Fourier transforms of spherical distributions on compact symmetric spaces
In our previous articles  and  we studied Fourier series on a symmetric space M = U/K of the compact type. 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 these results to K-invariant distributions onM.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. Finally, we characterize the singular support of a distribution in terms of its Fourier transform, and we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator. The extension from symmetric spaces of compact type to all compact symmetric spaces is sketched in an appendix.
Publication Source (Journal or Book title)
Ólafsson, G., & Schlichtkrull, H. (2011). Fourier transforms of spherical distributions on compact symmetric spaces. Mathematica Scandinavica, 109 (1), 93-113. https://doi.org/10.7146/math.scand.a-15179