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.

翻译：我们研究了任意紧致李群上函数的三阶矩。利用表示论技术，我们将经典傅里叶理论中带限函数的概念推广到紧致群$SU(n)、SO(n)、Sp(n)$上的函数。随后我们证明：对于一般带限函数，其三阶矩或其傅里叶等价形式——双谱——可在单一酉矩阵平移的意义上确定原函数。此外，当$G=SU(n)$或$G=SO(2n+1)$时，我们证明三阶矩能确定带限函数的$G$-轨道。作为推论，我们得到这些群的一类有限维表示，其一般向量的轨道可由三阶矩确定。当$G=SO(3)$时，这一结论为冷冻电镜领域提供了相关理论依据，这也是我们研究该问题的原始动机。