  In our previous articles "A local Paley-Wiener theorem for compact symmetric
spaces", Adv. Math. 218 (2008), 202--215, and "Fourier series on compact
symmetric spaces" (submitted) we studied Fourier series on a compact symmetric
space M=U/K. In particular, we proved a Paley-Wiener type theorem for the
smooth functions on M, which have sufficiently small support and are
K-invariant, respectively K-finite. In this article we extend those results to
K-invariant distributions on M. We show that the Fourier transform of a
distribution, which is supported in a sufficiently small ball around the base
point, extends to a holomorphic function of exponential type. We describe the
image of the Fourier transform in the space of holomorphic functions. We
characterize the singular support of a distribution in terms of its Fourier
transform. Finally, we use the Paley-Wiener theorem to characterize the
distributions of small support, which are in the range of a given invariant
differential operator.
