Math 7390

Harmonic Analysis - I
T-TH: 12:10-1:30 pm Lockett 239
INSTRUCTOR Gestur Olafsson
Office 322 Lockett
Office Hours T-TH: 11:00-12:00, 3:00-4:00 pm, and by request
Phone 578-1608 and 225-337-2206 (cell)
Text Noncommutative Harmonic Analysis, An Introduction by R. Fabec and G. Olafsson

You can download the lecture note at Noncommutative Harmonic Analsysis using your departmental login and password. The aim is to cover variuous material from Capter 1-4. If there is time, then we will also introduce material from other chapters.

There will be 2-3 homework due at least every second week and one longer list of homeworks around middterm. There will be a take home final due on Monday, Dec. 6, at 7:30 pm. You can replace the final by a 45 min. lecture on a selected topic.

Important Dates:

  1. Mid-semester exams are October 11-16 and the grades are due 9:00 on October 19.
  2. Fall holidays begins on October 21 and classes resume on October 25.
  3. Thanksgiving holidays begins 12:30 pm on November 24.
  4. Classes end on December 4.

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 Hn=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.

Professor He will offer a more advanced course during the spring 2011.