  Many practical sampling patterns for function approximation on the rotation
group utilizes regular samples on the parameter axes. In this paper, we relate
the mutual coherence analysis for sensing matrices that correspond to a class
of regular patterns to angular momentum analysis in quantum mechanics and
provide simple lower bounds for it. The products of Wigner d-functions, which
appear in coherence analysis, arise in angular momentum analysis in quantum
mechanics. We first represent the product as a linear combination of a single
Wigner d-function and angular momentum coefficients, otherwise known as the
Wigner 3j symbols. Using combinatorial identities, we show that under certain
conditions on the bandwidth and number of samples, the inner product of the
columns of the sensing matrix at zero orders, which is equal to the inner
product of two Legendre polynomials, dominates the mutual coherence term and
fixes a lower bound for it. In other words, for a class of regular sampling
patterns, we provide a lower bound for the inner product of the columns of the
sensing matrix that can be analytically computed. We verify numerically our
theoretical results and show that the lower bound for the mutual coherence is
larger than Welch bound. Besides, we provide algorithms that can achieve the
lower bound for spherical harmonics.
