  We develop a sampling scheme on the sphere that permits accurate computation
of the spherical harmonic transform and its inverse for signals band-limited at
$L$ using only $L^2$ samples. We obtain the optimal number of samples given by
the degrees of freedom of the signal in harmonic space. The number of samples
required in our scheme is a factor of two or four fewer than existing
techniques, which require either $2L^2$ or $4L^2$ samples. We note, however,
that we do not recover a sampling theorem on the sphere, where spherical
harmonic transforms are theoretically exact. Nevertheless, we achieve high
accuracy even for very large band-limits. For our optimal-dimensionality
sampling scheme, we develop a fast and accurate algorithm to compute the
spherical harmonic transform (and inverse), with computational complexity
comparable with existing schemes in practice. We conduct numerical experiments
to study in detail the stability, accuracy and computational complexity of the
proposed transforms. We also highlight the advantages of the proposed sampling
scheme and associated transforms in the context of potential applications.
