  We present a theory of ultradistributional boundary values for harmonic
functions defined on the Euclidean unit ball. We also give a characterization
of ultradifferentiable functions and ultradistributions on the sphere in terms
of their spherical harmonic expansions. To this end, we obtain explicit
estimates for partial derivatives of spherical harmonics, which are of
independent interest and refine earlier estimates by Calderon and Zygmund. We
apply our results to characterize the support of ultradistributions on the
sphere via Abel summability of their spherical harmonic expansions.
