  This paper presents theoretical analysis and software implementation for real
harmonics analysis on the special orthogonal group. Noncommutative harmonic
analysis for complex-valued functions on the special orthogonal group has been
studied extensively. However, it is customary to treat real harmonic analysis
as a special case of complex harmonic analysis, and there have been limited
results developed specifically for real-valued functions. Here, we develop a
set of explicit formulas for real-valued irreducible unitary representations on
the special orthogonal group, and provide several operational properties, such
as derivatives, sampling, and Clebsch-Gordon coefficients. Furthermore, we
implement both of complex and real harmonics analysis on the special orthogonal
group into an open source software package that utilizes parallel processing
through the OpenMP library. The efficacy of the presented results are
illustrated by benchmark studies and an application to spherical shape
matching.
