- The student is familiar with the fundaments of the theory of Fourier series, of the Fourier transform, and of the finite Fourier theory.
- The student knows the most important theorems, like the inversion theorem, the Parseval/Plancherel formula and Poisson summation.
- The student is able to recognize situations where the theory applies and can use it for solution of a broad class of partial differential equations.
- The student has some understanding of the connection between Fourier theory and character theory of abelian groups.
This course is based on a recent book by Elias Stein, one of the best known exports in the field of Fourier analysis, also known as harmonic analysis. Fourier analysis begins with the idea of Fourier that "every" periodic function is an (infinite) linear combination of sine and cosine functions, but quite some work is required in order to make this idea precise. These Fourier series, as well as the Fourier transform, have many applications to (partial) differential equations and even to number theory! We will also have a look at Fourier theory on finite abelian groups (which in principle is part of algebra), which has many important applications.