Completeness of the Bispectrum on Compact Groups

This paper derives completeness properties of the bispectrum for compact groups and their homogeneous spaces. The bispectrum is the Fourier transform of the triple correlation, just as the magnitude-squared spectrum is the Fourier transform of the autocorrelation. The bispectrum has been applied in time series analysis to measure non-Gaussianity and non-linearity. It has also been applied to provide orientation and position independent character recognition, as well as to analyze statistical properties of the cosmic microwave background radiation; in both cases, the data may be defined on a sphere. On the real line, it is known that the bispectrum is not only invariant under translation of the underlying function, but in many cases of interest, it is also complete, in that the function may be recovered uniquely up to a translation from its bispectrum. This paper extends the completeness theory of the bispectrum to compact groups and their homogeneous spaces, including the sphere. The main result, which depends on Tannaka-Krein duality theory, shows that every function whose Fourier coefficient matrices are always nonsingular is completely determined by its bispectrum, up to a single group action. Furthermore, algorithms are described for reconstructing functions defined on SU(2) and SO(3) from their bispectra.