Analysis Seminar: "Time-Frequency Analysis and the Dark Side of Representation Theory"

Gerald Folland, University of Washington

Abstract: Translations and modulations --- that is, the operators $T_xf(t) = f(t+x)$ and $M_yf(t) = e^{2\pi iyt}f(t)$ --- are basic ingredients of Fourier analysis on $L^2(\mathbb R)$.  There has been a lot of research involving the study of a discrete set of translations and modulations, $T_{ja}$ and $M_{kb}$, where $a$ and $b$ are fixed and  $j$ and $k$ range over the integers.  The group of operators generated by these is a unitary representation of the so-called discrete Heisenberg group.  How does it decompose into irreducible representations?  When $ab$ is rational, the solution to this problem is a nice exercise in Fourier analysis, but when $ab$ is irrational, it provides concrete examples of several of the pathological phenomena that occur in the representation theory of non-Abelian discrete groups.  We shall discuss both aspects of this situation.

Host: Steven Krantz