transform - Including Schwart space, Sobolev spaces and Sobolev embedding
  
  
• Seven lectures on differential operators with constant coefficients, fundamental solutions and hypoellipticity
  
  
• Four lectures on operators, trace class, Hilbert-Schmid

To pass the course each student was required to carry out one of the projects which were described in the second week of classes. The following projects were suggested to the students:

1. Radon-Nikodym theorem
   
   
2. Kuiper's theorem: The group of unitary operators on a (separable infinite dimensional) Hilbert space is contractible
   
   
3. Seeley's extension theorem
   
   
4. Gibb's phenomenon
   
   
5. Surjectivity of any non-trivial constant coefficient differential operator, P: S' (Rⁿ) → S' (Rⁿ)
   
   
6. Every elliptic differential operator with constant coefficients is surjective as a map on C^∞(U), for any open set U ⊂ Rⁿ
   
   
7. Lidskii's theorem on trace class operators on L²(Rⁿ)

The final grade was based on the homework and the project. There were no tests or examinations.