Math 7390

The subject proper of harmonic analysis is to decompose functions into ''simpler'' functions, where the meaning of simple depends on the context we are working in. In the theory of differential equations that means to write an arbitrary functions as a superposition of eigenfunctions. If we have a symmetry group acting on the system, then we would like to write an arbitary function as a sum of functions that transforms in a simple and controllable way under the symmetry group. The simplest example is the use of polar coordinates and radial functions for rotation symmetric equations. Sometimes, we are working with general spaces than Rⁿ. Here anlysis meets with Lie groups, geometry, representation theory and harmonic analysis to form abstract harmonic analysis.

The course begins with a short overview of classical Fourier analysis on the torus and Rⁿ. This leads us to topics like:

1. Periodic functions and Fourier series;
2. Convergence of Fourier series;
3. Spaces of functions on Rⁿ. In particular, we will discuss the space of compactly supported functions, functions of compact support and the algebraic structure of those spaces, i.e., convolution.
4. The Fourier transform of rapidly decreasing functions and L²-functions, inversion formula and Plancherel theorems.
5. Introduction to distribution theory and the continuous linear functionals on function spaces. How to differentiate distributions. The Fourier transform of distributions.
6. Application of the Fourier transform to differential equations. In particular we will discuss the heat equation and the wave equation.
7. Hermite functions and polynomials.
8. At the end, we will also discuss some other integral transforms. In partiuclar, we will discuss the continuous wavelet transform, derive a Plancherel formula and an inversion formula.

We can view Rⁿ as a set or as amanifold. But we can also view it as an abelian group. In that sense Rⁿ  is a part of abelian harmonic analysis. The simplest example of  nonabelian harmonic analysis is the Heisenberg group H_n=R²ⁿ⁺¹  (with a new group multiplication). The Heisenberg group is also a simple example of a Lie group and of a topological group . There are several other well known examples of topological groups like the group of rotations SO(n) , the group of all invertible matrices GL (n,R). Depending on the time and interest we will at the end discuss some advanced topics related to topological groups.

Representations of topological groups are central in several branches of mathematics: In number theory and the study of authomorphic functions and forms, in geometry as a tool to construct important vector bundles and differential operators, and in the study of Riemannian symmetric spaces. Finally, those are important tools in analysis, in particular analysis on some special homogeneous manifolds like the sphere, Grassmanians, the upper half plane and its generalizations. Representations even shows up in branches of applied mathematics as generalizations of the windowed Fourier transform and wavelets. Several examples of those applications in analysis and geometry will be discussed in the class.

The second part of this class takes plase in the spring 2008, where the weight is on topological groups, homogeneous spaces and representation theory. The instructore will be Mark Davidson.