Teaching

Summer Semester 2023-24:  Harmonic Analysis (Graduate)

General course information: 

Lectures: Tu 09:00-11:00, We 09:00-11:00, A3.301

Exercises [Lukas Langen]: Thu 16:00-18:00, A3.301

Start of the exercises: 

Second week of the lectures

Contents:

After a general introduction to the theory of locally compact groups and some basics about homogeneous spaces, the first part of the course will be devoted to analysis on locally compact abelian groups. This theory includes the theory of Fourier series and the Fourier transform on R^n as special cases. In particular, positive definite functions, Plancherel's theorem, and Pontryagin duality will be discussed.

The second part of the course is dedicated to the theory of Gelfand pairs and their spherical functions. We will deal with homogeneous spaces that are characterized by commutative convolution algebras. Important examples include spheres and hyperbolic spaces. This theory interacts with methods of representation theory, the basics of which we will also cover. 

Prerequisites: Knowledge of functional analysis to the extent of a one-semester introduction.

References:

• G.B. Folland, A course in abstract harmonic analysis. CRC Press 1995

• W. Rudin, Fourier Analysis on Groups. New edition, Wiley Verlag 1990

• J. Faraut, Analyse harmonique sur les pairs de Guelfand et les espaces hyperboliques. Analyse harmonique, Chap. IV, C.I.M.P.A. Nice, 1982  

• G. van Dijk, Introduction to Harmonic Analysis and Generalized Gelfand pairs, de Gruyter Verlag 2009