Title

Fourier transforms of spherical distributions on compact symmetric spaces

Document Type

Article

Publication Date

1-1-2011

Abstract

In our previous articles [27] and [28] 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)

Mathematica Scandinavica

First Page

93

Last Page

113

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