  We study the third moment for functions on arbitrary compact Lie groups. We
use techniques of representation theory to generalize the notion of
band-limited functions in classical Fourier theory to functions on the compact
groups $SU(n), SO(n), Sp(n)$. We then prove that for generic band-limited
functions the third moment or, its Fourier equivalent, the bispectrum
determines the function up to translation by a single unitary matrix. Moreover,
if $G=SU(n)$ or $G=SO(2n+1)$ we prove that the third moment determines the
$G$-orbit of a band-limited function. As a corollary we obtain a large class of
finite-dimensional representations of these groups for which the third moment
determines the orbit of a generic vector. When $G=SO(3)$ this gives a result
relevant to cryo-EM which was our original motivation for studying this
problem.
