Orbit Recovery for Band-Limited Functions

.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.Keywordsorbit recoverybispectrumMSC codes94A1222D10