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：我们研究了任意紧李群上函数的第三矩。利用表示论技术，我们将经典傅里叶分析中带限函数的概念推广到紧群 $\SU(n), \SO(n), \Sp(n)$ 上的函数。随后证明，对于一般的带限函数，其第三矩（或傅里叶等价形式下的双谱）可由单个酉矩阵的平移唯一确定该函数。此外，当 $G=\SU(n)$ 或 $G=\SO(2n+1)$ 时，我们证明第三矩可确定带限函数的 $G$-轨道。作为推论，我们得到这些群的一类有限维表示，其中第三矩能确定一般向量的轨道。当 $G=\SO(3)$ 时，该结果给出了与冷冻电镜（cryo-EM）相关的结论——这最初正是我们研究该问题的动机。