Harmonic analysis

Lecturer: Henrik Stetkær

Content:
Harmonic analysis is analysis of functions defined on a group G, where the analysis is done with due respect to the underlying group structure. The group structure comes into play, because the functions will be written as a superposition of the continuous group characters. So group characters will be our fundamental building blocks. If G=(R, +) or T (the circle group), then these building blocks are the exponential functions exp , and
exp , and so the harmonic analysis becomes the classical Fourier analysis.
We shall start the course by proving uniqueness and existence of the so-called Haar measure on locally compact groups. This measure plays the role for general groups that the Lebesgue measure plays for the special case of the group R. By help of the Haar measure we shall make a detailed analysis of functions on two special types of groups, namely abelian groups and compact groups. The latter type will show that characters do not suffice, which leads us further to the theory of representations of groups and *-algebras on Hilbert spaces.

Prerequisites: Corresponding to the course Analysis II from the Fall term 2000.

Literature: Notes, or possibly Gerald B. Folland, A Course in Abstract Harmonic Analysis, CRC PRESS. Boca Raton, Ann Arbor, London, Tokyo 1995.