  We propose a Gauss-Legendre quadrature based sampling on the rotation group
for the representation of a band-limited signal such that the Fourier transform
(FT) of a signal can be exactly computed from its samples. Our figure of merit
is the sampling efficiency, which is defined as a ratio of the degrees of
freedom required to represent a band-limited signal in harmonic domain to the
number of samples required to accurately compute the FT. The proposed sampling
scheme is asymptotically as efficient as the most efficient scheme developed
very recently. For the computation of FT and inverse FT, we also develop fast
algorithms of complexity similar to the complexity attained by the fast
algorithms for the existing sampling schemes. The developed algorithms are
stable, accurate and do not have any pre-computation requirements. We also
analyse the computation time and numerical accuracy of the proposed algorithms
and show, through numerical experiments, that the proposed Fourier transforms
are accurate with errors on the order of numerical precision.
